Secondary Invariants for Frechet Algebras

and Quasihomomorphisms

Denis Perrot

Received: March 3, 2008 Revised: June 30, 2008

Communicated by Joachim Cuntz

Abstract. A Fréchet algebra endowed with a multiplicatively convex topology has two types of invariants: homotopy invariants (topological K-theory and periodic cyclic homology) and secondary invariants (multiplicative K-theory and the non-periodic versions of cyclic homology). The aim of this paper is to establish a Riemann-Roch-Grothendieck theorem relating direct images for homotopy and secondary invariants of Fréchet m-algebras under finitely summable quasihomomorphisms.

2000 Mathematics Subject Classification: 19D55, 19K56, 46L80, 46L87. Keywords and Phrases: K-theory, bivariant cyclic cohomology, index theory.

1 Introduction

For a noncommutative space described by an associative Fréchet algebra 𝒜 over ℂ, we distinguish two types of invariants. The first type are (smooth) homotopy invariants, for example topological K-theory [27] and periodic cyclic homology [5]. The other type are secondary invariants; they are no longer stable under homotopy and carry a finer information about the “geometry” of the space 𝒜. Typical examples of secondary invariants are algebraic K-theory [29] (which will not be used here), multiplicative K-theory [17] and the unstable versions of cyclic homology [18]. The aim of this paper is to define push-forward maps for homotopy and secondary invariants between two Fréchet algebras 𝒜 and ℬ, induced by a smooth finitely summable quasihomomorphism [8]. The compatibility between the different types of invariants is expressed through a noncommutative Riemann-Roch-Grothendieck theorem (Theorem 6.3). The present p aper is the first part of a series on secondary characteristic classes. In the second part we will show how to obtain local formulas for push-forward maps, following a general principle inspired by renormalization which establishes the link with chiral anomalies in quantum field theory [25]; in order to keep a reasonable size to the present paper, these methods will be published in a separate survey with further examples [26].

We deal with Fréchet algebras endowed with a multiplicatively convex topology, or Fréchet m-algebras for short. These algebras can be presented as inverse limits of sequences of Banach algebras, and as a consequence many constructions valid for Banach algebras carry over Fréchet m-algebras. In particular Phillips [27] defines topological K-theory groups K_{n}^{top}(𝒜) for any such algebra 𝒜 and n ∈ ℤ. The fundamental properties of interest for us are (smooth) homotopy invariance and Bott periodicity, i.e. K_{n+2}^{top}(𝒜) ≅ K_{n}^{top}(𝒜). Hence there are essentially two topological K-theory groups for any Fréchet m-algebra, K_{0}^{top}(𝒜) whose elements are roughly represented by idempotents in the stabilization of 𝒜 by the algebra 𝒦 of ”smooth compact operators”, and K_{1}^{top}(𝒜) whose elements are represented by invertibles. Fréchet m-algebras naturally arise in many situations related to differential geometry, commutative or not, and the formulation of index problems. In the latter situation one usually encounters an algebra ℐ of ”finitely summable operators”, for us a Fréchet m-algebra provided with a continuous trace on its p-th power for some p ≥ 1. A typical example is the Schatten class ℐ = ℒ^{p}(H) of p-summable operators on an infinite-dimensional separable Hilbert space H. 𝒜 can be stabilized by the completed projective tensor product ℐ ˆ⊗𝒜 and its topological K-theory K_{n}^{top}(ℐ ˆ⊗𝒜) is the natural receptacle for indices. Other important topological invariants of 𝒜 (as a locally convex algebra) are provided by the periodic cyclic homology groups HP_{n}(𝒜), which is the correct version sharing the properties of smooth homotopy invariance and periodicity mod 2 with topological K-theory [5]. For any finitely summable algebra ℐ the Chern character K_{n}^{top}(ℐ ˆ⊗𝒜) → HP_{n}(𝒜) allows to obtain cohomological formulations of index theorems. If one wants to go beyond differential topology and detect secondary invariants as well, which are no longer stable under homotopy, one has to deal with algebraic K-theory [29] and the unstable versions of cyclic homology [18]. In principle the algebraic K-theory groups K_{n}^{alg}(𝒜) defined for any n ∈ ℤ provide interesting secondary invariants for any ring 𝒜, but are very hard to calculate. It is also unclear if algebraic K-theory can be linked to index theory in a way consistent with topological K-theory, and in particular if it is possible to construct direct images of algebraic K-theory classes in a reasonable context. Instead, we will generalize slightly an idea of Karoubi [16, 17] and define for any Fréchet m-algebra 𝒜 the multiplicative K-theory groups MK_{n}^{ℐ} (𝒜), n ∈ ℤ, indexed by a given finitely summable Fréchet m-algebra ℐ. Depending on the parity of the degree n, multiplicative K-theory classe s are represented by idempotents or invertibles in certain extensions of ℐ ˆ⊗𝒜, together with a transgression of their Chern character in certain quotient complexes. Multiplicative K-theory is by definition a mixture of the topological K-theory K_{n}^{top}(ℐ ˆ⊗𝒜) and the non-periodic cyclic homology HC_{n}(𝒜). It provides a “good” approximation of algebraic K-theory but is much more tractable. In addition, the Jones-Goodwillie Chern character in negative cyclic homology K_{n}^{alg}(𝒜) → HN_{n}(𝒜) factors through multiplicative K-theory. The precise relations between topological, multiplicative K-theory and the various versions of cyclic homology are encoded in a commutative diagram whose rows are long exact sequences of abelian groups

[figure: K_{n+1}^{top}(ℐ ˆ⊗𝒜); HC_{n−1}(𝒜); δ; MK_{n}^{ℐ} (𝒜); K_{n}^{top}(ℐ ˆ⊗𝒜); HP_{n+1}(𝒜); S; HC_{n−1}(𝒜); B̃; HN_{n}(𝒜); I; HP_{n}(𝒜)] (1)

The particular case ℐ = ℂ was already considered by Karoubi [16, 17] after the construction by Connes and Karoubi of regulator maps on algebraic K-theory [6]. The incorporation of a finitely summable algebra ℐ is rather straightforward. This diagram describes the primary and secondary invariants associated to the noncommutative “manifold” 𝒜. We mention that the restriction to Fréchet m-algebras is mainly for convenience. In principle these constructions could be extended to all locally convex algebras over ℂ, however the subsequent results, in particular the proof of the Riemann-Roch-Grothendieck theorem would become much more involved.

If now 𝒜 and ℬ are two Fréchet m-algebras, it is natural to consider the adequate “morphisms” mapping the primary and secondary invariants from 𝒜 to ℬ. Let ℐ be a p-summable Fréchet m-algebra. By analogy with Cuntz’ description of bivariant K-theory for C^{∗}-algebras [8], if ℰ ▷ ℐ ˆ⊗ℬ denotes a Fréchet m-algebra containing ℐ ˆ⊗ℬ as a (not necessarily closed) two-sided ideal, we define a p-summable quasihomomorphism from 𝒜 to ℬ as a continuous homomorphism

ρ : 𝒜 → ℰ^{s} ▷ ℐ^{s} ˆ⊗ℬ ,

where ℰ^{s} and ℐ^{s} are certain ℤ_{2}-graded algebras obtained from ℰ and ℐ by a standard procedure. Quasihomomorphisms come equipped with a parity (even or odd) depending on the construction of ℰ^{s} and ℐ^{s}. In general, we may suppose that the parity is p mod 2. We say that ℐ is multiplicative if it is provided with a homomorphism ⊠ : ℐ ˆ⊗ℐ → ℐ, possibly defined up to adjoint action of multipliers on ℐ, and compatible with the trace. A basic example of multiplicative p-summable algebra is, once again, the Schatten class ℒ^{p}(H). Then it is easy to show that such a quasihomomorphism induces a pushforward map in topological K-theory ρ_{!} : K_{n}^{top}(ℐ ˆ⊗𝒜) → K_{n−p}^{top}(ℐ ˆ⊗ℬ), whose degree coincides with the parity of the quasihomomorphism. This is what one expects from bivariant K-theory and is not really new. Our goal is to extend this map to the entire diagram (1). Direct images for the unstable versions of cyclic homology are necessarily induced by a bivariant non-periodic cyclic cohomology class ch^{p}(ρ) ∈ HC^{p}(𝒜 , ℬ). This bivariant Chern character exists only under certain admissibility properties about the algebra ℰ (note that it is sufficient for ℐ to be (p + 1)-summable instead of p-summable). In particular, the bivariant Chern caracter constructed by Cuntz for any quasihomomorphism in [9, 10] cannot be used here because it provides a bivariant periodic cyclic cohomology class, which does not detect the secondary invariants of 𝒜 and ℬ. We give the precise definition of an admissible quasihomomorphism and construct the bivariant Chern character ch^{p}(ρ) in section 3, on the basis of previous works [23]. An analogous construction was obtained by Nistor [20, 21] or by Cuntz and Quillen [12]. However the bivariant Chern character of [23] is related to other constructions involv ing the heat operator and can be used concretely for establishing local index theorems, see for example [24]. The pushforward map in topological K-theory combined with the bivariant Chern character leads to a pushforward map in multiplicative K-theory ρ_{!} : MK_{n}^{ℐ} (𝒜) → MK_{n−p}^{ℐ}(ℬ). Our first main result is the following non-commutative version of the Riemann-Roch-Grothendieck theorem (see Theorem 6.3 for a precise statement):

Theorem 1.1 Let ρ : 𝒜 → ℰ^{s} ▷ ℐ^{s} ˆ⊗ℬ be an admissible quasihomomorphism of parity p mod 2. Suppose that ℐ is (p + 1)-summable in the even case and p-summable in the odd case. Then one has a graded-commutative diagram

[figure: K_{n+1}^{top}(ℐ ˆ⊗𝒜); ρ_{!}; HC_{n−1}(𝒜); ch^{p}(ρ); MK_{n}^{ℐ} (𝒜); ρ_{!}; K_{n}^{top}(ℐ ˆ⊗𝒜); ρ_{!}; K_{n+1−p}^{top}(ℐ ˆ⊗ℬ); HC_{n−1−p}(ℬ); MK_{n−p}^{ℐ}(ℬ); K_{n−p}^{top}(ℐ ˆ⊗ℬ)]

compatible with the cyclic homology SBI exact sequences after taking the Chern characters K_{∗}^{top}(ℐ ˆ⊗·) → HP_{∗} and MK_{∗}^{ℐ} → HN_{∗}.

At this point it is interesting to note that the pushforward maps ρ_{!} and the bivariant Chern character ch^{p}(ρ) enjoy some invariance properties with respect to equivalence relations among quasihomomorphisms. Two types of equivalence relations are defined: smooth homotopy and conjugation by invertibles. The second relation corresponds to “compact perturbation” in Kasparov KK-theory for C^{∗}-algebras [2]. In the latter situation, the M_{2}- stable version of conjugation essentially coincide with homotopy, at least for separable 𝒜 and σ-unital ℬ. For Fréchet algebras however, M_{2}-stable conjugation is strictly stronger than homotopy as an equivalence relation. This is indeed in the context of Fréchet algebras that secondary invariants appear. The pushforward maps in topological K-theory and periodic cyclic homology are invariant under homotopy of quasihomomorphisms. The maps in multiplicative K-theory and the non-perio dic versions of cyclic homology HC_{∗} and HN_{∗} are only invariant under conjugation and not homotopy. Also note that in contrast with the C^{∗}-algebra situation, the Kasparov product of two quasihomomorphisms ρ : 𝒜 → ℰ^{s} ▷ ℐ^{s} ˆ⊗ℬ and ρ′ : ℬ → ℱ^{s} ▷ ℐ^{s} ˆ⊗𝒞 is not defined as a quasihomomorphism from 𝒜 to 𝒞. The various bivariant K-theories constructed for m-algebras [9, 10] or even for general bornological algebras [11] cannot be used here, again because they are homotopy invariant by construction. We leave the construction of a bivariant K-theory compatible with secondary invariants as an open problem.

In the last part of the paper we illustrate the Riemann-Roch-Grothendieck theorem by constructing assembly maps for certain crossed product algebras. If Γ is a discrete group acting on a Fréchet m-algebra 𝒜, under certain conditions the crossed product 𝒜 ⋊ Γ is again a Fréchet m-algebra and one would like to obtain multiplicative K-theory classes out of a geometric model inspired by the Baum-Connes construction [1]. Thus let P →^{Γ} M be a principal Γbundle over a compact manifold M, and denote by 𝒜_{P} the algebra of smooth sections of the associated 𝒜 -bundle. If D is a K-cycle for M represented by a pseudodifferential operator, we obtain a quasihomomorphism from 𝒜_{P} to 𝒜 ⋊ Γ and hence a map

MK_{n}^{ℐ} (𝒜_{P}) → MK_{n−p}^{ℐ}(𝒜 ⋊ Γ)

for suitable p and Schatten ideal ℐ. In general this map cannot exhaust the entire multiplicative K-theory of the crossed product but nevertheless interesting secondary invariants arise in this way. In the case where 𝒜 is the algebra of smooth functions on a compact manifold, 𝒜_{P} is commutative and its secondary invaiants are closely related to (smooth) Deligne cohomology. From this point of view the pushforward map in multiplicative K-theory should be considered as a non-commutative version of “integrating Deligne classes along the fibers” of a submersion. We perform the computations for the simple example provided by the noncommutative torus.

The paper is organized as follows. In section 2 we review the Cuntz-Quillen formulation of (bivariant) cyclic cohomology [12] in terms of quasi-free extensions for m-algebras. Nothing is new but we take the opportunity to fix the notations and recall a proof of generalized Goodwillie theorem. In section 3 we define quasihomomorphisms and construct the bivariant Chern character. The formulas are identical to those found in [23] but in addition we carefully establish their adic properties and conjugation invariance. In section 4 we recall Phillips’ topological K-theory for Fréchet m-algebras, and introduce the periodic Chern character K_{n}^{top}(ℐ ˆ⊗𝒜) → HP_{n}(𝒜) when ℐ is a finitely summable algebra. The essential point here is to give explicit and simple formulas for subsequent use. Section 5 is devoted to the definition of the multiplicative K-theory groups MK_{n}^{ℐ} (𝒜) and the proof of the long exact seque nce relating them with topological K-theory and cyclic homology. We also construct the negative Chern character MK_{n}^{ℐ} (𝒜) → HN_{n}(𝒜) and show the compatibility with the SBI exact sequence. Direct images of topological and multiplicative K-theory under quasihomomorphisms are constructed in section 6 and the Riemann-Roch-Grothendieck theorem is proved. The example of assembly maps and crossed products is treated in section 7.

2 Cyclic homology

Cyclic homology can be defined for various classes of associative algebras over ℂ, in particular complete locally convex algebras. For us, a locally convex algebra 𝒜 has a topology induced by a family of continuous seminorms p : 𝒜 → ℝ_{+}, for which the multiplication 𝒜 × 𝒜 → 𝒜 is jointly continuous. Hence for any seminorm p there exists a seminorm q such that p(a_{1}a_{2}) ≤ q(a_{1})q(a_{2}), ∀a_{i} ∈ 𝒜. For technical reasons however we shall restrict ourselves to multiplicatively convex algebras [5], whose topology is generated by a family of submultiplicative seminorms

p(a_{1}a_{2}) ≤ p(a_{1})p(a_{2}) ∀a_{i} ∈ 𝒜 .

A complete multiplicatively convex algebra is called m-algebra, and may equivalently be described as a projective limit of Banach algebras. The unitalization 𝒜^{+} = ℂ ⊕ 𝒜 of an m-algebra 𝒜 is again an m-algebra, for the seminorms p̃(λ1 + a) = |λ| + p(a), ∀λ ∈ ℂ, a ∈ 𝒜. In the same way, if ℬ is another m-algebra, the direct sum 𝒜 ⊕ ℬ is an m-algebra for the topology generated by the seminorms (p⊕q)(a, b) = p(a)+q(b), where p is a seminorm on 𝒜 and q a seminorm on ℬ. Also, the algebraic tensor product 𝒜 ⊗ ℬ may be endowed with the projective topology induced by the seminorms

(p ⊗ q)(c) = inf { ∑_{i=1}^{n} p(a_{i})q(b_{i}) such that c = ∑_{i=1}^{n} a_{i} ⊗ b_{i} ∈ 𝒜 ⊗ ℬ} . (2)

The completion 𝒜 ˆ⊗ℬ = 𝒜 ˆ⊗_{π}ℬ of the algebraic tensor product under this family of seminorms is the projective tensor product of Grothendieck [14], and is again an m-algebra. Cyclic homology, cohomology and bivariant cyclic cohomology for m-algebras can be defined either within the cyclic bicomplex formalism of Connes [5], or the X-complex of Cuntz and Quillen [12]. We will make an extensive use of both formalisms throughout this paper. In general, we suppose that all linear maps or homomorphims between m-algebras are continuous, tensor products are completed projective tensor products, and extensions of m-algebras 0 → ℐ → ℛ → 𝒜 → 0 always admit a continuous linear splitting σ : 𝒜 → ℛ.

2.1 Cyclic bicomplex

Non-commutative differential forms. Let 𝒜 be an m-algebra. The space of non-commutative differential forms over 𝒜 is the algebraic direct sum Ω𝒜 = ⨁_{n≥0} Ω^{n}𝒜 of the n-forms subspaces Ω^{n}𝒜 = 𝒜^{+} ˆ⊗𝒜^{ˆ⊗n} for n ≥ 1 and Ω^{0}𝒜 = 𝒜, where 𝒜^{+} is the unitalization of 𝒜. Each of the subspaces Ω^{n}𝒜 is complete but we do not complete the direct sum. It is customary to use the differential notation a_{0}da_{1} … da_{n} (resp. da_{1} … da_{n}) for the string a_{0} ⊗ a_{1} … ⊗ a_{n} (resp. 1 ⊗ a_{1} … ⊗ a_{n}). A continuous differential d : Ω^{n}𝒜 → Ω^{n+1}𝒜 is uniquely specified by d(a_{0}da_{1} … da_{n}) = da_{0}da_{1} … da_{n} and d^{2} = 0. A continuous and associative product Ω^{n}𝒜 × Ω^{m}𝒜 → Ω^{n+m}𝒜 is defined as usual and fulfills the Leibniz rule d(ω_{1}ω_{2}) = dω_{1}ω_{2} + (−)^{|ω_{1}|}ω_{1}dω_{2}, where |ω_{1}| is the degree of ω_{1}. This tu rns Ω𝒜 into a differential graded (DG) algebra. On Ω𝒜 are defined various operators. First of all, the Hochschild boundary map b : Ω^{n+1}𝒜 → Ω^{n}𝒜 reads b(ωda) = (−)^{n}[ω, a] for ω ∈ Ω^{n}𝒜, and b = 0 on Ω^{0}𝒜 = 𝒜. One easily shows that b is continuous and b^{2} = 0, hence Ω𝒜 is a complex graded over ℕ. The Hochschild homology of 𝒜 (with coefficients in the bimodule 𝒜) is the homology of this complex:

HH_{n}(𝒜) = H_{n}(Ω𝒜 , b) ,  ∀n ∈ ℕ .  (3)

Then the Karoubi operator κ : Ω^{n}𝒜 → Ω^{n}𝒜 is defined by 1 − κ = db + bd. Therefore κ is continuous and commutes with b and d. One has κ(ω da) = (−)^{n}da ω for any ω ∈ Ω^{n}𝒜 and a ∈ 𝒜. The last operator is Connes’ B : Ω^{n}𝒜 → Ω^{n+1}𝒜, equal to (1 + κ + … + κ^{n})d on Ω^{n}𝒜. It is also continuous and verifies B^{2} = 0 = Bb+bB and Bκ = κB = B. Thus Ω𝒜 endowed with the two anticommuting differentials (b, B) becomes a bicomplex. It splits as a direct sum Ω𝒜 = Ω𝒜^{+} ⊕ Ω𝒜^{−} of even and odd degree differential forms, hence is a ℤ_{2}-graded complex for the total boundary map b + B. However its homology is trivial [18]. The various versions of cyclic homology are defined using the natural filtrations on Ω𝒜. Following Cuntz and Quillen [12], we define the Hodge filtration on Ω𝒜 as the decreasing family of ℤ_{2}-graded subcomplexes for the total boundary b + B

F^{n}Ω𝒜 = bΩ^{n+1}𝒜 ⊕ ⨁_{k>n} Ω^{k}𝒜 , ∀n ∈ ℤ ,

with the convention that F^{n}Ω𝒜 = Ω𝒜 for n < 0. The completion of Ω𝒜 is defined as the projective limit of ℤ_{2}-graded complexes

Ω̂𝒜 = \underleftarrow{lim}_{n} Ω𝒜 /F^{n}Ω𝒜 = ∏_{n≥0} Ω^{n}𝒜 .  (4)

Hence Ω̂𝒜 = Ω̂^{+}𝒜 ⊕ Ω̂^{−}𝒜 is a ℤ_{2}-graded complex endowed with the total boundary map b + B. It is itself filtered by the decreasing family of ℤ_{2}-graded subcomplexes F^{n}Ω̂𝒜 = Ker(Ω̂𝒜 → Ω𝒜 /F^{n}Ω𝒜), which may be written

F^{n}Ω̂𝒜 = bΩ^{n+1}𝒜 ⊕ ∏_{k>n} Ω^{k}𝒜 ,  ∀n ∈ ℤ .  (5)

In particular the quotient Ω̂𝒜 /F^{n}Ω̂𝒜 is a ℤ_{2}-graded complex isomorphic to Ω𝒜 /F^{n}Ω𝒜, explicitly

Ω̂𝒜 /F^{n}Ω̂𝒜 = ⨁_{k=0}^{n−1} Ω^{k}𝒜 ⊕ Ω^{n}𝒜 /b(Ω^{n+1}𝒜) ,  (6)

and it vanishes for n < 0. As a topological vector space, Ω̂𝒜 /F^{n}Ω̂𝒜 may fail to be separated because the image b(Ω^{n+1}𝒜) is not closed in general.

Definition 2.1 In any degree n ∈ ℤ, the periodic, non-periodic and negative cyclic homologies are respectively the (b + B)-homologies

HP_{n}(𝒜) = H_{n+2ℤ}(Ω̂𝒜) ,
HC_{n}(𝒜) = H_{n+2ℤ}(Ω̂𝒜 /F^{n}Ω̂𝒜) ,  (7)
HN_{n}(𝒜) = H_{n+2ℤ}(F^{n−1}Ω̂𝒜) .

Hence HP_{n}(𝒜) ≅ HP_{n+2}(𝒜) is 2-periodic, HC_{n}(𝒜) = 0 for n < 0 and HN_{n}(𝒜) ≅ HP_{n}(𝒜) for n ≤ 0. By construction these cyclic homology groups fit into a long exact sequence

… ⟶ HP_{n+1}(𝒜) ⟶^{S} HC_{n−1}(𝒜) ⟶^{B} HN_{n}(𝒜) ⟶^{I} HP_{n}(𝒜) ⟶ …  (8)

where S is induced by projection, I by inclusion, and the connecting map corresponds to the operator B. The link between cyclic and Hochshild homology may be obtained through non-commutative de Rham homology [16], defined as

HD_{n}(𝒜) := H_{n+2ℤ}(Ω̂𝒜 /F^{n+1}Ω̂𝒜) , ∀n ∈ ℤ .  (9)

This yields for any n ∈ ℤ a short exact sequence of ℤ_{2}-graded complexes

0 ⟶ G_{n}(𝒜) ⟶ Ω̂𝒜 /F^{n}Ω̂𝒜 ⟶ Ω̂𝒜 /F^{n−1}Ω̂𝒜 ⟶ 0 ,

where G_{n} is Ω^{n}𝒜 /bΩ^{n+1}𝒜 in degree n mod 2, and bΩ^{n}𝒜 in degree n − 1 mod 2. One has H_{n+2ℤ}(G_{n}) = HH_{n}(𝒜) and H_{n−1+2ℤ}(G_{n}) = 0, so that the associated six-term cyclic exact sequence in homology reduces to

0 → HD_{n−1}(𝒜) → HC_{n−1}(𝒜) → HH_{n}(𝒜) → HC_{n}(𝒜) → HD_{n−2}(𝒜) → 0 ,

and Connes’s SBI exact sequence [4] for cyclic homology is actually obtained by splicing together the above sequences for all n ∈ ℤ:

… ⟶ HC_{n+1}(𝒜) ⟶^{S} HC_{n−1}(𝒜) ⟶^{B} HH_{n}(𝒜) ⟶^{I} HC_{n}(𝒜) ⟶ … (10)

Hence the non-commutative de Rham homology group HD_{n}(𝒜) may be identified with the image of the periodicity shift S : HC_{n+2}(𝒜) → HC_{n}(𝒜). Clearly the exact sequence (8) can be transformed to (10) by taking the natural maps HP_{n}(𝒜) → HC_{n}(𝒜) and HN_{n}(𝒜) → HH_{n}(𝒜).

Passing to the dual theory, let Hom(Ω̂𝒜 , ℂ) be the ℤ_{2}-graded complex of linear maps Ω̂𝒜 → ℂ which are continuous for the adic topology on Ω̂𝒜 induced by the Hodge filtration. It is concretely described as the direct sum

Hom(Ω̂𝒜 , ℂ) = ⨁_{n≥0} Hom(Ω^{n}𝒜 , ℂ) ,

where Hom(Ω^{n}𝒜 , ℂ) is the space of continuous linear maps Ω^{n}𝒜 → ℂ. The space Hom(Ω̂𝒜 , ℂ) is endowed with the transposed of the boundary operator b + B on Ω̂𝒜. Then the periodic cyclic cohomology of 𝒜 is the cohomology of this complex:

HP^{n}(𝒜) = H^{n+2ℤ}(Hom(Ω̂𝒜 , ℂ)) .  (11)

One defines analogously the non-periodic and negative cyclic cohomologies which fit into an IBS long exact sequence.

2.2 X-complex and quasi-free algebras

We now turn to the description of the X-complex. It first appeared in the coalgebra context in Quillen’s work [28], and subsequently was used by Cuntz and Quillen in their formulation of cyclic homology [12]. Here we recall the X-complex construction for m-algebras. Let ℛ be an m-algebra. The space of non-commutative one-forms Ω^{1}ℛ is a ℛbimodule, hence we can take its quotient Ω^{1}ℛ_{♮} by the subspace of commutators [ℛ, Ω^{1}ℛ] = bΩ^{2}ℛ. Ω^{1}ℛ_{♮} may fail to be separated in general. However, it is automatically separated when ℛ is quasi-free, see below. In order to avoid confusions in the subsequent notations, we always write a one-form x_{0}dx_{1} ∈ Ω^{1}ℛ with a bold d when dealing with the X-complex of ℛ. The latter is the ℤ_{2}-graded complex [12]

X(ℛ) : ℛ ⇄_{\overline{b}}^{♮d} Ω^{1}ℛ_{♮} ,  (12)

where ℛ = X_{+}(ℛ) is located in even degree and Ω^{1}ℛ_{♮} = X_{−}(ℛ) in odd degree. The class of the generic element (x_{0}dx_{1} mod [,]) ∈ Ω^{1}ℛ_{♮} is usually denoted by ♮x_{0}dx_{1}. The map ♮d : ℛ → Ω^{1}ℛ_{♮} thus sends x ∈ ℛ to ♮dx. Also, the Hochschild boundary b : Ω^{1}ℛ → ℛ vanishes on the commutator subspace [ℛ, Ω^{1}ℛ], hence passes to a well-defined map \overline{b} : Ω^{1}ℛ_{♮} → ℛ. Explicitly the image of ♮x_{0}dx_{1} by \overline{b} is the commutator [x_{0}, x_{1}]. These maps are continuous and satisfy ♮d∘\overline{b} = 0 and \overline{b}∘♮d = 0, so that (X(ℛ), ♮d⊕\overline{b}) indeed defines a ℤ_{2}-graded complex. We mention that everything can be formulated when ℛ itself is a ℤ_{2}-graded algebra: we just have to replace everywhere the ordinary commutators by graded commutators, and the differentials anticommute with elements of odd degree. In particular one gets \overline{b}♮xdy = (−)^{|x|}[x, y], where |x| is the degree of x and [x, y] is the graded commutator. The X-complex is obviously a functor from m-algebras to ℤ_{2}-graded complexes: if ρ : ℛ → 𝒮 is a continuous homomorphism, it induces a chain map of even degree X(ρ) : X(ℛ) → X(𝒮), by setting X(ρ)(x) = ρ(x) and X(ρ)(♮x_{0}dx_{1}) = ♮ ρ(x_{0})dρ(x_{1}). In fact the X-complex may be identified with the quotient of the (b + B)- complex Ω̂ℛ by the subcomplex F^{1}Ω̂ℛ = bΩ^{2}ℛ ⊕ ∏_{k≥2} Ω^{k}ℛ of the Hodge filtration, i.e. there is an exact sequence

0 → F^{1}Ω̂ℛ → Ω̂ℛ → X(ℛ) → 0 .

It turns out that the X-complex is especially designed to compute the cyclic homology of algebras for which the subcomplex F^{1}Ω̂ℛ is contractible. This led Cuntz and Quillen to the following definition:

Definition 2.2 ([12]) An m-algebra ℛ is called quasi-free if there exists a continuous linear map ϕ : ℛ → Ω^{2}ℛ with property

ϕ(xy) = ϕ(x)y + xϕ(y) + dxdy , ∀x, y ∈ ℛ .  (13)

We refer to [12, 19] for many other equivalent definitions of quasi-free algebras. Let us just observe that a quasi-free algebra has dimension ≤ 1 with respect to Hochschild cohomology. Indeed, the map ϕ allows to contract the Hochschild complex of ℛ in dimensions > 1, and this contraction carries over to the cyclic bicomplex. First, the linear map

σ : Ω^{1}ℛ_{♮} → Ω^{1}ℛ , ♮xdy ↦ xdy + b(xϕ(y))

is well-defined because it vanishes on the commutator subspace [ℛ, Ω^{1}ℛ] = bΩ^{2}ℛ by the algebraic property of ϕ. Hence σ is a continuous linear splitting of the exact sequence 0 → bΩ^{2}ℛ → Ω^{1}ℛ → Ω^{1}ℛ_{♮} → 0. By the way, this implies that for a quasi-free algebra ℛ, the topological vector space Ω^{1}ℛ splits into the direct sum of two closed subspaces bΩ^{2}ℛ and Ω^{1}ℛ_{♮}. Then, we extend ϕ to a continuous linear map ϕ : Ω^{n}ℛ → Ω^{n+2}ℛ in all degrees n ≥ 1 by the formula

ϕ(x_{0}dx_{1} … dx_{n}) = ∑_{i=0}^{n}(−)^{ni}ϕ(x_{i})dx_{i+1} … dx_{n}dx_{0} … dx_{i−1} .

The following proposition gives a chain map γ : X(ℛ) → Ω̂ℛ which is inverse to the natural projection π : Ω̂ℛ → X(ℛ) up to homotopy. Remark that the infinite sum (1 − ϕ)^{−1} := ∑_{n=0}^{∞} ϕ^{n} makes sense as a linear map ℛ → Ω̂^{+}ℛ or Ω^{1}ℛ → Ω̂^{−}ℛ.

Proposition 2.3 Let ℛ be a quasi-free m-algebra. Then

i) The map γ : X(ℛ) → Ω̂ℛ defined for x, y ∈ ℛ by

γ(x) = (1 − ϕ)^{−1}(x)  (14)
γ(♮xdy) = (1 − ϕ)^{−1}(xdy + b(xϕ(y)))

is a chain map of even degree from the X-complex to the (b + B)-complex. ii) Let π : Ω̂ℛ → X(ℛ) be the natural projection. There is a contracting homotopy of odd degree h : Ω̂ℛ → Ω̂ℛ such that

π ∘ γ = Id on X(ℛ) ,
γ ∘ π = Id + [b + B, h] on Ω̂ℛ .

Hence X(ℛ) and Ω̂ℛ are homotopy equivalent.

Proof: See the proof of [22], Proposition 4.2. There the result was stated in the particular case of a tensor algebra ℛ = T 𝒜, but the general case of a quasi-free algebra is strictly identical (with the tensor algebra the image of γ actually lands in the subcomplex ΩT 𝒜 ⊂ Ω̂T 𝒜 for a judicious choice of ϕ, but for generic quasi-free algebras it is necessary to take the completion Ω̂ℛ of Ωℛ).

Extensions. Let 𝒜 be an m-algebra. By an extension of 𝒜 we mean an exact sequence of m-algebras 0 → ℐ → ℛ → 𝒜 → 0 provided with a continuous linear splitting 𝒜 → ℛ, and the topology of the ideal ℐ is induced by its inclusion in ℛ. Hence as a topological vector space ℛ is the direct sum of the closed subspaces ℐ and 𝒜. By convention, the powers ℐ^{n} of the ideal ℐ will always denote the image in ℛ of the n-th tensor power ℐ ˆ⊗ … ˆ⊗ℐ by the multiplication map. For n ≤ 0, we define ℐ^{0} as the algebra ℛ. Now let us suppose that all the powers ℐ^{n} are closed and direct summands in ℛ (this is automatically satisfied if ℛ is quasi-free). Then the quotients ℛ/ℐ^{n} are m-algebras and give rise to an inverse system with surjective homomorphisms

0 ← 𝒜 = ℛ/ℐ ← ℛ/ℐ^{2} ← … ← ℛ/ℐ^{n} ← …

We denote by ℛ̂ = \underleftarrow{lim}_{n} ℛ/ℐ^{n} the projective limit and view it as a proalgebra indexed by the directed set ℤ (see [19]). Since the bicomplex of noncommutative differential forms Ω̂ℛ and the X-complex X(ℛ) are functorial in ℛ, we can define Ω̂ℛ̂ and X(ℛ̂) as the ℤ_{2}-graded pro-complexes

Ω̂ℛ̂ = \underleftarrow{lim}_{n} Ω̂(ℛ/ℐ^{n}) = \underleftarrow{lim}_{m,n} Ω(ℛ/ℐ^{n})/F^{m}Ω(ℛ/ℐ^{n}) ,
X(ℛ̂) = \underleftarrow{lim}_{n} X(ℛ/ℐ^{n}) ,

endowed respectively with the total boundary maps b + B and ∂ = ♮d ⊕ \overline{b}. When ℛ is quasi-free, a refinement of Proposition 2.3 yields a chain map γ : X(ℛ̂) → Ω̂ℛ̂ inverse to the projection π : Ω̂ℛ̂ → X(ℛ̂) up to homotopy, which we call a generalized Goodwillie equivalence:

Proposition 2.4 Let 0 → ℐ → ℛ → 𝒜 → 0 be an extension of m-algebras, with ℛ quasi-free. Then the chain map γ : X(ℛ) → Ω̂ℛ extends to a homotopy equivalence of pro-complexes X(ℛ̂) → Ω̂ℛ̂.

Proof: We recall the proof because it will be useful for establishing Proposition 3.10. Let us introduce the following decreasing filtration of the space Ω^{m}ℛ by the subspaces H^{k}Ω^{m}ℛ, k ∈ ℤ:

H^{k}Ω^{m}ℛ = ∑_{k_{0}+...+k_{m}≥k} ℐ^{k_{0}}dℐ^{k_{1}} … dℐ^{k_{m}} .

Clearly H^{k+1}Ω^{m}ℛ ⊂ H^{k}Ω^{m}ℛ, and for k ≤ 0 H^{k}Ω^{m}ℛ = Ω^{m}ℛ. Morally, H^{k}Ω^{m}ℛ contains at least k powers of the ideal ℐ. The direct sum ⨁_{m} H^{k}Ω^{m}ℛ is stable by the operators d, b, κ, B for any k. We have to establish how k changes when the linear map ϕ : Ω^{m}ℛ → Ω^{m+2}ℛ is applied. First consider ϕ : ℛ → Ω^{2}ℛ. If x_{1}, …, x_{k} denote k elements in ℛ, one has by the algebraic property of ϕ (see [12])

ϕ(x_{1} … x_{k}) = ∑_{i=1}^{k} x_{1} … x_{i−1}ϕ(x_{i})x_{i+1} … x_{k}

+ ∑_{1≤i<j≤k} x_{1} … x_{i−1}dx_{i}x_{i+1} … x_{j−1}dx_{j}x_{j+1} … x_{k} .

Taking the elements x_{i} in the ideal ℐ and using that ϕ(ℐ) ⊂ Ω^{2}ℛ yields

ϕ(ℐ^{k}) ⊂ ∑_{i=1}^{k} ℐ^{i−1}dℛdℛℐ^{k−i} + ∑_{1≤i<j≤k} ℐ^{i−1}dℐ ℐ^{j−i−1}dℐ ℐ^{k−j} .

Therefore ϕ(ℐ^{k}) ⊂ H^{k−1}Ω^{2}ℛ for any k. Now from the definition of ϕ on Ω^{m}ℛ, one has

ϕ(ℐ^{k_{0}}dℐ^{k_{1}} … dℐ^{k_{m}}) ⊂ ∑_{l=0}^{m} ϕ(ℐ^{k_{l}})dℐ^{k_{l+1}} … dℐ^{k_{l−1}} ⊂ H^{k−1}Ω^{m+2}ℛ

whenever k = k_{0} + … + k_{m}, hence ϕ(H^{k}Ω^{m}ℛ) ⊂ H^{k−1}Ω^{m+2}ℛ. Now let us evaluate the even part of the chain map γ : ℛ → Ω̂^{+}ℛ. The part of γ landing in Ω^{2m}ℛ is the m-th power ϕ^{m}. One gets ϕ^{m}(ℐ^{k}) ⊂ H^{k−m}Ω^{2m}ℛ, hence ϕ^{m} sends the quotient algebra ℛ/ℐ^{k} to Ω^{2m}(ℛ/ℐ^{n}) provided (1 + 2m)n ≤ k − m (indeed 1 + 2m is the maximal number of factors ℛ in the tensor product Ω^{2m}ℛ). Passing to the projective limits, ϕ^{m} induces a well-defined map ℛ̂ → Ω^{2m}ℛ̂, and summing over all degrees 2m yields γ : ℛ̂ → Ω̂^{+}ℛ̂. Let us turn to the odd part of the chain map γ : Ω^{1}ℛ_{♮} → Ω̂^{−}ℛ. By construction, it is the composition of the linear map σ : Ω^{1}ℛ_{♮} → Ω^{1}ℛ with all the powers ϕ^{m} : Ω^{1}ℛ → Ω^{2m+1}ℛ. One has ♮(ℐ^{k}dℛ + ℛd(ℐ^{k})) ⊂ ♮(ℐ^{k}dℛ + ℐ^{k−1}dℐ), and by the definition of σ,

σ ♮(ℐ^{k}dℛ + ℐ^{k−1}dℐ)  ⊂ ℐ^{k}dℛ + ℐ^{k−1}dℐ + b(ℐ^{k}ϕ(ℛ) + ℐ^{k−1}ϕ(ℐ))
⊂ H^{k}Ω^{1}ℛ + bH^{k−1}Ω^{2}ℛ ⊂ H^{k−1}Ω^{1}ℛ .

Therefore (ϕ^{m} ∘ σ)♮(ℐ^{k}dℛ + ℐ^{k−1}dℐ) ⊂ H^{k−m−1}Ω^{2m+1}ℛ. Since ℛ is the direct sum of ℛ/ℐ^{k} and ℐ^{k} as a topological vector space, the quotient Ω^{1}(ℛ/ℐ^{k})_{♮} coincides with Ω^{1}ℛ/(ℐ^{k}dℛ + ℛd(ℐ^{k}) + [ℛ, Ω^{1}ℛ]), and the map ϕ^{m} ∘ σ : Ω^{1}(ℛ/ℐ^{k})_{♮} → Ω^{2m+1}(ℛ/ℐ^{n}) is well-defined provided (2m + 2)n ≤ k − m − 1. Thus passing to the projective limits induces a map Ω^{1}ℛ̂_{♮} → Ω^{2m+1}ℛ̂, and summing over m yields γ : Ω^{1}ℛ̂_{♮} → Ω̂^{−}ℛ̂. Finally, the contracting homotopy h of Proposition 2.3 is also constructed from ϕ (see [22] Proposition 4.2), hence extends to a contracting homotopy h : Ω̂ℛ̂ → Ω̂ℛ̂. The relations π ∘ γ = Id on X(ℛ̂) and γ ∘ π = Id + [b + B, h] on Ω̂ℛ̂ follow immediately.

Adic filtration. Suppose that ℐ is a (not necessarily closed) two-sided ideal in ℛ, provided with its own topology of m-algebra for which the inclusion ℐ → ℛ is continuous and the multiplication map ℛ^{+} × ℐ × ℛ^{+} → ℐ is jointly continuous. As usual we define the powers ℐ^{n} as the two-sided ideals corresponding to the image in ℛ of the n-fold tensor products ℐ ˆ⊗ … ˆ⊗ℐ under multiplication. Following [12], we introduce the adic filtration of X(ℛ) by the subcomplexes

F_{ℐ}^{2n}X(ℛ) : ℐ^{n+1} + [ℐ^{n}, ℛ] ⇄ ♮ℐ^{n}d ℛ  (15)
F_{ℐ}^{2n+1}X(ℛ) : ℐ^{n+1} ⇄ ♮(ℐ^{n+1}d ℛ + ℐ^{n}d ℐ) ,

where the commutator [ℐ^{n}, ℛ] is by definition the image of ℐ^{n}dℛ under the Hochschild operator b, and ℐ^{n} is defined as the unitalized algebra ℛ^{+} for n ≤ 0. This is a decreasing filtration because F_{ℐ}^{n+1}X(ℛ) ⊂ F_{ℐ}^{n}X(ℛ), and for n < 0 one has F_{ℐ}^{n}X(ℛ) = X(ℛ). Denote by X_{n}(ℛ, ℐ) = X(ℛ)/F_{ℐ}^{n}X(ℛ) the quotient complex. It is generally not separated. One gets in this way an inverse system of ℤ_{2}-graded complexes {X_{n}(ℛ, ℐ)}_{n∈ℤ} with projective limit X̂(ℛ, ℐ).

Now suppose that we start from an extension of m-algebras 0 → ℐ → ℛ → 𝒜 → 0 with continuous linear splitting, and assume that any power ℐ^{n} is a direct summand in ℛ. Then, the sequence X_{n}(ℛ, ℐ) is related to the Xcomplexes of the quotient m-algebras ℛ/ℐ^{n}:

0 ← X_{0}(ℛ, ℐ) = 𝒜 /[𝒜 , 𝒜] ← X_{1}(ℛ, ℐ) = X(𝒜) ← …
… ← X(ℛ/ℐ^{n−1}) ← X_{2n−1}(ℛ, ℐ) ← X_{2n}(ℛ, ℐ) ← X(ℛ/ℐ^{n}) ← …

Hence the projective limit of the system {X_{n}(ℛ, ℐ)}_{n∈ℤ} is isomorphic to the X-complex of the pro-algebra ℛ̂:

X̂(ℛ, ℐ) = \underleftarrow{lim}_{n} X_{n}(ℛ, ℐ) = \underleftarrow{lim}_{n} X(ℛ/ℐ^{n}) = X(ℛ̂) .  (16)

The pro-complex X̂(ℛ, ℐ) is naturally filtered by the family of subcomplexes F^{n}X̂(ℛ, ℐ) = Ker(X̂ → X_{n}). If 0 → 𝒥 → 𝒮 → ℬ → 0 is another extension of m-algebras with continuous linear splitting, then the space of linear maps between the two pro-complexes X̂(ℛ, ℐ) and X̂(𝒮 , 𝒥), or between X̂ and X̂′ for short, is given by

Hom(X̂, X̂′) = \underleftarrow{lim}_{m} ( \underrightarrow{lim}_{n} Hom(X_{n}, X′_{m})) ,  (17)

where Hom(X_{n}, X′_{m}) is the space of continuous linear maps between the ℤ_{2}- graded complexes X_{n}(ℛ, ℐ) and X_{m}(𝒮 , 𝒥). Thus Hom(X̂, X̂′) is a ℤ_{2}- graded complex. It corresponds to the space of linear maps {f : X̂ → X̂′ | ∀k, ∃n : f(F^{n}X̂) ⊂ F^{k}X̂′}; the boundary of an element f of parity |f| is given by the graded commutator ∂ ∘ f − (−)^{|f|}f ∘ ∂ with the bounary maps ∂ = ♮d⊕\overline{b} on X̂ and X̂′. Hom(X̂, X̂′) itself is filtered by the subcomplexes of linear maps of order ≤ n for any n ∈ ℕ:

Hom^{n}(X̂, X̂′) = {f : X̂ → X̂′ | ∀k, f(F^{k+n}X̂) ⊂ F^{k}X̂′} .  (18)

These Hom-complexes will be used in the various definitions of bivariant cyclic cohomology, once the relation between the adic filtration over the X-complex of a quasi-free algebra ℛ and the Hodge filtration of the cyclic bicomplex over the quotient algebra 𝒜 = ℛ/ℐ is established.

2.3 The tensor algebra

Taking ℛ as the tensor algebra of an m-algebra 𝒜 provides the link with cyclic homology [9, 12]. The (non-unital) tensor algebra T 𝒜 is the completion of the algebraic direct sum ⨁_{n≥1} 𝒜^{ˆ⊗n} with respect to the family of seminorms

p̂ = ⨁_{n≥1} p^{⊗n} = p ⊕ (p ⊗ p) ⊕ (p ⊗ p ⊗ p) ⊕ …,

where p runs through all the submultiplicative seminorms on 𝒜. Of course p^{⊗n} is the projective seminorm on 𝒜^{⊗n} defined by a generalization of (2). These seminorms are submultiplicative with respect to the tensor product 𝒜^{ˆ⊗n} × 𝒜^{ˆ⊗m} → 𝒜^{ˆ⊗n+m} and therefore the completion T 𝒜 is an m-algebra. It is free, hence quasi-free: a linear map ϕ : T 𝒜 → Ω^{2}T 𝒜 with the property ϕ(xy) = ϕ(x)y + xϕ(y) + dxdy may be canonically constructed by setting ϕ(a) = 0 on the generators a ∈ 𝒜 ⊂ T 𝒜, and then recursively ϕ(a_{1} ⊗ a_{2}) = da_{1}da_{2}, ϕ(a_{1} ⊗ a_{2} ⊗ a_{3}) = (da_{1}da_{2})a_{3} + d(a_{1} ⊗ a_{2})da_{3}, and so on... The multiplication map T 𝒜 → 𝒜, sending a_{1}⊗…⊗a_{n} to the product a_{1} … a_{n}, is continuous and we denote by J𝒜 its kernel. Since the inclusion σ_{𝒜} : 𝒜 → T 𝒜 is a continuous linear splitting of the multiplication map, the two-sided ideal J𝒜 is a direct summand in T 𝒜. This implies a linearly split quasi-free extension 0 → J𝒜 → T 𝒜 → 𝒜 → 0. It is the universal free extension of 𝒜 in the following sense: let 0 → ℐ → ℛ → 𝒜 → 0 be any other extension (ℛ is not necessarily quasi-free), provided with a continuous linear splitting σ : 𝒜 → ℛ. Then one gets a commutative diagram

[figure: 0; J𝒜; ρ_{∗}; T 𝒜; ρ_{∗}; 𝒜; 0; 0; ℐ; ℛ; 𝒜; σ; 0]

where ρ_{∗} : T 𝒜 → ℛ is the continuous algebra homomorphism obtained by setting ρ_{∗}(a) = σ(a) on the generators a ∈ 𝒜 ⊂ T 𝒜. Moreover, the homomorphism ρ_{∗} is independent of the linear splitting σ up to homotopy (two splittings can always be connected by a linear homotopy). As remarked by Cuntz and Quillen [12], the tensor algebra is closely related to a deformation of the algebra of even-degree noncommutative differential forms Ω^{+}𝒜. Endow the space Ω^{+}𝒜 with the Fedosov product

ω_{1} ⊙ ω_{2} := ω_{1}ω_{2} − dω_{1}dω_{2} , ω_{i} ∈ Ω^{+}𝒜 .  (19)

Then (Ω^{+}𝒜 , ⊙) is a dense subalgebra of T 𝒜, with the explicit correspondence

Ω^{+}𝒜 ∋ a_{0}da_{1} … da_{2n} ⟷ a_{0} ⊗ ω(a_{1}, a_{2}) ⊗ … ⊗ ω(a_{2n−1}, a_{2n}) ∈ T 𝒜 .

It turns out that the Fedosov product ⊙ extends to the projective limit Ω̂^{+}𝒜 and the latter is isomorphic to the pro-algebra

T̂𝒜 = lim_{⟵}_{n} T 𝒜 /(J𝒜)^{n} .  (20)

Moreover, Ω̂𝒜 and X(T̂𝒜) = X̂(T 𝒜 , J𝒜) are isomorphic as ℤ_{2}-graded provector spaces [12], and this isomorphism identifies the Hodge filtration F^{n}Ω̂𝒜 with the adic filtration F^{n}X̂(T 𝒜 , J𝒜). By a fundamental result of Cuntz and Quillen, all these identifications are homotopy equivalences of pro-complexes, i.e. the boundary b + B on Ω̂𝒜 corresponds to the boundary ♮d ⊕ \overline{b} on X̂(T 𝒜 , J𝒜) up to homotopy and rescaling (see [12]). Hence the periodic and negative cyclic homologies of 𝒜 may be computed respectively by X̂(T 𝒜 , J𝒜) and F^{n}X̂(T 𝒜 , J𝒜). Also, the non-periodic cyclic homology of 𝒜 may be computed by the quotient complex X_{n}(T 𝒜 , J𝒜) which is homotopy equivalent to the complex Ω̂𝒜 /F^{n}Ω̂𝒜. More generally the same result holds if tensor algebra T 𝒜 is replaced by any quasi-free extension of 𝒜. Indeed if 0 → ℐ → ℛ → 𝒜 → 0 is a quasi-free extension with c ontinuous linear splitting, the classifying homomorphism ρ_{∗} : T 𝒜 → ℛ induces a map of pro-complexes X(ρ_{∗}) : X̂(T 𝒜 , J𝒜) → X̂(ℛ, ℐ) compatible with the adic filtrations induced by the ideals J𝒜 and ℐ. It turns out to be a homotopy equivalence, irrespective to the choice of ℛ:

Theorem 2.5 (Cuntz-Quillen [12]) For any linearly split extension of malgebras 0 → ℐ → ℛ → 𝒜 → 0 with ℛ quasi-free, one has isomorphisms

HP_{n}(𝒜) = H_{n+2ℤ}(X̂(ℛ, ℐ)) ,
HC_{n}(𝒜) = H_{n+2ℤ}(X_{n}(ℛ, ℐ)) ,  (21)
HD_{n}(𝒜) = H_{n+2ℤ}(X_{n+1}(ℛ, ℐ)) ,
HN_{n}(𝒜) = H_{n+2ℤ}(F^{n−1}X̂(ℛ, ℐ)) .

These filtrations also allow to define various versions of bivariant cyclic cohomology, which may be formulated either within the X-complex framework or by means of the (b + B)-complex of differential forms.

Definition 2.6 ([12]) Let 𝒜 and ℬ be m-algebras, and choose arbitrary (linearly split) quasi-free extensions 0 → ℐ → ℛ → 𝒜 → 0 and 0 → 𝒥 → 𝒮 → ℬ → 0. The bivariant periodic cyclic cohomology of 𝒜 and ℬ is the homology of the ℤ_{2}-graded complex (17) of linear maps between the pro-complexes X̂(ℛ, ℐ) and X̂(𝒮 , 𝒥):

HP^{n}(𝒜 , ℬ) = H_{n+2ℤ}(Hom(X̂(ℛ, ℐ), X̂(𝒮 , 𝒥))) , ∀n ∈ ℤ .  (22)

For any n ∈ ℤ, the non-periodic cyclic cohomology group HC^{n}(𝒜 , ℬ) of degree n is the homology, in degree n mod 2, of the ℤ_{2}-graded subcomplex (18) of linear maps of order ≤ n:

HC^{n}(𝒜 , ℬ) = H_{n+2ℤ}(Hom^{n}(X̂(ℛ, ℐ), X̂(𝒮 , 𝒥))) .  (23)

The embedding Hom^{n} ↪ Hom^{n+2} induces, for any n, the S-operation in bivariant cyclic cohomology S : HC^{n}(𝒜 , ℬ) → HC^{n+2}(𝒜 , ℬ), and Hom^{n} ↪ Hom yields a natural map HC^{n}(𝒜 , ℬ) → HP^{n}(𝒜 , ℬ).

Of course the bivariant periodic theory has period two: HP^{n+2} = HP^{n}. Let us look at particular cases. The algebra ℂ is quasi-free hence X̂(T ℂ, Jℂ) is homotopically equivalent to X(ℂ) : ℂ ⇄ 0, and the periodic cyclic homology of ℂ is simply HP_{0}(ℂ) = ℂ and HP_{1}(ℂ) = 0. This implies that for any m-algebra 𝒜, we get the usual isomorphisms HP^{n}(ℂ, 𝒜) ≅ HP_{−n}(𝒜) and HP^{n}(𝒜 , ℂ) ≅ HP^{n}(𝒜) in any degree n. For the non-periodic theory, one has the isomorphism HC^{n}(ℂ, 𝒜) ≅ HN_{−n}(𝒜) with negative cyclic homology, and HC^{n}(𝒜 , ℂ) ≅ HC^{n}(𝒜) is the non-periodic cyclic cohomology of Connes [4]. Finally, since any class φ ∈ HC^{p}(𝒜 , ℬ) is represented by a chain map sending the subcomplex F^{n}X̂(T 𝒜 , J𝒜) to F^{n−p}X̂(T ℬ, Jℬ) for any n ∈ ℤ, it is not difficult to check that φ induces a transformation of degree −p between the SBI exact sequences for 𝒜 and ℬ, i.e. a graded-commutative diagram

\xymatrix{HP_{n+1}(𝒜) \ar[r]^{S} \ar[d]^{φ} & HC_{n−1}(𝒜) \ar[r]^{B} \ar[d]^{φ} & HN_{n}(𝒜) \ar[r]^{I} \ar[d]^{φ} & HP_{n}(𝒜) \ar[d]^{φ} \\ HP_{n−p+1}(ℬ) \ar[r]^{S} & HC_{n−p−1}(ℬ) \ar[r]^{B} & HN_{n−p}(ℬ) \ar[r]^{I} & HP_{n−p}(ℬ)}

The graded-commutativity comes from the fact that the middle square is actually anticommutative when φ is of odd degree, for in this case the connecting morphism B anticommutes with the chain map representing φ.

3 Quasihomomorphisms and Chern character

In this section we define quasihomomorphisms for metrizable (or Fréchet) malgebras and construct a bivariant Chern character. The topology of a Fréchet m-algebra is defined by a countable family of submultiplicative seminorms, and can alternatively be considered as the projective limit of a sequence of Banach algebras [27]. In particular, the projective tensor product of two Fréchet malgebras is again a Fréchet m-algebra. We say that a Fréchet m-algebra ℐ is p-summable (with p ≥ 1 an integer), if there is a continuous trace Tr : ℐ^{p} → ℂ on the pth power of ℐ. Recall that by definition, ℐ^{p} is the image in ℐ of the p-th completed tensor product ℐ ˆ⊗ … ˆ⊗ℐ under the multiplication map. Hence the trace is understood as a continuous linear map ℐ ˆ⊗ … ˆ⊗ℐ → ℂ, and the tracial property means that it vanishes on the image of 1 − λ, where the operator λ is the backward cyclic permutation λ(i_{1} ⊗ … ⊗ i_{p}) = i_{p} ⊗ i_{1} … ⊗ i_{p−1}. In the low degree p = 1 we interpret the trace just as a linear map ℐ → ℂ vanishing on the subspace of commutators [ℐ , ℐ] := bΩ^{1}ℐ. Now consider any Fréchet m-algebra ℬ and form the completed tensor product ℐ ˆ⊗ℬ. Suppose that ℰ is a Fréchet m-algebra containing ℐ ˆ⊗ℬ as a (not necessarily closed) two-sided ideal, in the sense that the inclusion ℐ ˆ⊗ℬ → ℰ is continuous. The left and right multiplication maps ℰ ×ℐ ˆ⊗ℬ×ℰ → ℐ ˆ⊗ℬ are then automatically jointly continuous (see [7]). As in [13], we define the semidirect sum ℰ ⋉ ℐ ˆ⊗ℬ as the algebra modeled on the vector space ℰ ⊕ ℐ ˆ⊗ℬ, where the product is such that as many elements as possible are put in the summand ℐ ˆ⊗ℬ. The semi-direct sum is a Fréchet algebra but it may fail to be multiplicatively convex in general. The situation when ℰ ⋉ ℐ ˆ⊗ℬ is a Fréchet m-algebra will be depicted as

ℰ ▷ ℐ ˆ⊗ℬ  (24)

to stress the analogy with [8]. The definition of quasihomomorphisms involves a ℤ_{2}-graded version of ℰ ▷ ℐ ˆ⊗ℬ, depending only on a choice of parity (even or odd). It is constructed as follows:

1) Even case: Define ℰ_{+}^{s} as the Fréchet m-algebra ℰ ⋉ ℐ ˆ⊗ℬ. It is endowed with a linear action of the group ℤ_{2} by automorphisms: the image of an element (a, b) ∈ ℰ ⊕ ℐ ˆ⊗ℬ under the generator F of the group is (a + b, −b). We define the ℤ_{2}-graded algebra ℰ^{s} as the crossed product ℰ_{+}^{s} ⋊ ℤ_{2}. Hence ℰ^{s} splits as the direct sum ℰ_{+}^{s} ⊕ ℰ_{−}^{s} where ℰ_{+}^{s} is the subalgebra of even degree elements and ℰ_{−}^{s} = F ℰ_{+}^{s} is the odd subspace. This definition is rather abstract but there is a concrete description of ℰ^{s} in terms of 2 × 2 matrices. Consider M_{2}(ℰ) = M_{2}(ℂ) ˆ⊗ℰ as a ℤ_{2}-graded algebra with grading operator ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ). Thus diagonal elements are of even degree and off-diagonal elements are odd. ℰ^{s} can be identified with a (nonclosed) subalgebra of M_{2}(ℂ) ˆ⊗ℰ in the following way. Any element x+F y ∈ ℰ^{s} may be decomposed to its even and odd parts x, y ∈ ℰ ⊕ℐ ˆ⊗ℬ, with x = (a, b) and y = (c, d). Then x + F y is represented by the matrix

x + F y = (\begin{matrix}a + b & c \\ c + d & a \end{matrix}) with a, c ∈ ℰ , b, d ∈ ℐ ˆ⊗ℬ .

The action of ℤ_{2} on ℰ_{+}^{s} is implemented by the adjoint action of the following odd-degree multiplier of M_{2}(ℰ):

F = (\begin{matrix}0 & 1 \\ 1 & 0 \end{matrix}) ∈ M_{2}(ℂ) , F^{2} = 1 .  (25)

Denote by ℐ^{s} = ℐ_{+}^{s} ⊕ ℐ_{−}^{s} the ℤ_{2}-graded algebra M_{2}(ℂ) ˆ⊗ℐ, with ℐ_{+}^{s} the subalgebra of diagonal elements and ℐ_{−}^{s} the off-diagonal subspace. We thus have an inclusion of ℐ^{s} ˆ⊗ℬ as a (non-closed) two-sided ideal in ℰ^{s}, with ℰ^{s} ▷ ℐ^{s} ˆ⊗ℬ. The commutator [F, ℰ_{+}^{s}] is contained in ℐ_{−}^{s} ˆ⊗ℬ. Finally, we denote by tr_{s} the supertrace of even degree on M_{2}(ℂ):

tr_{s} : M_{2}(ℂ) → ℂ , tr_{s} (\begin{matrix}a′ & c \\ c′ & a \end{matrix}) = a′ − a .

2) Odd case: Now regard M_{2}(ℰ) as a trivially graded algebra and define ℰ_{+}^{s} as the (non-closed) subalgebra

ℰ_{+}^{s} = (\begin{matrix}ℰ & ℐ ˆ⊗ℬ \\ ℐ ˆ⊗ℬ & ℰ \end{matrix} )  (26)

provided with its own topology of complete m-algebra. Let C_{1} = ℂ ⊕ εℂ be the complex Clifford algebra of the one-dimensional euclidian space. C_{1} is the ℤ_{2}-graded algebra generated by the unit 1 ∈ ℂ in degree zero and ε in degree one with ε^{2} = 1. We define the ℤ_{2}-graded algebra ℰ^{s} as the tensor product C_{1} ˆ⊗ℰ_{+}^{s}. Hence ℰ^{s} = ℰ_{+}^{s} ⊕ ℰ_{−}^{s} where ℰ_{+}^{s} is the subalgebra of even degree and ℰ_{−}^{s} = εℰ_{+}^{s} is the odd subspace. Similarly, define ℐ^{s} = M_{2}(C_{1}) ˆ⊗ℐ = ℐ_{+}^{s}⊕ℐ_{−}^{s}. Then ℐ^{s} ˆ⊗ℬ is a (non-closed) two-sided ideal of ℰ^{s} and one has ℰ^{s} ▷ ℐ^{s} ˆ⊗ℬ. The matrix

F = ε (\begin{matrix}1 & 0 \\ 0 & −1 \end{matrix}) ∈ M_{2}(C_{1}) , F^{2} = 1  (27)

is an odd multiplier of ℰ^{s} and the commutator [F, ℰ_{+}^{s}] is contained in ℐ_{−}^{s} ˆ⊗ℬ. Finally, we define a supertrace tr_{s} of odd degree on C_{1} by sending the generators 1 to 0 and ε to ±\sqrt{2i}. The normalization ±\sqrt{2i} is chosen for compatibility with Bott periodicity, see [22]. We will choose conventionally the “sign” as −\sqrt{2i} in order to simplify the subsequent formulas. One thus has

tr_{s} : M_{2}(C_{1}) → ℂ , tr_{s} (\begin{matrix}a + εa′ & b + εb′ \\ c + εc′ & d + εd′ \end{matrix}) = −\sqrt{2i} (a′ + d′) .

The objects F, ℰ^{s} and ℐ^{s} are defined in such a way that we can handle the even and odd case simultaneously. This allows to give the following synthetic definition of quasihomomorphisms.

Definition 3.1 Let 𝒜, ℬ, ℐ, ℰ be Fréchet m-algebras. Assume that ℐ is p-summable and ℰ ▷ ℐ ˆ⊗ℬ. A quasihomomorphism from 𝒜 to ℬ is a continuous homomorphism

ρ : 𝒜 → ℰ^{s} ▷ ℐ^{s} ˆ⊗ℬ  (28)

sending 𝒜 to the even degree subalgebra ℰ_{+}^{s}. The quasihomomorphism comes equipped with a degree (even or odd) depending on the degree chosen for the above construction of ℰ^{s}. In particular, the linear map a ∈ 𝒜 ↦ [F, ρ(a)] ∈ ℐ_{−}^{s} ˆ⊗ℬ is continuous.

In other words, a quasihomomorphism of even degree ρ = ( \begin{matrix}ρ_{+} & 0 \\ 0 & ρ_{−} \end{matrix} ) is a pair of homomorphisms (ρ_{+}, ρ_{−}) : 𝒜 ⇉ ℰ such that the difference ρ_{+}(a) − ρ_{−}(a) lies in the ideal ℐ ˆ⊗ℬ for any a ∈ 𝒜. A quasihomomorphism of odd degree is a homomorphism ρ : 𝒜 → M_{2}(ℰ) such that the off-diagonal elements land in ℐ ˆ⊗ℬ.

The Cuntz-Quillen formalism for bivariant cyclic cohomology HC^{n}(𝒜 , ℬ) requires to work with quasi-free extensions of 𝒜 and ℬ. Hence let us suppose that we choose such extensions of m-algebras

0 → 𝒢 → ℱ → 𝒜 → 0 , 0 → 𝒥 → ℛ → ℬ → 0 ,

with ℱ and ℛ quasi-free. We always take ℱ = T 𝒜 as the universal free extension of 𝒜, but we leave the possibility to take any quasi-free extension ℛ for ℬ since the tensor algebra T ℬ will not be an optimal choice in general. The first step toward the bivariant Chern character is to lift a given quasihomomorphism ρ : 𝒜 → ℰ^{s} ▷ ℐ^{s} ˆ⊗ℬ to a quasihomomorphism from ℱ to ℛ, compatible with the filtrations by the ideals 𝒢 ⊂ ℱ, 𝒥 ⊂ ℛ. This requires to fix some admissibility conditions on the intermediate algebra ℰ:

Definition 3.2 Let 0 → 𝒥 → ℛ → ℬ → 0 be a quasi-free extension of ℬ, and let ℐ be p-summable with trace Tr : ℐ^{p} → ℂ. We say that ℰ ▷ ℐ ˆ⊗ℬ is provided with an ℛ-admissible extension if there are algebras ℳ ▷ ℐ ˆ⊗ℛ and 𝒩 ▷ ℐ ˆ⊗𝒥 and a commutative diagram of extensions

[figure: 0; 𝒩; ℳ; ℰ; 0; 0; ℐ ˆ⊗𝒥; ℐ ˆ⊗ℛ; ℐ ˆ⊗ℬ; 0]  (29)

with the following properties: i) Any power 𝒩 ^{n} is a direct summand in ℳ (as a topological vector space);

ii) For any degree n ≥ max(1, p−1), the linear map (ℐ ˆ⊗ℛ)^{n}d(ℐ ˆ⊗ℛ) → Ω^{1}ℛ_{♮} induced by the trace ℐ^{n+1} → ℂ factors through the quotient

♮(ℐ ˆ⊗ℛ)^{n}d(ℐ ˆ⊗ℛ) = (ℐ ˆ⊗ℛ)^{n}d(ℐ ˆ⊗ℛ) mod [ℳ , Ω^{1}ℳ] ,

and the chain map Tr : F_{ℐ ˆ⊗ℛ}^{2n+1}X(ℳ) → X(ℛ) thus obtained is of order zero with respect to the adic filtration induced by the ideals 𝒩 ⊂ ℳ and 𝒥 ⊂ ℛ.

In the following we will say that ℰ is ℛ-admissible, keeping in mind that the extension ℳ is given. Condition i) is automatically satisfied for example if ℳ is a quasi-free algebra (this will not always be the case). The chain map Tr of condition ii) is constructed as follows. For n ≥ 1 one has the inclusion ♮(ℐ ˆ⊗ℛ)^{n+1}dℳ ⊂ ♮(ℐ ˆ⊗ℛ)^{n}d(ℐ ˆ⊗ℛ), so that the subcomplex of the ℐ ˆ⊗ℛadic filtration reads

F_{ℐ ˆ⊗ℛ}^{2n+1}X(ℳ) : (ℐ ˆ⊗ℛ)^{n+1} ⇄ ♮(ℐ ˆ⊗ℛ)^{n}d(ℐ ˆ⊗ℛ) .

Then, the trace ℐ^{n+1} → ℂ induces a partial trace (ℐ ˆ⊗ℛ)^{n+1} → ℛ. The latter combined with ♮(ℐ ˆ⊗ℛ)^{n}d(ℐ ˆ⊗ℛ) → Ω^{1}ℛ_{♮} yields a linear map in any degree n ≥ max(1, p − 1)

Tr : F_{ℐ ˆ⊗ℛ}^{2n+1}X(ℳ) → X(ℛ) ,  (30)

compatible with the inclusions F_{ℐ ˆ⊗ℛ}^{2n+3}X(ℳ) ⊂ F_{ℐ ˆ⊗ℛ}^{2n+1}X(ℳ). The trace over ℐ^{n+1} ensures that (30) is a chain map. It is not obvious, however, that it is automatically of degree zero with respect to the 𝒩 -adic and 𝒥 -adic filtrations, i.e. that the intersection F_{ℐ ˆ⊗ℛ}^{2n+1}X(ℳ) ∩ F_{𝒩}^{k} X(ℳ) is mapped to F_{𝒥}^{k} X(ℛ) for any k ∈ ℤ. This should be imposed as a condition. Remark that the case p = 1, n = 0 is pathological, since there is no canonical way to map the space ♮((ℐ ˆ⊗ℛ)dℳ + ℳ^{+}d(ℐ ˆ⊗ℛ)) to Ω^{1}ℛ_{♮} using only the trace over ℐ. In this situation, it seems preferable to impose directly the existence of a chain map Tr : F_{ℐ ˆ⊗ℛ}^{1} X(ℳ) → X(ℛ) in the definition of admissibility.

Example 3.3 When 𝒜 is arbitrary and ℬ = ℂ, a p-summable quasihomomorphism represents a K-homology class of 𝒜 in the sense of [4, 5]. Here we take ℐ = ℒ^{p}(H) as the Schatten ideal of p-summable operators on a separable infinite-dimensional Hilbert space H. Recall that ℐ is a two-sided ideal in the algebra of all bounded operators ℒ = ℒ (H). ℐ is a Banach algebra for the norm ∥x∥_{p} = (Tr(|x|^{p}))^{1/p}, ℒ is provided with the operator norm, and the products ℐ × ℒ × ℐ → ℐ are jointly continuous. Since ℒ and ℐ are Banach algebras, the semi-direct sum ℒ ⋉ℐ is automatically a Banach algebra and we can write ℒ ▷ ℐ. A p-summable K-homology class of even degree is represented by a pair of continuous homomorphisms (ρ_{+}, ρ_{−}) : 𝒜 ⇉ ℒ such that the difference ρ_{+} − ρ_{−} lands to ℐ. We get in this way an even degree quasihomomorph ism ρ : 𝒜 → ℒ^{s} ▷ ℐ^{s}. Here it is important to note that by a slight modification of the intermediate algebra ℒ, it is always possible to consider ℐ as a closed ideal [13]. Indeed if we define ℰ as the Banach algebra

ℰ = ℒ ⋉ ℐ

then one clearly has ℰ ▷ ℐ and ℐ is closed in ℰ by construction. The pair of homomorphisms (ρ_{+}, ρ_{−}) : 𝒜 ⇉ ℒ may be replaced with a new pair (ρ′_{+}, ρ′_{−}) : 𝒜 ⇉ ℰ by setting ρ′_{+}(a) = (ρ_{−}(a), ρ_{+}(a) − ρ_{−}(a)) and ρ′_{−}(a) = (ρ_{−}(a), 0) in ℒ ⊕ ℐ. The above K-homology class is then represented by the new quasihomomorphism ρ′ : 𝒜 → ℰ^{s} ▷ ℐ^{s}. In the odd case, a p-summable K-homology class is represented by a continuous homomorphism ρ : 𝒜 → ( \begin{matrix}ℒ & ℐ \\ ℐ & ℒ \end{matrix} ), which can be equivalently described as a homomorphism ρ′ : 𝒜 → M_{2}(ℰ) with off-diagonal elements in ℐ. Concerning cyclic homology, the algebra ℂ is quasi-free, hence the quasi-free extension ℛ = ℂ and 𝒥 = 0 computes the cyclic homology of ℂ. Therefore by choosing ℳ = ℰ and 𝒩 = 0, the algebra ℰ ▷ ℐ is ℂ-admissible (condition ii) is trivial since Ω^{1}ℂ_{♮} = 0).

Example 3.4 More generally, if ℐ is a p-summable Fréchet m-algebra contained as a (not necessarily closed) two-sided ideal in a unital Fréchet m-algebra ℒ, with ℒ ▷ ℐ, a p-summable quasihomomorphism from 𝒜 to ℬ could be constructed from the generic intermediate algebra ℰ = ℒ ˆ⊗ℬ, provided that the map ℐ ˆ⊗ℬ → ℰ is injective. If 0 → 𝒥 → ℛ → ℬ → 0 is a quasi-free extension of ℬ, the choice ℳ = ℒ ˆ⊗ℛ and 𝒩 = ℒ ˆ⊗𝒥 shows that ℰ is ℛ-admissible provided that the maps ℐ ˆ⊗ℛ → ℳ and ℐ ˆ⊗𝒥 → 𝒩 are injective. In fact it is easy to get rid of these injectivity conditions by redefining the algebra

ℰ = (ℒ ⋉ ℐ) ˆ⊗ℬ ,

which contains ℐ ˆ⊗ℬ as a closed ideal. Then ℰ becomes automatically ℛadmissible by taking ℳ = (ℒ ⋉ ℐ) ˆ⊗ℛ and 𝒩 = (ℒ ⋉ ℐ) ˆ⊗𝒥 (remark that (ℒ ⋉ ℐ)^{n} = ℒ ⋉ ℐ for any n because ℒ is unital, hence 𝒩 ^{n} is a direct summand in ℳ). The chain map Tr : F_{ℐ ˆ⊗ℛ}^{2n+1}X(ℳ) → X(ℛ) is obtained by multiplying all the factors in ℒ and ℐ, and taking the trace on ℐ^{n+1}, while the compatibility between the 𝒩 -adic and 𝒥 -adic filtrations is obvious. Although interesting examples arise under this form (see section 7), the algebra ℰ cannot be decomposed into a tensor product with ℬ in all situations.

Example 3.5 An important example where ℰ cannot be taken in the previous form is provided by the Bott element of the real line. Here 𝒜 = ℂ and ℬ = C^{∞}(0, 1) is the algebra of smooth functions f : [0, 1] → ℂ such that f and all its derivatives vanish at the endpoints 0 and 1. Take ℐ = ℂ as a 1-summable algebra, and ℰ = C^{∞}[0, 1] is the algebra of smooth functions f : [0, 1] → ℂ with the derivatives vanishing at the endpoints, while f itself takes arbitrary values at 0 and 1. ℬ and ℰ provided with their usual Fréchet topology are m-algebras, and one has ℰ ▷ ℬ. The Bott element is represented by the quasihomomorphism of odd degree

ρ : ℂ → ℰ^{s} ▷ ℐ^{s} ˆ⊗ℬ ,

where ℐ^{s} = M_{2}(C_{1}) and ℰ^{s} ⊂ M_{2}(C_{1}) ˆ⊗ℰ by construction. The homomorphism ρ : ℂ → ℰ_{+}^{s} is built from an arbitrary real-valued function ξ ∈ ℰ such that ξ(0) = 0, ξ(1) = π/2, and sends the unit e ∈ ℂ to the matrix

ρ(e) = R^{−1} (\begin{matrix}e & 0 \\ 0 & 0 \end{matrix}) R , R = (\begin{matrix}cos ξ & sin ξ \\ − sin ξ & cos ξ \end{matrix}) .

The algebra ℬ is quasi-free hence we can choose ℛ = ℬ, 𝒥 = 0 as quasifree extension. The cyclic homology of ℬ is therefore computed by X(ℬ). Moreover, setting ℳ = ℰ and 𝒩 = 0 shows that ℰ is ℬ-admissible. Indeed, Ω^{1}ℬ_{♮} is contained in the space Ω^{1}(0, 1) of ordinary (commutative) complexvalued smooth one-forms over [0, 1] vanishing at the endpoints with all their derivatives. The chain map Tr : F_{ℬ}^{2n+1}X(ℰ) → X(ℬ) is thus well-defined for any n ≥ 1, and just amounts to project noncommutative forms over ℰ to ordinary (commutative) differential forms over [0, 1].

We shall now construct the bivariant Chern character of a given p-summable quasihomomorphism ρ : 𝒜 → ℰ^{s}▷ℐ^{s} ˆ⊗ℬ. We take the universal free extension 0 → J𝒜 → T 𝒜 → 𝒜 → 0 for 𝒜, and choose some quasi-free extension 0 → 𝒥 → ℛ → ℬ → 0 for ℬ with the property that the algebra ℰ ▷ ℐ ˆ⊗ℬ is ℛ-admissible. The bivariant Chern character should be represented by a chain map between the complexes X(T 𝒜) and X(ℛ), compatible with the adic filtrations induced by the ideals J𝒜 and 𝒥 (section 2). Our task is thus to lift the quasihomomorphism to the quasi-free algebras T 𝒜 and ℛ. First, the admissibility condition provides a diagram of extensions (29). From ℳ ▷ℐ ˆ⊗ℛ define the ℤ_{2}-graded algebra ℳ^{s} in complete analogy with ℰ^{s}: depending on the degree of the quasihomomorphism, ℳ^{s} is a subalgebra of M_{2}(ℂ) ˆ⊗ℳ (even case) or M_{2}(C_{1}) ˆ⊗ℳ (odd case), with commutator property [F, ℳ_{+}^{s}] ⊂ ℐ_{−}^{s} ˆ⊗ℛ. Also, from 𝒩 ▷ ℐ ˆ⊗𝒥 define 𝒩 ^{s} as the ℤ_{2}-graded subalgebra of M_{2}(ℂ) ˆ⊗𝒩 or M_{2}(C_{1}) ˆ⊗𝒩 with commutator [F, 𝒩_{+}^{s}] ⊂ ℐ_{−}^{s} ˆ⊗𝒥. The algebras ℰ^{s}, ℳ^{s} and 𝒩 ^{s} are gifted with a differential of odd degree induced by the graded commutator [F, ] (its square vanishes because F^{2} = 1). Then we get an extension of ℤ_{2}-graded differential algebras

0 → 𝒩 ^{s} → ℳ^{s} → ℰ^{s} → 0 .

The restriction to the even-degree subalgebras yields an extension of trivially graded algebras 0 → 𝒩_{+}^{s} → ℳ_{+}^{s} → ℰ_{+}^{s} → 0, split by a continuous linear map σ : ℰ_{+}^{s} → ℳ_{+}^{s} (recall the splitting is our basic hypothesis about extensions of m-algebras). The universal properties of the tensor algebra T 𝒜 then allows to extend the homomorphism ρ : 𝒜 → ℰ_{+}^{s} to a continuous homomorphism ρ_{∗} : T 𝒜 → ℳ_{+}^{s} by setting ρ_{∗}(a_{1} ⊗ … ⊗ a_{n}) = σρ(a_{1}) ⊗ … ⊗ σρ(a_{n}):

\xymatrix{0 \ar[r] & J𝒜 \ar[r] \ar[d]_{ρ_{∗}} & T 𝒜 \ar[r] \ar[d]_{ρ_{∗}} & 𝒜 \ar[r] \ar[d]^{ρ} & 0 \\ 0 \ar[r] & 𝒩_{+}^{s} \ar[r] & ℳ_{+}^{s} \ar[r] & ℰ_{+}^{s} \ar[r] \ar@{.>}[l]_{σ} & 0}  (31)

A priori ρ_{∗} depends on the choice of linear splitting σ, but in a way which will not affect the cohomology class of the bivariant Chern character. This construction may be depicted in terms of a p-summable quasihomomorphism ρ_{∗} : T 𝒜 → ℳ^{s} ▷ ℐ^{s} ˆ⊗ℛ, compatible with the adic filtration by the ideals in the sense that J𝒜 is mapped to 𝒩 ^{s} ▷ ℐ^{s} ˆ⊗𝒥. Hence, ρ_{∗} extends to a quasihomomorphism of pro-algebras

ρ_{∗} : T̂𝒜 → ℳ̂^{s} ▷ ℐ^{s} ˆ⊗ℛ̂ ,  (32)

where T̂𝒜, ℳ̂^{s} and ℛ̂ are the adic completions of T 𝒜, ℳ^{s} and ℛ with respect to the ideals J𝒜, 𝒩 ^{s} and 𝒥. Next, depending on the degree of the quasihomomorphism, the even supertrace tr_{s} : M_{2}(ℂ) → ℂ or the odd supertrace tr_{s} : M_{2}(C_{1}) → ℂ yields a chain map X(ℳ^{s}) → X(ℳ) by setting αx ↦ tr_{s}(α)x and ♮ αxd(βy) ↦ ±tr_{s}(αβ)♮xdy for any x, y ∈ ℳ and α, β ∈ M_{2}(ℂ) or M_{2}(C_{1}). The sign ± is the parity of the matrix β, which has to move across the differential d. Hence composing with the chain map Tr : F_{ℐ ˆ⊗ℛ}^{2n+1}X(ℳ) → X(ℛ) guaranteed by the admissibility condition, we obtain for any integer n ≥ max(1, p − 1) a supertrace chain map

τ : F_{ℐ^{s} ˆ⊗ℛ}^{2n+1} X(ℳ^{s}) →^{tr_{s}} F_{ℐ ˆ⊗ℛ}^{2n+1}X(ℳ) →^{Tr} X(ℛ) ,  (33)

of order zero with respect to the 𝒩 ^{s}-adic filtration on X(ℳ^{s}) and the 𝒥 - adic filtration on X(ℛ). The parity of τ corresponds to the parity of the quasihomomorphism. This allows to construct a chain map χ^{n} : Ω̂ℳ_{+}^{s} → X(ℛ) from the (b + B)-complex of the algebra ℳ_{+}^{s}, in any degree n ≥ p having the same parity as the supertrace τ. Observe that the linear map x ∈ ℳ_{+}^{s} ↦ [F, x] ∈ ℐ_{−}^{s} ˆ⊗ℛ is continuous by construction.

Proposition 3.6 Let ρ : 𝒜 → ℰ^{s} ▷ ℐ^{s} ˆ⊗ℬ be a p-summable quasihomomorphism of parity p mod 2, with ℛ-admissible algebra ℰ. Given any integer n ≥ p of the same parity, consider two linear maps χ_{0}^{n} : Ω^{n}ℳ_{+}^{s} → ℛ and χ_{1}^{n} : Ω^{n+1}ℳ_{+}^{s} → Ω^{1}ℛ_{♮} defined by

χ_{0}^{n}(x_{0}dx_{1} … dx_{n}) = (−)^{n} \frac{Γ(1 + \frac{n}{2})}{(n + 1)!} ∑_{λ∈S_{n+1}} ε(λ) τ(x_{λ(0)}[F, x_{λ(1)}] … [F, x_{λ(n)}])

(34)

χ_{1}^{n}(x_{0}dx_{1} … dx_{n+1}) = (−)^{n} \frac{Γ(1 + \frac{n}{2})}{(n + 1)!} ∑_{i=1}^{n+1} τ♮(x_{0}[F, x_{1}] … dx_{i} … [F, x_{n+1}])

where S_{n+1} is the cyclic permutation group of n + 1 elements and ε is the signature. Then χ_{0}^{n} and χ_{1}^{n} define together a chain map χ^{n} : Ω̂ℳ_{+}^{s} → X(ℛ) of parity n mod 2, i.e. fulfill the relations

χ_{0}^{n}B = 0 , ♮dχ_{0}^{n} − (−)^{n}χ_{1}^{n}B = 0 , \overline{b}χ_{1}^{n} − (−)^{n}χ_{0}^{n}b = 0 , χ_{1}^{n}b = 0 . (35)

Moreover χ^{n} is invariant under the Karoubi operator κ acting on Ω^{n}ℳ_{+}^{s} and Ω^{n+1}ℳ_{+}^{s}.

Proof: This follows from purely algebraic manipulations, using the following general properties:

- The graded commutator [F, ] is a differential and τ([F, ]) = 0;

- dF = 0 so that [F, ] and d are anticommuting differentials;

- τ♮ is a supertrace. The computation is lengthy but straightforward.

Thus we have attached to a p-summable quasihomomorphism ρ : 𝒜 → ℰ^{s} ▷ ℐ^{s} ˆ⊗ℬ of parity p mod 2 a sequence of cocycles χ^{n} (n ≥ p) of the same parity in the ℤ_{2}-graded complex Hom(Ω̂ℳ_{+}^{s}, X(ℛ)). They are in fact all cohomologous, and the proposition below gives an explicit transgression formula in terms of the eta-cochain:

Proposition 3.7 Let ρ : 𝒜 → ℰ^{s} ▷ ℐ^{s} ˆ⊗ℬ be a p-summable quasihomomorphism of parity p mod 2, with ℛ-admissible algebra ℰ. Given any integer n ≥ p + 1 of parity opposite to p, consider two linar maps η_{0}^{n} : Ω^{n}ℳ_{+}^{s} → ℛ and η_{1}^{n} : Ω^{n+1}ℳ_{+}^{s} → Ω^{1}ℛ_{♮} defined by

η_{0}^{n}(x_{0}dx_{1} … dx_{n}) = \frac{Γ(\frac{n+1}{2})}{(n + 1)!} \frac{1}{2}τ(F x_{0}[F, x_{1}] … [F, x_{n}] +

∑_{i=1}^{n}(−)^{ni}[F, x_{i}] … [F, x_{n}]F x_{0}[F, x_{1}] … [F, x_{i−1}])

η_{1}^{n}(x_{0}dx_{1} … dx_{n+1}) =  (36)

\frac{Γ(\frac{n+1}{2})}{(n + 2)!} ∑_{i=1}^{n+1} \frac{1}{2}τ♮(ix_{0}F + (n + 2 − i)F x_{0})[F, x_{1}] … dx_{i} … [F, x_{n+1}] .

Then η_{0}^{n} and η_{1}^{n} define together a cochain η^{n} ∈ Hom(Ω̂ℳ_{+}^{s}, X(ℛ)) of parity n mod 2, whose coboundary equals the difference of cocycles

χ^{n−1} − χ^{n+1} = (♮d ⊕ \overline{b})η^{n} − (−)^{n}η^{n}(b + B) .

Expressed in terms of components this amounts to the identities

χ_{0}^{n−1} = −(−)^{n}η_{0}^{n}B ,  \overline{b}η_{1}^{n} − (−)^{n}(η_{0}^{n}b + η_{0}^{n+2}B) = 0 ,  (37)
χ_{1}^{n−1} = ♮dη_{0}^{n} − (−)^{n}η_{1}^{n}B , ♮dη_{0}^{n+2} − (−)^{n}(η_{1}^{n}b + η_{1}^{n+2}B) = 0 .

Proof: Direct computation.

Remark 3.8 Using a trick of Connes [4], we may replace the chain map τ by τ′ = \frac{1}{2}τ(F [F, ]). This allows to improve the summability condition by requiring the quasihomomorphism to be only (p + 1)-summable instead of p-summable, while the condition on the degree remains n ≥ p for χ^{n} and n ≥ p + 1 for η^{n}. It is traightforward to write down the new formulas for χ^{n} and observe that it involves exactly n+1 commutators [F, x]. These formulas were actually obtained in [23] in a more general setting where we allow dF ≠ 0.

Definition 3.9 The bivariant Chern character of the quasihomomorphism ρ : 𝒜 → ℰ^{s} ▷ ℐ^{s} ˆ⊗ℬ is represented in any degree n ≥ p by the composition of chain maps

ch^{n}(ρ) : X(T 𝒜) ⟶^{γ} Ω̂T 𝒜 ⟶^{ρ_{∗}} Ω̂ℳ_{+}^{s} \underrightarrow{χ^{n}} X(ℛ) ,  (38)

where γ : X(T 𝒜) → Ω̂T 𝒜 is the Goodwillie equivalence constructed in section 2 for any quasi-free algebra and ρ_{∗} : T 𝒜 → ℳ_{+}^{s} is the classifying homomorphism. In the same way, define a transgressed cochain in Hom(X(T 𝒜), X(ℛ)) by means of the eta-cochain in any degree:

/ ch^{n}(ρ) : X(T 𝒜) ⟶^{γ} Ω̂T 𝒜 ⟶^{ρ_{∗}} Ω̂ℳ_{+}^{s} \underrightarrow{η^{n}} X(ℛ) .  (39)

It fulfills the transgression property ch^{n}(ρ) − ch^{n+2}(ρ) = [∂, / ch^{n+1}(ρ)] where ∂ is the X-complex boundary map.

Recall that γ(x) = (1 − ϕ)^{−1}(x) and γ(♮xdy) = (1 − ϕ)^{−1}(xdy + b(xϕ(y))) for any x, y ∈ T 𝒜, where the map ϕ : Ω^{n}T 𝒜 → Ω^{n+2}T 𝒜 is uniquely defined from its restriction to zero-forms. Its existence is guaranteed by the fact that T 𝒜 is a free algebra. Several choices are possible, but conventionally we always take ϕ : T 𝒜 → Ω^{2}T 𝒜 as the canonical map obtained by setting ϕ(a) = 0 on the generators a ∈ 𝒜 ⊂ T 𝒜, and then extended to all T 𝒜 by the algebraic property ϕ(xy) = ϕ(x)y + xϕ(y) + dxdy. Of course ch^{n}(ρ) and / ch^{n}(ρ) are not very interesting a priori, because the Xcomplex of the non-completed tensor algebra T 𝒜 is contractible. However, taking into account the adic filtrations induced by the ideals J𝒜 ⊂ T 𝒜 and 𝒥 ⊂ ℛ yields non-trivial bivariant objects. By virtue of Remark 3.8 we suppose from now on that ℐ is (p + 1)-summable.

Proposition 3.10 Let ρ : 𝒜 → ℰ^{s} ▷ ℐ^{s} ˆ⊗ℬ be a (p + 1)-summable quasihomomorphism of parity p mod 2 with ℛ-admissible extension ℰ, and let n ≥ p be an integer of the same parity. The composites χ^{n}ρ_{∗}γ and η^{n+1}ρ_{∗}γ are linear maps X(T 𝒜) → X(ℛ) verifying the adic properties

χ^{n}ρ_{∗}γ : F_{J𝒜}^{k} X(T 𝒜) → F_{𝒥}^{k−n}X(ℛ) ,
η^{n+1}ρ_{∗}γ : F_{J𝒜}^{k} X(T 𝒜) → F_{𝒥}^{k−n−2}X(ℛ) ,

for any k ∈ ℤ. Consequently the composite ch^{n}(ρ) = χ^{n}ρ_{∗}γ defines a cocycle of parity n mod 2 in the complex Hom^{n}(X̂(T 𝒜 , J𝒜), X̂(ℛ, 𝒥)) and the Chern character is a bivariant cyclic cohomology class of degree n:

ch^{n}(ρ) ∈ HC^{n}(𝒜 , ℬ) , ∀n ≥ p .  (40)

Moreover, the transgression relation ch^{n}(ρ) − ch^{n+2}(ρ) = [∂, / ch^{n+1}(ρ)] holds in the complex Hom^{n+2}(X̂(T 𝒜 , J𝒜), X̂(ℛ, 𝒥)), which implies

ch^{n+2}(ρ) ≡ Sch^{n}(ρ) in HC^{n+2}(𝒜 , ℬ) .  (41)

In particular the cocycles ch^{n}(ρ) for different n define the same periodic cyclic cohomology class ch(ρ) ∈ HP^{n}(𝒜 , ℬ).

Proof: Let us denote by 0 → 𝒢 → ℱ → 𝒜 → 0 the universal extension 0 → J𝒜 → T 𝒜 → 𝒜 → 0. Recall that ℳ^{s} and its ideal 𝒩 ^{s} are ℤ_{2}- graded differential algebras on which the graded commutator [F, ] acts as a differential of odd degree. Moreover the commutation relations [F, ℳ_{+}^{s}] ⊂ ℐ_{−}^{s} ˆ⊗ℛ and [F, 𝒩_{+}^{s}] ⊂ ℐ_{−}^{s} ˆ⊗𝒥 hold. Now, we have to investigate the adic behaviour of the Goodwillie equivalence γ : X(ℱ) → Ω̂ℱ with respect to the filtration F_{𝒢}^{k}X(ℱ). The first step in that direction was actually done in the proof of Proposition 2.4, where the following filtration of the subspaces Ω^{n}ℱ was introduced:

H^{k}Ω^{n}ℱ = ∑_{k_{0}+...+k_{n}≥k} 𝒢^{k_{0}}d𝒢^{k_{1}} … d𝒢^{k_{n}} ⊂ Ω^{n}ℱ .

Let us look at the image of the latter filtration under the maps χ^{n}ρ_{∗} and η^{n}ρ_{∗} : Ω̂ℱ → X(ℛ) given by Eqs. (34, 36). We know that the homomorphism ρ_{∗} : ℱ → ℳ_{+}^{s} respects the ideals 𝒢 and 𝒩_{+}^{s}. Hence if x_{0}, …, x_{n} denote n + 1 elements in 𝒢^{k_{0}}, …, 𝒢^{k_{n}} respectively, with k_{0} + … + k_{n} ≥ k, then x_{λ(0)}[F, x_{λ(1)}] … [F, x_{λ(n)}] ∈ (𝒩 ^{s})^{k} for any permutation λ ∈ S_{n+1}. Hence applying the supertrace τ, which is a chain map of order zero with respect to the 𝒩 ^{s}-adic and 𝒥 -adic filtrations on X(ℳ^{s}) and X(ℛ), yields (from now on we omit to write the homomorphism ρ_{∗})

χ_{0}^{n}(H^{k}Ω^{n}ℱ) ⊂ 𝒥^{k} .  (42)

In the same way, for n + 1 elements x_{0}, …, x_{n+1} in 𝒢^{k_{0}}, …, 𝒢^{k_{n+1}}, the oneform ♮x_{0}[F, x_{1}] … dx_{i} … [F, x_{n+1}] involves k_{0} + … + k_{n+1} ≥ k powers of the ideal 𝒩 ^{s}, hence lies in the subspace ♮((𝒩 ^{s})^{k}dℳ^{s} + (𝒩 ^{s})^{k−1}d𝒩 ^{s}). Thus applying the supertrace τ one gets

χ_{1}^{n}(H^{k}Ω^{n+1}ℱ) ⊂ ♮(𝒥^{k}dℛ + 𝒥^{k−1}d𝒥) .  (43)

Proceeding in exactly the same fashion with the maps η_{0}^{n} : Ω^{n}ℱ → ℛ and η_{1}^{n} : Ω^{n+1}ℱ → Ω^{1}ℛ_{♮}, it is clear that

η_{0}^{n}(H^{k}Ω^{n}ℱ) ⊂ 𝒥^{k} .  (44)
η_{1}^{n}(H^{k}Ω^{n+1}ℱ) ⊂ ♮(𝒥^{k}dℛ + 𝒥^{k−1}d𝒥) .  (45)

However these estimates are not optimal concerning the component χ_{1}^{n}. We need a refinement of the H-filtration. For any k ∈ ℤ, n ≥ 0, let us define the subspaces

G^{k}Ω^{n}ℱ = ∑_{k_{0}+...+k_{n}≥k} 𝒢^{k_{0}}(dℱ)𝒢^{k_{1}}(dℱ) … 𝒢^{k_{n−1}}(dℱ)𝒢^{k_{n}} + H^{k+1}Ω^{n}ℱ .

Then for fixed n, G^{∗}Ω^{n}ℱ is a decreasing filtration of Ω^{n}ℱ, and by convention G^{k}Ω^{n}ℱ = Ω^{n}ℱ for k ≤ 0. One has G^{k}Ω^{n}ℱ ⊂ H^{k}Ω^{n}ℱ. Now observe the following. Since [F, ] and d are derivations, the map x_{0}dx_{1} … dx_{i} … dx_{n+1} ↦ ♮x_{0}[F, x_{1}] … dx_{i} … [F, x_{n+1}] has the property that

𝒢^{k_{0}}(dℱ)𝒢^{k_{1}}(dℱ) … (dℱ)𝒢^{k_{i}} … (dℱ)𝒢^{k_{n+1}} → ♮(𝒩 ^{s})^{k_{0}}[F, ℳ^{s}](𝒩 ^{s})^{k_{1}}[F, ℳ^{s}] … (dℳ^{s})(𝒩 ^{s})^{k_{i}} … [F, ℳ^{s}](𝒩 ^{s})^{k_{n+1}}

⊂ ♮(𝒩 ^{s})^{k}dℳ^{s} ,

and because χ_{1}^{n}(H^{k+1}Ω^{n+1}ℱ) ⊂ ♮(𝒥^{k+1}dℛ + 𝒥^{k}d𝒥) ⊂ ♮𝒥^{k}dℛ, one gets the crucial estimate

χ_{1}^{n}(G^{k}Ω^{n+1}ℱ) ⊂ ♮𝒥^{k}dℛ .  (46)

Now we have to understand the way γ sends the X-complex filtration

F_{𝒢}^{2k}X(ℱ) : 𝒢^{k+1} + [𝒢^{k}, ℱ] ⇄ ♮𝒢^{k}dℱ
F_{𝒢}^{2k+1}X(ℱ) : 𝒢^{k+1} ⇄ ♮(𝒢^{k+1}dℱ + 𝒢^{k}d𝒢) ,

to the filtration G^{∗}Ω^{n}ℱ, in all degrees n. Recall that γ(x)|_{Ω^{2n}ℱ} = ϕ^{n}(x) and γ(♮xdy)|_{Ω^{2n+1}ℱ} = ϕ^{n}(xdy + b(xϕ(y))) for any x, y ∈ ℱ, where the map ϕ : Ω^{n}ℱ → Ω^{n+2}ℱ is obtained from its restriction to zero-forms as

ϕ(x_{0}dx_{1} … dx_{n}) = ∑_{i=0}^{n}(−)^{ni}ϕ(x_{i})dx_{i+1} … dx_{n}dx_{0} … dx_{i−1} .

Note the following important properties of ϕ. Firstly, it is invariant under the Karoubi operator κ : Ω^{n}ℱ → Ω^{n}ℱ in the sense that ϕ ∘ κ = ϕ, and vanishes on the image of the boudaries d, B : Ω^{n}ℱ → Ω^{n+1}ℱ. Secondly, the relation ϕb − bϕ = B holds on Ω^{n}ℱ whenever n ≥ 1 (see [22] §4). Since we have to apply successive powers of ϕ on the filtration G^{k}Ω^{n}ℱ, the computation will be greatly simplified by exploiting κ-invariance. Define the linear map ϕ̃: Ω^{n}ℱ → Ω^{n+2}ℱ by

ϕ̃(x_{0}dx_{1} … dx_{n}) = ∑_{i=0}^{n}(−)^{i}dx_{0} … dx_{i−1}ϕ(x_{i})dx_{i+1} … dx_{n} .

Then ϕ̃ coincides with ϕ modulo the image of 1 − κ. In particular the relation ϕ^{n} = ϕ ∘ ϕ̃^{n−1} holds. The advantage of the map ϕ̃ stems from the fact that it does not involve cyclic permutations of the elements x_{i}, and verifies the following optimal compatibility with the G-filtration

ϕ̃(G^{k}Ω^{n}ℱ) ⊂ G^{k−1}Ω^{n+2}ℱ ∀k, n ≥ 0 ,

whereas the map ϕ is only compatible with the (coarser) H-filtration:

ϕ(H^{k}Ω^{n}ℱ) ⊂ H^{k−1}Ω^{n+2}ℱ ∀k, n ≥ 0 .

We shall now evaluate the image of the filtration F_{𝒢}^{k}X(ℱ) under the map γ : X(ℱ) → Ω̂ℱ. Firstly, one has γ(𝒢^{k+1}) ∩ Ω^{2n}ℱ = ϕ^{n}(𝒢^{k+1}). But 𝒢^{k+1} ⊂ G^{k+1}Ω^{0}ℱ and ϕ^{n} = ϕ ∘ ϕ̃^{n−1}, hence

γ(𝒢^{k+1}) ∩ Ω^{2n}ℱ ⊂ ϕ(G^{k−n+2}Ω^{2n−2}ℱ) .  (47)

Secondly, the image of ♮𝒢^{k}dℱ in Ω^{2n+1}ℱ is given by ϕ^{n}(𝒢^{k}dℱ +b(𝒢^{k}ϕ(ℱ))). One has b(𝒢^{k}ϕ(ℱ)) ⊂ [𝒢^{k}dℱ , ℱ] ⊂ 𝒢^{k}dℱ, hence we only need to compute ϕ^{n}(𝒢^{k}dℱ) ⊂ ϕ^{n}(G^{k}Ω^{1}ℱ), and

γ(♮𝒢^{k}dℱ) ∩ Ω^{2n+1}ℱ ⊂ ϕ(G^{k−n+1}Ω^{2n−1}ℱ) .  (48)

Thirdly, [𝒢^{k}, ℱ] = \overline{b}♮𝒢^{k}dℱ so that γ([𝒢^{k}, ℱ]) = (b + B)γ(♮𝒢^{k}dℱ) because γ is a chain map. Therefore, the image of [𝒢^{k}, ℱ] restricted to Ω^{2n}ℱ is contained in Bϕ^{n−1}(𝒢^{k}dℱ) + bϕ^{n}(𝒢^{k}dℱ). We may estimate coarsly the first term as Bϕ^{n−1}(𝒢^{k}dℱ) ⊂ Bϕ^{n−1}(H^{k}Ω^{1}ℱ) ⊂ H^{k−n+1}Ω^{2n}ℱ, and the second term as bϕϕ̃^{n−1}(𝒢^{k}dℱ) ⊂ bϕ(G^{k−n+1}Ω^{2n−1}ℱ). Hence

γ([𝒢^{k}, ℱ]) ∩ Ω^{2n}ℱ ⊂ H^{k−n+1}Ω^{2n}ℱ + bϕ(G^{k−n+1}Ω^{2n−1}ℱ) .  (49)

Fourthly, the image of ♮𝒢^{k}d𝒢 in Ω^{2n+1}ℱ is given by ϕ^{n}(𝒢^{k}d𝒢 + b(𝒢^{k}ϕ(𝒢))). We estimate coarsly ϕ^{n}(𝒢^{k}d𝒢) ⊂ ϕ^{n}(H^{k+1}Ω^{1}ℱ) ⊂ H^{k−n+1}Ω^{2n+1}ℱ. Then, one has 𝒢^{k}ϕ(𝒢) ⊂ 𝒢^{k}dℱ dℱ ⊂ G^{k}Ω^{2}ℱ, and using repeatedly the relations ϕb − bϕ = B, ϕB = 0 gives ϕ^{n}b(𝒢^{k}ϕ(𝒢)) ⊂ bϕ^{n}(G^{k}Ω^{2}ℱ) + Bϕ^{n−1}(G^{k}Ω^{2}ℱ) ⊂ bϕ(G^{k−n+1}Ω^{2n}ℱ) + BH^{k−n+1}Ω^{2n}ℱ. Thus

γ(♮𝒢^{k}d𝒢) ∩ Ω^{2n+1}ℱ ⊂ H^{k−n+1}Ω^{2n+1}ℱ + bϕ(G^{k−n+1}Ω^{2n}ℱ) .  (50)

Now everything is set to evaluate the adic behaviour of the composites χ^{n}γ and η^{n}γ. We shall deal only with even degrees, the odd case is similar.

Hence let us start with the map χ_{0}^{2n}γ : ℱ → ℛ. For any k ∈ ℤ, Eq. (47) gives χ_{0}^{2n}γ(𝒢^{k+1}) ⊂ χ_{0}^{2n} ∘ ϕ(G^{k−n+2}Ω^{2n−2}ℱ). But χ_{0}^{2n} is κ-invariant, hence χ_{0}^{2n} ∘ ϕ = χ_{0}^{2n} ∘ ϕ̃. Therefore, χ_{0}^{2n}γ(𝒢^{k+1}) ⊂ χ_{0}^{2n}(G^{k−n+1}Ω^{2n}ℱ) ⊂ 𝒥^{k−n+1} using G^{k−n+1}Ω ⊂ H^{k−n+1}Ω and (42). Now we look at its companion χ_{1}^{2n}γ : Ω^{1}ℱ_{♮} → Ω^{1}ℛ_{♮}. From (48) one gets χ_{1}^{2n}(♮𝒢^{k}dℱ) ⊂ χ_{1}^{2n} ∘ ϕ(G^{k−n+1}Ω^{2n−1}ℱ). But χ_{1}^{2n} is also κ-invariant and χ_{1}^{2n} ∘ ϕ = χ_{1}^{2n} ∘ ϕ̃, thus χ_{1}^{2n}γ(♮𝒢^{k}dℱ) ⊂ χ_{1}^{2n}(G^{k−n}Ω^{2n+1}ℱ) ⊂ ♮𝒥^{k−n}dℛ by (46). This allows to estimate the image of [𝒢^{k}, ℱ] = \overline{b}♮𝒢^{k}dℱ under the chain map χ^{2n}γ. Indeed χ_{0}^{2n}γ(\overline{b}♮𝒢^{k}dℱ) = \overline{b}χ_{1}^{2n}γ(♮𝒢^{k}dℱ) ⊂ \overline{b}♮𝒥^{k−n}dℛ, so that χ_{0}^{2n}γ([𝒢^{k}, ℱ]) ⊂ [𝒥^{k−n}, ℛ]. Collecting these results shows the effect of the map χ^{2n}γ on the adic filtration in even degree:

\{ \begin{matrix}χ_{0}^{2n}γ : & 𝒢^{k+1} + [𝒢^{k}, ℱ] & ⟶ & 𝒥^{k−n+1} + [𝒥^{k−n}, ℛ] \\ χ_{1}^{2n}γ : & ♮𝒢^{k}dℱ & ⟶ & ♮𝒥^{k−n}dℛ \end{matrix}

hence χ^{2n}γ : F_{𝒢}^{2k}X(ℱ) → F_{𝒥}^{2k−2n}X(ℛ). To understand the effect on the filtration in odd degree, one has to evaluate χ_{1}^{2n}γ on ♮𝒢^{k}d𝒢. From (50), one gets χ_{1}^{2n}γ(♮𝒢^{k}d𝒢) ⊂ χ_{1}^{2n}(H^{k−n+1}Ω^{2n+1}ℱ + bϕ(G^{k−n+1}Ω^{2n}ℱ)). But (35) shows χ_{1}^{2n} ∘ b = 0, and (43) implies χ_{1}^{2n}γ(♮𝒢^{k}d𝒢) ⊂ ♮(𝒥^{k−n+1}dℛ + 𝒥^{k−n}d𝒥). One thus gets the adic behaviour of the chain map χ^{2n}γ on the filtration of odd degree:

\{ \begin{matrix}χ_{0}^{2n}γ : & 𝒢^{k+1} & ⟶ & 𝒥^{k−n+1} \\ χ_{1}^{2n}γ : & ♮(𝒢^{k+1}dℱ + 𝒢^{k}d𝒢) & ⟶ & ♮(𝒥^{k−n+1}dℛ + 𝒥^{k−n}d𝒥) \end{matrix}

hence χ^{2n}γ : F_{𝒢}^{2k+1}X(ℱ) → F_{𝒥}^{2k−2n+1}X(ℛ) and χ^{2n}γ is a map of order 2n. Using similar methods, one shows that χ^{2n+1}γ is of order 2n + 1. Now we investigate the eta-cochain. Consider η_{0}^{2n}γ : ℱ → ℛ.

(47) gives η_{0}^{2n}γ(𝒢^{k+1}) ⊂ η_{0}^{2n}ϕ(G^{k−n+2}Ω^{2n−2}ℱ). However η^{2n} is not κ-invariant, so that we cannot replace ϕ by ϕ̃. We are forced to consider ϕ(G^{k−n+2}Ω^{2n−2}ℱ) ⊂ H^{k−n+1}Ω^{2n}ℱ and consequently η_{0}^{2n}γ(𝒢^{k+1}) ⊂ 𝒥^{k−n+1} by (44). Similarly, (49) implies η_{0}^{2n}γ([𝒢^{k}, ℱ]) ⊂ η_{0}^{2n}(H^{k−n+1}Ω^{2n}ℱ) + η_{0}^{2n}bϕ(G^{k−n+1}Ω^{2n−1}ℱ) ⊂ η_{0}^{2n}(H^{k−n}Ω^{2n}ℱ) hence η_{0}^{2n}γ([𝒢^{k}, ℱ]) ⊂ 𝒥^{k−n}. Its companion η_{1}^{2n}γ : Ω^{1}ℱ_{♮} → Ω^{1}ℛ_{♮} evaluated on ♮𝒢^{k}dℱ uses equation (48) again with ϕ(G^{k−n+1}Ω^{2n−1}ℱ) ⊂ H^{k−n}Ω^{2n+1}ℱ, so that η_{1}^{2n}γ(♮𝒢^{k}dℱ) ⊂ η_{1}^{2n}(H^{k−n}Ω^{2n+1}ℱ) ⊂ ♮(𝒥^{k−n}dℛ + 𝒥^{k−n−1}d𝒥) by (45). This shows the effect of η^{2n}γ on the filtration of even degree

\{ \begin{matrix}η_{0}^{2n}γ : & 𝒢^{k+1} + [𝒢^{k}, ℱ] & ⟶ & 𝒥^{k−n} \\ η_{1}^{2n}γ : & ♮𝒢^{k}dℱ & ⟶ & ♮(𝒥^{k−n}dℛ + 𝒥^{k−n−1}d𝒥) \end{matrix}

hence η^{2n}γ : F_{𝒢}^{2k}X(ℱ) → F_{𝒥}^{2k−2n−1}X(ℛ). For the odd filtration, let us compute from (50) η_{1}^{2n}γ(♮𝒢^{k}d𝒢) ⊂ η_{1}^{2n}(H^{k−n+1}Ω^{2n+1}ℱ) + η_{1}^{2n}bϕ(G^{k−n+1}Ω^{2n}ℱ). But the identities (37) show that η_{1}^{2n}b = χ_{1}^{2n+1}, hence using κ-invariance one gets η_{1}^{2n}bϕ(G^{k−n+1}Ω^{2n}ℱ) ⊂ χ_{1}^{2n+1}ϕ̃(G^{k−n+1}Ω^{2n}ℱ) ⊂ χ_{1}^{2n+1}(G^{k−n}Ω^{2n+2}ℱ).

Therefore, (45) and (46) imply η_{1}^{2n}γ(♮𝒢^{k}d𝒢) ⊂ ♮𝒥^{k−n}dℛ. These results give the adic behaviour of η^{2n}γ with respect to the odd filtration

\{ \begin{matrix}η_{0}^{2n}γ : & 𝒢^{k+1} & ⟶ & 𝒥^{k−n+1} \\ η_{1}^{2n}γ : & ♮(𝒢^{k+1}dℱ + 𝒢^{k}d𝒢) & ⟶ & ♮𝒥^{k−n}dℛ \end{matrix}

hence η^{2n}γ : F_{𝒢}^{2k+1}X(ℱ) → F_{𝒥}^{2k−2n}X(ℛ) and η^{2n}γ is a map of order 2n + 1. Similarly, one shows that η^{2n+1}γ is of order 2n + 2.

Note that the chain maps γ and χ^{n} extend to the adic completions of all the algebras involved, so that from now on we will consider the bivariant Chern character ch^{n}(ρ) ∈ Hom^{n}(X̂(T 𝒜 , J𝒜), X̂(ℛ, 𝒥)) as a chain map of pro-complexes

ch^{n}(ρ) : X(T̂𝒜) ⟶^{γ} Ω̂T̂𝒜 ⟶^{ρ_{∗}} Ω̂ℳ̂_{+}^{s} \underrightarrow{χ^{n}} X(ℛ̂) .  (51)

We would like to introduce some equivalence relations among quasihomomorphisms, and discuss the corresponding invariance properties of the Chern character. The first equivalence relation is (smooth) homotopy. It involves the algebra C^{∞}[0, 1] of smooth functions f : [0, 1] → ℂ, such that all the derivatives of order ≥ 1 vanish at the enpoints 0 and 1, while the values of f itself remain arbitrary. We have already seen that C^{∞}[0, 1] endowed with its usual Fréchet topology is an m-algebra. It is moreover nuclear [14], so that its projective tensor product 𝒜 ˆ⊗C^{∞}[0, 1] with any m-algebra 𝒜 is isomorphic to the algebra of smooth 𝒜 -valued functions over [0, 1], with all derivatives of order ≥ 1 vanishing at the endpoints. We will usually denote by 𝒜 [0, 1] this m-algebra. The second equivalence relation of interest among quasihomomorphisms is conjugation by an invertible element of the unitalized algebra (ℰ_{+}^{s})^{+}.

Definition 3.11 Let ρ_{0} : 𝒜 → ℰ^{s} ▷ ℐ^{s} ˆ⊗ℬ and ρ_{1} : 𝒜 → ℰ^{s} ▷ ℐ^{s} ˆ⊗ℬ be two quasihomomorphisms with same parity. They are called

i) homotopic if there exists a quasihomomorphism ρ : 𝒜 → ℰ [0, 1]^{s} ▷ ℐ^{s} ˆ⊗ℬ[0, 1] such that evaluation at the endpoints gives ρ_{0} and ρ_{1};

ii) conjugate if there exists an invertible element in the unitalized algebra U ∈ (ℰ_{+}^{s})^{+} with U − 1 ∈ ℰ_{+}^{s}, such that ρ_{1} = U^{−1}ρ_{0}U as a homomorphism 𝒜 → ℰ_{+}^{s}.

Remark that the commutators [F, U] and [F, U^{−1}] always lie in the ideal ℐ^{s} ˆ⊗ℬ ⊂ ℰ^{s}. When the algebra ℐ is M_{2}-stable (i.e. M_{2}(ℐ) ≅ ℐ), two conjugate quasihomomorphisms are also homotopic, but the converse is not true. Hence conjugation is strictly stronger than homotopy as an equivalence relation. The former is an analogue of “compact perturbation” of quasihomomorphisms in Kasparov’s bivariant K-theory for C^{∗}-algebras, see [2]. The proposition below describes the compatibility between these equivalence relations and the bivariant Chern character.

Proposition 3.12 Let ρ_{0} : 𝒜 → ℰ^{s} ▷ ℐ^{s} ˆ⊗ℬ and ρ_{1} : 𝒜 → ℰ^{s} ▷ ℐ^{s} ˆ⊗ℬ be two (p + 1)-summable quasihomomorphisms of parity p mod 2, with ℰ admissible with respect to a quasi-free extension ℛ of ℬ. Let n ≥ p be any integer of the same parity.

i) If ρ_{0} and ρ_{1} are homotopic, then Sch^{n}(ρ_{0}) ≡ Sch^{n}(ρ_{1}) in HC^{n+2}(𝒜 , ℬ). In particular ch^{n}(ρ_{0}) ≡ ch^{n}(ρ_{1}) whenever n ≥ p + 2.

ii) If ρ_{0} and ρ_{1} are conjugate, then ch^{n}(ρ_{0}) ≡ ch^{n}(ρ_{1}) in HC^{n}(𝒜 , ℬ) for all n ≥ p.

Proof: First observe that if ρ : 𝒜 → ℰ^{s} ▷ℐ^{s} ˆ⊗ℬ is a quasihomomorphism with ℛ-admissible algebra ℰ, the lifting homomorphism ρ_{∗} : T 𝒜 → ℳ_{+}^{s} factors through the tensor algebra T ℰ_{+}^{s} by virtue of the commutative diagram

[figure: 0; J𝒜; φ; T 𝒜; φ; 𝒜; ρ; 0; 0; Jℰ_{+}^{s}; T ℰ_{+}^{s}; ℰ_{+}^{s}; 0; 0; 𝒩_{+}^{s}; ℳ_{+}^{s}; ℰ_{+}^{s}; σ; 0]

where the homomorphism φ : T 𝒜 → T ℰ_{+}^{s} is φ(a_{1} ⊗ … ⊗ a_{n}) = ρ(a_{1}) ⊗ … ⊗ ρ(a_{n}), and the arrow T ℰ_{+}^{s} → ℳ_{+}^{s} maps a tensor product e_{1} ⊗…⊗e_{n} ∈ T ℰ_{+}^{s} to the product σ(e_{1}) … σ(e_{n}). By the naturality of the Goodwillie equivalences γ_{𝒜} : X(T 𝒜) → Ω̂T 𝒜 and γ_{ℰ_{+}^{s}} : X(T ℰ_{+}^{s}) → Ω̂T ℰ_{+}^{s}, one immediately sees that the bivariant Chern character coincides with the composition of chain maps

ch^{n}(ρ) : X(T 𝒜) ⟶^{X(φ)} X(T ℰ_{+}^{s}) ⟶^{γ_{ℰ^{s}_{+}}} Ω̂T ℰ_{+}^{s} \underrightarrow{χ^{n}} X(ℛ) .

Hence, all the information about the homomorphism ρ : 𝒜 → ℰ_{+}^{s} is concentrated in the chain map X(φ) : X(T 𝒜) → X(T ℰ_{+}^{s}). This will simplify the comparison of Chern characters associated to homotopic or conjugate quasihomomorphisms.

i) Homotopy: the cocycles ch^{n}(ρ_{0}) and ch^{n}(ρ_{1}) differ only by the chain maps X(φ_{i}) : X(T 𝒜) → X(T ℰ_{+}^{s}), i = 0, 1. We view ρ : 𝒜 → ℰ_{+}^{s}[0, 1] as a smooth family of homomorphisms ρ_{t} : 𝒜 → ℰ_{+}^{s} parametrized by t ∈ [0, 1], giving a homotopy between the two endpoints ρ_{0} and ρ_{1}. Cuntz and Quillen prove in [12] a Cartan homotopy formula which provides a transgression between the chain maps X(φ_{i}). At any point t ∈ [0, 1], denote by φ̇ = \frac{d}{dt}φ : T 𝒜 → T ℰ_{+}^{s} the derivative of the homomorphism φ_{t} with respect to t, and define a linear map ι : Ω^{m}T 𝒜 → Ω^{m−1}T ℰ_{+}^{s} by

ι(x_{0}dx_{1} … dx_{m}) = (φx_{0})(φ̇x_{1})d(φx_{2}) … d(φx_{m}) .

The tensor algebra T 𝒜 is quasi-free, hence consider any ϕ : T 𝒜 → Ω^{2}T 𝒜 verifying ϕ(xy) = ϕ(x)y + xϕ(y) + dxdy, and let h : X(T 𝒜) → X(T ℰ_{+}^{s}) be the linear map of odd degree

h(x) = ♮ιϕ(x) , h(♮xdy) = ι(xdy + b(xϕ(y)))

(the latter is well-defined on ♮xdy). Then Cuntz and Quillen show the following adic properties of h for any k ∈ ℤ,

h(F_{J𝒜}^{k} X(T 𝒜)) ⊂ F_{Jℰ_{+}^{s}}^{k−1}X(T ℰ_{+}^{s}) if φ̇(J𝒜) ⊂ Jℰ_{+}^{s} ,
h(F_{J𝒜}^{k} X(T 𝒜)) ⊂ F_{Jℰ_{+}^{s}}^{k} X(T ℰ_{+}^{s}) if φ̇(T 𝒜) ⊂ Jℰ_{+}^{s} ,

and moreover the transgression formula \frac{d}{dt}X(φ) = [∂, h] holds. Hence if we define by integration over [0, 1] the odd chain H = ∫_{0}^{1} dt h, one has

X(φ_{1}) − X(φ_{0}) = [∂, H]

in the complex Hom^{1}(X̂(T 𝒜 , J𝒜), X̂(T ℰ_{+}^{s}, Jℰ_{+}^{s})) in case φ̇(J𝒜) ⊂ Jℰ_{+}^{s}, or in the complex Hom^{0}(X̂(T 𝒜 , J𝒜), X̂(T ℰ_{+}^{s}, Jℰ_{+}^{s})) in case φ̇(T 𝒜) ⊂ Jℰ_{+}^{s}. For a general homotopy we are in the first case φ̇(J𝒜) ⊂ Jℰ_{+}^{s}. After composition by the chain map χ^{n}γ_{ℰ_{+}^{s}} ∈ Hom^{n}(X̂(T ℰ_{+}^{s}, Jℰ_{+}^{s}), X̂(ℛ, 𝒥)), this shows the transgression relation

ch^{n}(ρ_{1}) − ch^{n}(ρ_{0}) = (−)^{n}[∂, χ^{n}γ_{ℰ_{+}^{s}} H] ∈ Hom^{n+1}(X̂(T 𝒜 , J𝒜), X̂(ℛ, 𝒥)) ,

whence Sch^{n}(ρ_{1}) ≡ Sch^{n}(ρ_{0}) in HC^{n+2}(𝒜 , ℬ). The sign (−)^{n} comes from the parity of the chain map χ^{n}γ_{ℰ_{+}^{s}}.

ii) Conjugation: now φ_{0}, φ_{1} : T 𝒜 → T ℰ_{+}^{s} are the homomorphism lifts of ρ_{0} and ρ_{1} = U^{−1}ρ_{0}U. Introduce the pro-algebra T̂ℰ_{+}^{s} = \underleftarrow{lim}_{k} T ℰ_{+}^{s}/(Jℰ_{+}^{s})^{k} ≅ ∏_{k≥0} Ω^{2k}ℰ_{+}^{s}, and consider the invertible U ∈ (ℰ_{+}^{s})^{+} as an element Û of the unitalization (T̂ℰ_{+}^{s})^{+}, via the linear inclusion of zero-forms ℰ_{+}^{s} ↪ T̂ℰ_{+}^{s}. By proceeding as in [12], it turns out that Û is invertible, with inverse given by the series

Û^{−1} = ∑_{k≥0} U^{−1}(dU dU^{−1})^{k} ∈ (T̂ℰ_{+}^{s})^{+} .

Of course the image of Û^{−1} under the multiplication map (T̂ℰ_{+}^{s})^{+} → (ℰ_{+}^{s})^{+} is U^{−1}. We will show that φ_{1}, viewed as a homomorphism T 𝒜 → T̂ℰ_{+}^{s}, is homotopic to the homomorphism Û^{−1}φ_{0}Û. For any t ∈ [0, 1] define a linear map σ_{t} : 𝒜 → T̂ℰ_{+}^{s} by

σ_{t}(a) = (1 − t)ρ_{1}(a) + t Û^{−1}ρ_{0}(a)Û , ∀a ∈ 𝒜 ,

where ρ_{0}(a) and ρ_{1}(a) are considered as elements of the subspace of zeroforms ℰ_{+}^{s} ↪ T̂ℰ_{+}^{s}. Thus σ_{t} is a linear lifting of the constant homomorphism ρ_{1} : 𝒜 → ℰ_{+}^{s}. Then use the universal property of the tensor algebra T 𝒜 to build a smooth family of homomorphisms φ(t) : T 𝒜 → T̂ℰ_{+}^{s} by means of the commutative diagram with exact rows

\xymatrix{0 \ar[r] & J𝒜 \ar[r] \ar[d]_{φ(t)} & T 𝒜 \ar[r] \ar[d]_{φ(t)} & 𝒜 \ar[r] \ar[d]^{ρ_{1}} \ar@{.>}[dl]_{σ_{t}} & 0 \\ 0 \ar[r] & Ĵℰ_{+}^{s} \ar[r] & T̂ℰ_{+}^{s} \ar[r] & ℰ_{+}^{s} \ar[r] & 0}

By construction one has φ(0) = φ_{1}, φ(1) = Û^{−1}φ_{0}Û and the derivative φ̇ sends T 𝒜 to the ideal Ĵℰ_{+}^{s}. Hence from the Cartan homotopy formula of part i) we deduce that the chain maps X(φ_{1}) and X(Û^{−1}φ_{0}Û) are cohomologous in the complex Hom^{0}(X̂(T 𝒜 , J𝒜), X̂(T ℰ_{+}^{s}, Jℰ_{+}^{s})). Then we have to show that X(Û^{−1}φ_{0}Û) and X(φ_{0}) are cohomologous. Consider the following linear map of odd degree h : X(T 𝒜) → X(T̂ℰ_{+}^{s}) ≅ X̂(T ℰ_{+}^{s}, Jℰ_{+}^{s}) defined by

h(x) = ♮(Û^{−1}φ_{0}(x)dÛ) , h(♮xdy) = 0 .

It is easy to see that h defines a cochain of order zero, i.e. lies in the complex Hom^{0}(X̂(T 𝒜 , J𝒜), X̂(T ℰ_{+}^{s}, Jℰ_{+}^{s})). Moreover, one has the transgression relation X(Û^{−1}φ_{0}Û)−X(φ_{0}) = [∂, h]. Indeed (we replace φ_{0}(x) by x for notational simplicity)

X(Û^{−1}φ_{0}Û)(x) − X(φ_{0})(x) = Û^{−1}xÛ − x = [Û^{−1}x, Û]
= \overline{b}♮(Û^{−1}xdÛ) = \overline{b}h(x) ,

X(Û^{−1}φ_{0}Û)(♮xdy) − X(φ_{0})(♮xdy) = ♮Û^{−1}xÛd(Û^{−1}yÛ) − ♮xdy
= ♮(Û^{−1}xÛdÛ^{−1}yÛ + Û^{−1}xdyÛ + Û^{−1}xydÛ − xdy)
= ♮(−yxdÛÛ^{−1} + xydÛÛ^{−1}) = ♮Û^{−1}[x, y]dÛ
= h(\overline{b}♮xdy) ,

where in the second computation we use the identity dÛ^{−1} = −Û^{−1}dÛÛ^{−1} deduced from d1 = 0. This shows the equality of bivariant cyclic cohomology classes

X(φ_{1}) ≡ X(φ_{0}) ∈ HC^{0}(𝒜 , ℰ_{+}^{s}) ,

so that after composition with χ^{n}γ_{ℰ_{+}^{s}} ∈ HC^{n}(ℰ_{+}^{s}, ℬ), the equality ch^{n}(ρ_{1}) ≡ ch^{n}(ρ_{0}) holds in HC^{n}(𝒜 , ℬ).

Part ii) of the above proof also shows the independence of the cohomology class ch^{n}(ρ) ∈ HC^{n}(𝒜 , ℬ) with respect to the choice of linear splitting σ : ℰ_{+}^{s} → ℳ_{+}^{s} used to lift the homomorphism ρ, two such splittings being always homotopic. Then, from section 2 we know that any class in HC^{n}(𝒜 , ℬ) induces linear maps of degree −n between the cyclic homologies of 𝒜 and ℬ, compatible with the SBI exact sequence. Hence, if the quasihomomorphism is (p + 1)-summable and with parity p mod 2, the lowest degree representative of the Chern character ch^{p}(ρ) ∈ HC^{p}(𝒜 , ℬ) carries the maximal information. We collect these results in a theorem:

Theorem 3.13 Let ρ : 𝒜 → ℰ_{+}^{s} ▷ ℐ_{+}^{s} ˆ⊗ℬ be a (p + 1)-summable quasihomomorphism of parity p mod 2, with ℰ admissible with respect to a quasi-free extension of ℬ. The bivariant Chern character ch^{p}(ρ) ∈ HC^{p}(𝒜 , ℬ) induces a graded-commutative diagram

\xymatrix{HP_{n+1}(𝒜) \ar[r]^{S} \ar[d]^{ch^{p}(ρ)} & HC_{n−1}(𝒜) \ar[r]^{B} \ar[d]^{ch^{p}(ρ)} & HN_{n}(𝒜) \ar[r]^{I} \ar[d]^{ch^{p}(ρ)} & HP_{n}(𝒜) \ar[d]^{ch^{p}(ρ)} \\ HP_{n−p+1}(ℬ) \ar[r]^{S} & HC_{n−p−1}(ℬ) \ar[r]^{B} & HN_{n−p}(ℬ) \ar[r]^{I} & HP_{n−p}(ℬ)}

invariant under conjugation of quasihomomorphisms. Moreover the arrow in periodic cyclic homology HP_{n}(𝒜) → HP_{n−p}(ℬ) is invariant under homotopy of quasihomomorphisms.

Proof: The fact that Sch^{p}(ρ) ∈ HC^{p+2}(𝒜 , ℬ) is homotopy invariant shows its image in the periodic theory HP^{p}(𝒜 , ℬ) is homotopy invariant.

Example 3.14 When 𝒜 is arbitrary and ℬ = ℂ, we saw in Example 3.3 that a (p + 1)-summable quasihomomorphism ρ : 𝒜 → ℒ^{s} ▷ ℐ^{s}, represents a K-homology class of 𝒜. By hypothesis, the degree of the quasihomomorphism is p mod 2. The Chern character ch^{p}(ρ) ∈ HC^{p}(𝒜 , ℂ) ≅ HC^{p}(𝒜) is a cyclic cohomology class of degree p over 𝒜, represented by a chain map X̂(T 𝒜 , J𝒜) → ℂ vanishing on the subcomplex F^{p}X̂(T 𝒜 , J𝒜). Using the pro-vector space isomorphism X̂(T 𝒜 , J𝒜) ≅ Ω̂𝒜, one finds that ch^{p}(ρ) is non-zero only on the subspace of p-forms Ω^{p}𝒜, explicitly

ch^{p}(ρ)(a_{0}da_{1} … da_{p}) = \frac{c_{p}}{2} Tr_{s}(F [F, a_{0}] … [F, a_{p}]) ,

where Tr_{s} : (ℐ^{s})^{p+1} → ℂ is the supertrace of the (p + 1)-summable algebra ℐ and c_{p} is a constant depending on the degree. One has c_{p} = (−)^{n}(n!)^{2}/p! when p = 2n is even, and c_{p} = \sqrt{2πi} (−)^{n}/2^{p} when p = 2n + 1 is odd. This coincides with the Chern-Connes character [4, 5], up to a scaling factor accounting for the homotopy equivalence between the X-complex X̂(T 𝒜 , J𝒜) and the (b + B)- complex Ω̂𝒜.

Example 3.15 When 𝒜 = ℂ and ℬ = C^{∞}(0, 1), the Bott element (Example 3.5) represented by the odd 1-summable quasihomomorphism ρ : ℂ → ℰ^{s} ▷ ℐ^{s} ˆ⊗ℬ with ℬ-admissible extension ℰ = C^{∞}[0, 1], has a Chern character in HC^{1}(ℂ, ℬ) ≅ HP_{1}(ℬ). The periodic cyclic homology of ℬ is isomorphic to the de Rham cohomology of the open interval (0, 1), hence HP_{0}(ℬ) = 0 and HP_{1}(ℬ) = ℂ. Consequently, the Chern character ch^{1}(ρ) may be represented by a smooth one-form over [0, 1] vanishing at the endpoints. It involves a realvalued function ξ ∈ ℰ, with ξ(0) = 0 and ξ(1) = π/2, used in the construction of the homomorphism ρ : ℂ → ℰ_{+}^{s}. One explicitly finds

ch(ρ) = \sqrt{2πi} d(sin^{2} ξ) ,

so that its integral over the interval [0, 1] is normalized to \sqrt{2πi}, and of course does no depend on the chosen function ξ. This is due to the fact that quasihomomorphisms associated to different choices of ξ are homotopic.

4 Topological K-theory

We review here the topological K-theory of Fréchet m-algebras following Phillips [27], and construct various Chern character maps with value in cyclic homology. Topological K-theory for Fréchet m-algebras is defined in analogy with Banach algebras and fulfills the same properties of homotopy invariance, Bott periodicity and excision [27]. For our purposes, only homotopy invariance and Bott periodicity are needed. A basic example of Fréchet m-algebra is provided by the algebra 𝒦 of “smooth compact operators”. 𝒦 is the space of infinite matrices (A_{ij})_{i,j∈ℕ} with entries in ℂ and rapid decay, endowed with the family of submultiplicative norms

∥A∥_{n} = sup_{(i,j)∈ℕ^{2}}(1 + i + j)^{n}A_{ij} < ∞ ∀n ∈ ℕ .

The multiplication of matrices makes 𝒦 a Fréchet m-algebra. Moreover 𝒦 is nuclear as a locally convex vector space [14]. If 𝒜 is any Fréchet m-algebra, the completed tensor product 𝒦 ˆ⊗𝒜 is the smooth stabilization of 𝒜. Other important examples are the algebras C^{∞}[0, 1], resp. C^{∞}(0, 1), of smooth ℂvalued functions over the interval, with all derivatives of order ≥ 1, resp. ≥ 0, vanishing at the endpoints. As already mentioned in section 3, these are again nuclear Fréchet m-algebras and the completed tensor products 𝒜 [0, 1] = 𝒜 ˆ⊗C^{∞}[0, 1] and 𝒜 (0, 1) = 𝒜 ˆ⊗C^{∞}(0, 1) are isomorphic to the algebras of smooth 𝒜 -valued functions over the interval, with the appropriate vanishing boundary conditions. In particular S𝒜 := 𝒜 (0, 1) is the smooth suspension of 𝒜. We say that two idempotents e_{0}, e_{1} of an algebra 𝒜 are smoothly homotopic if there exists an id empotent e ∈ 𝒜 [0, 1] whose evaluation at the endpoints gives e_{0} and e_{1}. Similarly for invertible elements. The definition of topological K-theory involves idempotents and invertibles of the unitalized algebra (𝒦 ˆ⊗𝒜)^{+}. Choosing an isomorphism M_{2}(𝒦 ) ≅ 𝒦 makes (𝒦 ˆ⊗𝒜)^{+} a semigroup for the direct sum a ⊕ b = ( \begin{matrix}a & 0 \\ 0 & b \end{matrix} ). We denote by p_{0} the idempotent ( \begin{matrix}1 & 0 \\ 0 & 0 \end{matrix} ) of the matrix algebra M_{2}(𝒦 ˆ⊗𝒜)^{+}.

Definition 4.1 (Phillips [27]) Let 𝒜 be a Fréchet m-algebra. The topological K-theory of 𝒜 in degree zero and one is defined by

K_{0}^{top}(𝒜) = {set of smooth homotopy classes of idempotents e ∈ M_{2}(𝒦 ˆ⊗𝒜)^{+}
such that e − p_{0} ∈ M_{2}(𝒦 ˆ⊗𝒜) }
K_{1}^{top}(𝒜) = {set of smooth homotopy classes of invertibles g ∈ (𝒦 ˆ⊗𝒜)^{+}
such that g − 1 ∈ 𝒦 ˆ⊗𝒜 }

K_{0}^{top}(𝒜) and K_{1}^{top}(𝒜) are semigroups for the direct sum of idempotents and invertibles; in the case of idempotents, the direct sum e ⊕ e′ ∈ M_{4}(𝒦 ˆ⊗𝒜)^{+} has to be conjugated by the invertible matrix

c = (\begin{matrix}1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}) ∈ M_{4}(𝒦 ˆ⊗𝒜)^{+} , c^{−1} = c ,  (52)

in order to preserve the condition c(e ⊕ e′)c − p̃_{0} ∈ M_{4}(𝒦 ˆ⊗𝒜), with p̃_{0} the diagonal matrix diag(1, 1, 0, 0). The proof that K_{0}^{top}(𝒜) and K_{1}^{top}(𝒜) are actually abelian groups will be recalled in Lemma 5.2. The unit of K_{0}^{top}(𝒜) is the class of the idempotent p_{0} ∈ M_{2}(𝒦 ˆ⊗𝒜)^{+}, whereas the unit of K_{1}^{top}(𝒜) is represented by 1 ∈ (𝒦 ˆ⊗𝒜)^{+}. The fundamental property of topological K-theory is Bott periodicity [27]. Let S𝒜 = 𝒜 (0, 1) be the smooth suspension of 𝒜. Define two additive maps

α : K_{1}^{top}(𝒜) → K_{0}^{top}(S𝒜) ,  β : K_{0}^{top}(𝒜) → K_{1}^{top}(S𝒜)  (53)

as follows. First choose a real-valued function ξ ∈ C^{∞}[0, 1] such that ξ(0) = 0 and ξ(1) = π/2 (we recall that all the derivatives of ξ vanish at the endpoints). Let g ∈ (𝒦 ˆ⊗𝒜)^{+} represent an element of K_{1}^{top}(𝒜). Then the idempotent

α(g) = G^{−1}p_{0}G , α(g) − p_{0} ∈ M_{2}(𝒦 ˆ⊗S𝒜)

defines an element of K_{0}^{top}(S𝒜), where G : [0, 1] → M_{2}(𝒦 ˆ⊗𝒜)^{+} is the matrix function over [0, 1]

G = R^{−1} ( \begin{matrix}1 & 0 \\ 0 & g \end{matrix} ) R with R = ( \begin{matrix}cos ξ & sin ξ \\ − sin ξ & cos ξ \end{matrix} ) .

Now z = exp(4iξ) is a complex-valued invertible function over [0, 1] with winding number 1. The functions z − 1 and z^{−1} − 1 lie in C^{∞}(0, 1). Then for any idempotent e ∈ M_{2}(𝒦 ˆ⊗𝒜)^{+} representing a class in K_{0}^{top}(𝒜), we define the invertible element

β(e) = (1 + (z − 1)e)(1 + (z − 1)p_{0})^{−1} .

One has (1 + (z − 1)p_{0})^{−1} = (1 + (z^{−1} − 1)p_{0}), and the idempotent relations e^{2} = e, p_{0}^{2} = p_{0} imply β(e) = 1 + (z − 1)e(e − p_{0}) + (z^{−1} − 1)(p_{0} − e)p_{0}, which shows that β(e) − 1 is an element of the algebra M_{2}(𝒦 ˆ⊗S𝒜) ≅ 𝒦 ˆ⊗S𝒜, hence β(e) defines a class in K_{1}^{top}(S𝒜).

Proposition 4.2 (Bott periodicity [27]) The two maps defined above α : K_{1}^{top}(𝒜) → K_{0}^{top}(S𝒜) and β : K_{0}^{top}(𝒜) → K_{1}^{top}(S𝒜) are isomophisms of abelian groups.

Hence Bott periodicity implies K_{i}^{top}(S^{2}𝒜) = K_{i}^{top}(𝒜) for i = 0, 1, so that we may define topological K-theory groups in any degree n ∈ ℤ:

K_{n}^{top}(𝒜) = \{ \begin{matrix}K_{0}^{top}(𝒜) n even \\ K_{1}^{top}(𝒜) n odd . \end{matrix}  (54)

Following Cuntz and Quillen [12], we construct Chern characters with values in periodic cyclic homology K_{n}^{top}(𝒜) → HP_{n}(𝒜). Recall (section 2) that periodic cyclic homology is computed from any quasi-free extension 0 → 𝒥 → ℛ → 𝒜 → 0 by the pro-complex

X̂(ℛ, 𝒥) = X(ℛ̂) : ℛ̂ ⇄ Ω^{1}ℛ̂_{♮} ,

where the pro-algebra ℛ̂ = \underleftarrow{lim}_{n} ℛ/𝒥^{n} is the 𝒥 -adic completion of the quasifree algebra ℛ. In particular, the universal free extension 0 → J𝒜 → T 𝒜 → 𝒜 → 0 is quasi-free and the universal property of the tensor algebra leads to a classifying homomorphism T 𝒜 → ℛ compatible with the ideals J𝒜 and 𝒥 by means of the commutative diagram

[figure: 0; J𝒜; T 𝒜; 𝒜; 0; 0; 𝒥; ℛ; 𝒜; σ; 0]

for any choice of continuous linear section σ : 𝒜 → ℛ. The homomorphism T 𝒜 → ℛ thus extends to a homomorphism of pro-algebras T̂𝒜 → ℛ̂ and the induced morphism of complexes X(T̂𝒜) → X(ℛ̂) is a homotopy equivalence. The Chern character on topological K-theory requires to lift idempotents and invertible elements from the algebra 𝒦 ˆ⊗𝒜 to the pro-algebra

𝒦 ˆ⊗ℛ̂ = \underleftarrow{lim}_{n} 𝒦 ˆ⊗(ℛ/𝒥^{n}) .

If e ∈ M_{2}(𝒦 ˆ⊗𝒜)^{+} is an idempotent such that e − p_{0} ∈ M_{2}(𝒦 ˆ⊗𝒜), there always exists an idempotent lift ê ∈ M_{2}(𝒦 ˆ⊗ℛ̂)^{+} with ê − p_{0} ∈ M_{2}(𝒦 ˆ⊗ℛ̂), and two such liftings are always conjugate [12]. A concrete way to construct an idempotent lift is to work first with the tensor algebra and then push forward by the homomorphism 𝒦 ˆ⊗T̂𝒜 → 𝒦 ˆ⊗ℛ̂. Using the isomorphism of pro-vector spaces T̂𝒜 ≅ Ω̂^{+}𝒜, the following differential form of even degree defines an idempotent [12]

ê = e + ∑_{k≥1} \frac{(2k)!}{(k!)^{2}}(e − \frac{1}{2})(dede)^{k} ∈ M_{2}(𝒦 ˆ⊗T̂𝒜)^{+} ,  (55)

where concatenation products over M_{2}(𝒦 ) are taken. We will refer to (55) as the canonical lift of e, but it should be stressed that other choices are possible. Denoting also by ê its image in M_{2}(𝒦 ˆ⊗ℛ̂)^{+}, the Chern character of e is represented by the cycle of even degree

ch_{0}(ê) = Tr(ê − p_{0}) ∈ ℛ̂ ,  (56)

where the partial trace Tr : M_{2}(𝒦 ˆ⊗ℛ̂) → ℛ̂ comes from the usual trace of matrices with rapid decay. We will show below that the cyclic homology class of ch_{0}(ê) is invariant under smooth homotopies of ê. Moreover, the invariance of the trace under similarity implies that ch_{0} is additive. Next, if g ∈ (𝒦 ˆ⊗𝒜)^{+} is an invertible element such that g − 1 ∈ 𝒦 ˆ⊗𝒜, we have again to choose an invertible lift ĝ ∈ (𝒦 ˆ⊗ℛ̂)^{+} with ĝ − 1 ∈ 𝒦 ˆ⊗ℛ̂. It turns out that any lifting of g is invertible, and two such liftings are always homotopic [12]. A concrete way to construct an invertible lift is to use the linear inclusion of zero-forms 𝒦 ˆ⊗𝒜 ↪ 𝒦 ˆ⊗T̂𝒜 ≅ 𝒦 ˆ⊗Ω̂^{+}𝒜 and consider g as an element ĝ = g ∈ (𝒦 ˆ⊗T̂𝒜)^{+}. A simple computation shows that it is invertible, with inverse

ĝ^{−1} = ∑_{k≥0} g^{−1}(dg dg^{−1})^{k} ∈ (𝒦 ˆ⊗T̂𝒜)^{+} .  (57)

Here again we shall refer to the above ĝ as the canonical lift of g, but other choices are possible. Then denoting also by ĝ its image in (𝒦 ˆ⊗ℛ̂)^{+}, the Chern character of g is represented by the cycle of odd degree

ch_{1}(ĝ) = \frac{1}{\sqrt{2πi}} Tr♮ĝ^{−1}dĝ ∈ Ω^{1}ℛ̂_{♮} ,  (58)

with the trace map Tr : Ω^{1}(𝒦 ˆ⊗ℛ̂)_{♮} → Ω^{1}ℛ̂_{♮} . In this case also we will show that the cyclic homology class of ch_{1}(ĝ) is invariant under smooth homotopies of ĝ. Clearly ch_{1} is additive. The factor 1/\sqrt{2πi} is chosen for consistency with the bivariant Chern character. Note the following important property of idempotents and invertibles: two idempotents ê_{0}, ê_{1} ∈ M_{2}(𝒦 ˆ⊗ℛ̂)^{+} are homotopic if and only if their projections e_{0}, e_{1} ∈ M_{2}(𝒦 ˆ⊗𝒜)^{+} are homotopic, and similarly with invertibles [12]. Since the cyclic homology classes of the Chern characters ch_{0}(ê) and ch_{1}(ĝ) are homotopy invariant with respect to ê and ĝ, one gets well-defined additive maps ch_{0} : K_{0}^{top}(𝒜) → HP_{0}(𝒜) and ch_{1} : K_{1}^{top}(𝒜) → HP_{1}(𝒜) on the topological K-theory groups. They do not depend on the quasi-free extension ℛ since we know that the classifying homomorphism T̂𝒜 → ℛ̂ induces a homotopy equivalence of pro-complexes X(T̂𝒜) →^{∼} X(ℛ̂). To show the homotopy invariance of the Chern characters, we introduce the Cherns-Simons transgressions. Let ℛ̂[0, 1] be the tensor product ℛ̂ˆ⊗C^{∞}[0, 1], and let ê be any idempotent of M_{2}(𝒦 ˆ⊗ℛ̂[0, 1])^{+} with ê−p_{0} ∈ M_{2}(𝒦 ˆ⊗ℛ̂[0, 1]). Denote by s : C^{∞}[0, 1] → Ω^{1}[0, 1] the de Rham coboundary map with values in ordinary (commutative) one-forms over the interval. We then define the Chern-Simons form associated to ê as the chain of odd degree

cs_{1}(ê) = ∫_{0}^{1} Tr♮(−2ê + 1) sêdê ∈ Ω^{1}ℛ̂_{♮} ,  (59)

with obvious notations. Now let ĝ ∈ (𝒦 ˆ⊗ℛ̂[0, 1])^{+} be any invertible element such that ĝ − 1 ∈ 𝒦 ˆ⊗ℛ̂[0, 1]. The Chern-Simons form associated to ĝ is the chain of even degree

cs_{0}(ĝ) = \frac{1}{\sqrt{2πi}} ∫_{0}^{1} Tr(ĝ^{−1}sĝ) ∈ ℛ̂ .  (60)

Lemma 4.3 Let ê be an idempotent of the algebra M_{2}(𝒦 ˆ⊗ℛ̂[0, 1])^{+} with ê − p_{0} ∈ M_{2}(𝒦 ˆ⊗ℛ̂[0, 1]). Denote by ê_{0} and ê_{1} the idempotents of M_{2}(𝒦 ˆ⊗ℛ̂)^{+} obtained by evaluation at 0 and 1. Then one has

\overline{b}cs_{1}(ê) = ch_{0}(ê_{1}) − ch_{0}(ê_{0}) ∈ ℛ̂ .  (61)

Let ĝ ∈ (𝒦 ˆ⊗ℛ̂[0, 1])^{+} be an invertible element such that ĝ − 1 ∈ 𝒦 ˆ⊗ℛ̂[0, 1]. Denote by ĝ_{0} and ĝ_{1} the invertibles of (𝒦 ˆ⊗ℛ̂)^{+} obtained by evaluation at 0 and 1. Then one has

♮dcs_{0}(ĝ) = ch_{1}(ĝ_{1}) − ch_{1}(ĝ_{0}) ∈ Ω^{1}ℛ̂_{♮} .  (62)

Proof: First notice that the current ∫_{0}^{1} is odd, so that

\overline{b}cs_{1}(ê) = − ∫_{0}^{1} \overline{b}Tr♮(−2ê + 1) sêdê ,

and taking into account the fact that sê is also odd, one has

\overline{b}Tr♮(−2ê + 1) sêdê = −Tr[(−2ê + 1) sê, ê] = Tr((2ê − 1)sêê − êsê) ,

where we use the idempotent property ê^{2} = ê for the last equality. Since s is a derivation, one has sê = s(ê^{2}) = sêê + êsê and êsêê = 0, whence

\overline{b}cs_{1}(e) = ∫_{0}^{1} Tr(sêê + êsê) = ∫_{0}^{1} sTrê = Tr(ê_{1} − ê_{0}) = ch_{0}(ê_{1}) − ch_{0}(ê_{0}) ,

because ê_{0} and ê_{1} are the evaluations of ê respectively at 0 and 1. Let us proceed now with invertibles:

♮dcs_{0}(ĝ) = \frac{−1}{\sqrt{2πi}} ∫_{0}^{1} Tr♮d(ĝ^{−1}sĝ) ,

and because d is an odd derivation anticommuting with s, one has

Tr♮d(ĝ^{−1}sĝ) = Tr♮(dĝ^{−1}sĝ − ĝ^{−1}sdĝ) .

Then, d1 = 0 = s1 implies dĝ^{−1} = −ĝ^{−1}dĝĝ^{−1} and sĝ^{−1} = −ĝ^{−1}sĝĝ^{−1}. But Tr♮ is a supertrace, hence

Tr♮(−ĝ^{−1}dĝĝ^{−1}sĝ − ĝ^{−1}sdĝ) = Tr♮(ĝ^{−1}sĝĝ^{−1}dĝ − ĝ^{−1}sdĝ) = −sTr♮(ĝ^{−1}dĝ) .

By integration over the interval [0, 1], one gets

♮dcs_{0}(ĝ) = \frac{1}{\sqrt{2πi}}Tr♮(ĝ_{1}^{−1}dĝ_{1} − ĝ_{0}^{−1}dĝ_{0}) = ch_{1}(ĝ_{1}) − ch_{1}(ĝ_{0})

as wanted.

Hence the cyclic homology classes of the Chern characters are homotopy invariant as claimed. There is another consequence of the above lemma. Observe that the suspensions S𝒜 = 𝒜 (0, 1) and Sℛ̂ = ℛ̂(0, 1) are subalgebras of 𝒜 [0, 1] and ℛ̂[0, 1]. If e ∈ M_{2}(𝒦 ˆ⊗S𝒜)^{+} is an idempotent representing a class in K_{0}^{top}(S𝒜), and g ∈ (𝒦 ˆ⊗S𝒜)^{+} an invertible representing a class in K_{1}^{top}(S𝒜), we choose some lifts ê ∈ M_{2}(𝒦 ˆ⊗Sℛ̂)^{+} and ĝ ∈ (𝒦 ˆ⊗Sℛ̂)^{+}. Then cs_{1}(ê) and cs_{0}(ĝ) are closed and define homology classes in HP_{1}(𝒜) and HP_{0}(𝒜) respectively. The following lemma shows the compatibility with Bott periodicity:

Lemma 4.4 The Chern-Simons forms define additive maps cs_{1} : K_{0}^{top}(S𝒜) → HP_{1}(𝒜) and cs_{0} : K_{1}^{top}(S𝒜) → HP_{0}(𝒜). Moreover they are compatible with the Bott isomorphisms α : K_{1}^{top}(𝒜) → K_{0}^{top}(S𝒜) and β : K_{0}^{top}(𝒜) → K_{1}^{top}(S𝒜) and Chern characters, up to multiplication by a factor \sqrt{2πi}:

cs_{1} ∘ α ≡ \sqrt{2πi} ch_{1} : K_{1}^{top}(𝒜) → HP_{1}(𝒜) ,
cs_{0} ∘ β ≡ \sqrt{2πi} ch_{0} : K_{0}^{top}(𝒜) → HP_{0}(𝒜) .

Proof: Let ê be any idempotent of the pro-algebra M_{2}(𝒦 ˆ⊗Sℛ̂)^{+}. We have to prove the homotopy invariance of the cyclic homology class determined by the cycle

cs_{1}(ê) = ∫_{0}^{1} Tr♮(−2ê + 1) sêdê .

To this end, consider a smooth family of idempotents ê_{t} ∈ M_{2}(𝒦 ˆ⊗Sℛ̂)^{+} parametrized by t ∈ ℝ, such that ê_{t} − p_{0} ∈ M_{2}(𝒦 ˆ⊗Sℛ̂), ∀t. Denote by ˙ ê the derivative ∂ê/∂t. The idempotent property of the family ê implies the following identity:

\frac{∂}{∂t}Tr♮(−2ê + 1)sêdê = −♮dTr(ê( ˙êsê − sê˙ê)) − sTr♮ê( ˙êdê − dê˙ê) .

Since for any fixed t, the idempotent ê_{t} equals p_{0} at the boundaries of the suspended algebra M_{2}(𝒦 ˆ⊗Sℛ̂)^{+}, one gets ∫_{0}^{1} sTr♮ê( ˙êdê − dê˙ê) = 0 and

\frac{∂}{∂t} ∫_{0}^{1} Tr♮(−2ê + 1) sêdê = ♮d ∫_{0}^{1} Tr(ê( ˙êsê − sê˙ê)) .

This implies that the cyclic homology class of cs_{1}(ê) is a homotopy invariant of ê. Hence if ê lifts an idempotent e ∈ M_{2}(𝒦 ˆ⊗S𝒜)^{+}, the cyclic homology class of cs_{1}(ê) only depends on the homotopy class of e, and the map cs_{1} : K_{0}^{top}(S𝒜) → HP_{1}(𝒜) is well-defined. We now have to show the compatibility with Bott periodicity. Thus let g ∈ (𝒦 ˆ⊗𝒜)^{+} be an invertible such that g − 1 ∈ 𝒦 ˆ⊗𝒜, and let α(g) be the idempotent G^{−1}p_{0}G ∈ M_{2}(𝒦 ˆ⊗S𝒜)^{+} constructed by means of a rotation matrix

R = (\begin{matrix}cos ξ & sin ξ \\ − sin ξ & cos ξ \end{matrix}) , G = R^{−1} (\begin{matrix}1 & 0 \\ 0 & g \end{matrix}) R ,

where ξ ∈ C^{∞}[0, 1] is a real function with ξ(0) = 0 and ξ(1) = π/2. Then, it is clear that the idempotent ê = Ĝ^{−1}p_{0}Ĝ ∈ M_{2}(𝒦 ˆ⊗Sℛ̂)^{+} is a lifting of α(g), where the matrix Ĝ = R^{−1}( \begin{matrix}1 & 0 \\ 0 & ĝ \end{matrix} )R is built from any lifting ĝ ∈ (𝒦 ˆ⊗ℛ̂)^{+} of g. Hence, the cyclic homology class of cs_{1}(α̂(g)) is represented by cs_{1}(ê). A direct computation shows the equality

Tr♮(−2ê + 1) sêdê = s(cos ξ)Tr♮(−ĝ^{−1}dĝ + \frac{1}{2}dĝ − \frac{1}{2}dĝ^{−1}) ,

so that after integration over [0, 1] one gets, modulo boundaries ♮d(·)

cs_{1}(α̂(g)) ≡ Tr♮(ĝ^{−1}dĝ) mod ♮d ≡ \sqrt{2πi} ch_{1}(ĝ) mod ♮d .

Next, we turn to the map cs_{0}. If ĝ ∈ (𝒦 ˆ⊗Sℛ̂)^{+} is any invertible, one has

cs_{0}(ĝ) = \frac{1}{\sqrt{2πi}} ∫_{0}^{1} Tr(ĝ^{−1}sĝ) .

We have to show that the cyclic homology class of cs_{0}(ĝ) is a homotopy invariant of ĝ. To this end, consider a smooth one-parameter family of invertibles ĝ_{t} ∈ (𝒦 ˆ⊗Sℛ̂)^{+}. One has, with ˙ ĝ = ∂ĝ/∂t,

\frac{∂}{∂t}(ĝ^{−1}sĝ) = −ĝ^{−1} ˙ĝĝ^{−1}sĝ + ĝ^{−1}s ˙ĝ = [ĝ^{−1}sĝ, ĝ^{−1} ˙ĝ] + s(ĝ^{−1} ˙ĝ) .

Since Tr[ĝ^{−1}sĝ, ĝ^{−1} ˙ĝ] = −\overline{b}Tr♮ĝ^{−1}sĝd(ĝ^{−1} ˙ĝ), we get

\frac{∂}{∂t} ∫_{0}^{1} Tr(ĝ^{−1}sĝ) = \overline{b} ∫_{0}^{1} Tr♮ĝ^{−1}sĝd(ĝ^{−1} ˙ĝ) .

Hence the cyclic homology class of cs_{0}(ĝ) is homotopy invariant. In particular if ĝ lifts an invertible g ∈ (𝒦 ˆ⊗S𝒜)^{+}, the cyclic homology class of cs_{0}(ĝ) is a homotopy invariant of g and the map cs_{0} : K_{1}^{top}(S𝒜) → HP_{0}(𝒜) is well-defined. Now let e ∈ M_{2}(𝒦 ˆ⊗𝒜)^{+} be an idempotent, with e − p_{0} ∈ M_{2}(𝒦 ˆ⊗𝒜). Its image under the Bott map β is the invertible element β(e) ∈ (𝒦 ˆ⊗S𝒜)^{+} given by

β(e) = (1 + (z − 1)e)(1 + (z − 1)p_{0})^{−1} , where z = exp(4iξ). If ê ∈ M_{2}(𝒦 ˆ⊗ℛ̂)^{+} is any idempotent lift of e, it is clear that the invertible

ĝ = (1 + (z − 1)ê)(1 + (z − 1)p_{0})^{−1} ∈ (𝒦 ˆ⊗Sℛ̂)^{+}

is a lifting of β(e). Hence the cyclic homology class of cs_{0}(β̂(e)) is represented by cs_{0}(ĝ). Let us compute explicitly Tr(ĝ^{−1}sĝ):

Tr((1 + (z − 1)p_{0})(1 + (z − 1)ê)^{−1}s((1 + (z − 1)ê)(1 + (z − 1)p_{0})^{−1}))

= Tr((1 + (z^{−1} − 1)ê)sz ê − (1 + (z^{−1} − 1)p_{0})sz p_{0})
= Tr(ê − p_{0}) z^{−1}sz .

Since the integration of z^{−1}sz over the interval [0, 1] yields a factor 2πi, one is left with equivalences modulo boundaries \overline{b}(·)

cs_{0}(β̂(e)) ≡ \sqrt{2πi} Tr(ê − p_{0}) mod \overline{b} ≡ \sqrt{2πi} ch_{0}(ê) mod \overline{b}

as wanted.

Since our main motivation is index theory we will have to consider the stabilization of 𝒜 by a p-summable Fréchet m-algebra ℐ, that is, ℐ is provided with a continuous trace Tr : ℐ^{p} → ℂ as in section 3. Hence it will be convenient to define a Chern character K_{n}^{top}(ℐ ˆ⊗𝒜) → HP_{n}(𝒜). The difficulty of course is that the trace is not defined on the algebra 𝒦 ˆ⊗ℐ but only on its p-th power. To cope with this problem, we shall construct higher analogues of the Chern characters and Chern-Simons forms associated to idempotents and invertibles. Consider the following p-summable quasihomomorphism of even degree, from the algebra ℐ 𝒜 := ℐ ˆ⊗𝒜 to 𝒜:

ρ : ℐ 𝒜 → ℰ^{s} ▷ ℐ^{s}𝒜 ,  ℰ = ℐ^{+} ˆ⊗𝒜 ,  (63)

where ℐ^{+} is the unitalization of ℐ. Because ρ is of even degree, it is entirely specified by a pair of homomorphisms (ρ_{+}, ρ_{−}) : ℐ 𝒜 ⇉ ℰ which agree modulo the ideal ℐ 𝒜 ⊂ ℰ. Equivalently if we represent ℰ^{s} in the ℤ_{2}-graded matrix algebra M_{2}(ℰ) we can write ρ = ( \begin{matrix}ρ_{+} & 0 \\ 0 & ρ_{−} \end{matrix} ). By definition we set

ρ_{+} = Id : ℐ 𝒜 → ℐ 𝒜 ⊂ ℰ , ρ_{−} = 0 .

Let 0 → 𝒥 → ℛ → 𝒜 → 0 be any quasi-free extension of 𝒜, with continuous linear splitting σ : 𝒜 → ℛ. Then choosing ℳ = ℐ^{+} ˆ⊗ℛ and 𝒩 = ℐ^{+} ˆ⊗𝒥, one gets a commutative diagram

[figure: 0; 𝒩; ℳ; ℰ; 0; 0; ℐ 𝒥; ℐ ℛ; ℐ 𝒜; 0]

Since (ℐ^{+})^{n} = ℐ^{+} for any integer n, one sees that 𝒩 ^{n} = ℐ^{+} ˆ⊗𝒥^{n} is a direct summand in ℳ. Moreover, there is an obvious chain map Tr : F_{ℐ ℛ}^{2n+1}X(ℳ) → X(ℛ) for any n ≥ p − 1, obtained by taking the trace ℐ^{n+1} → ℂ. It follows that the algebra ℰ ▷ ℐ 𝒜 is ℛ-admissible (Definition 3.2), hence in all degrees 2n + 1 ≥ p the bivariant Chern characters ch^{2n}(ρ) ∈ HC^{2n}(ℐ 𝒜 , 𝒜) are defined and related by the S-operation in bivariant cyclic cohomology ch^{2n+2}(ρ) ≡ Sch^{2n}(ρ). We recall briefly the construction of ch^{2n}(ρ). By the universal properties of the tensor algebra T (ℐ 𝒜), the homomorphism ρ : ℐ 𝒜 → ( \begin{matrix}ℐ 𝒜 & 0 \\ 0 & 0 \end{matrix} ) ⊂ ℰ_{+}^{s} lifts to a classifying homomorphism ρ_{∗} through the commutative diagram (31)

[figure: 0; J(ℐ 𝒜); ρ_{∗}; T (ℐ 𝒜); ρ_{∗}; ℐ 𝒜; ρ; 0; 0; 𝒩_{+}^{s}; ℳ_{+}^{s}; ℰ_{+}^{s}; Id⊗σ; 0]

induced by the linear splitting. It extends to a homomorphism of pro-algebras ρ_{∗} : T̂(ℐ 𝒜) → ( \begin{matrix}ℐ ℛ̂ & 0 \\ 0 & 0 \end{matrix} ) ⊂ ℳ̂_{+}^{s}. The bivariant Chern character ch^{2n}(ρ) is the composite of the Goodwillie equivalence γ : X(T̂(ℐ 𝒜)) → Ω̂T̂(ℐ 𝒜) with the chain maps ρ_{∗} : Ω̂T̂(ℐ 𝒜) → Ω̂ℳ̂_{+}^{s} and χ^{2n} : Ω̂ℳ̂_{+}^{s} → X(ℛ̂). The two non-zero components of χ^{2n} are given by Eqs. (34) and defined on 2n and (2n + 1)-forms respectively:

χ_{0}^{2n} : Ω^{2n}ℳ̂_{+}^{s} → ℛ̂ , χ_{1}^{2n} : Ω^{2n+1}ℳ̂_{+}^{s} → Ω^{1}ℛ̂_{♮} .

The bivariant Chern character is designed to improve the summability degree and can be used to define the higher Chern characters of idempotents and invertibles via the composition

ch_{i}^{2n} : K_{i}^{top}(ℐ ˆ⊗𝒜) → HP_{i}(ℐ ˆ⊗𝒜) ⟶^{ch_{i}^{2n}(ρ)} HP_{i}(𝒜) ,  i = 0, 1 .  (64)

We shall now establish very explicit formulas for these higher characters. Let ê be an idempotent of the algebra M_{2}(𝒦 ˆ⊗ℐ ℛ̂)^{+}, such that ê − p_{0} ∈ M_{2}(𝒦 ˆ⊗ℐ ℛ̂). It is well-known (see for example [5]) that the differential forms

ch_{2n}(ê) = (−)^{n} \frac{(2n)!}{n!} Tr((ê − \frac{1}{2})(dêdê)^{n}) ∈ Ω^{2n}(ℐ ℛ̂) for n ≥ 1
ch_{0}(ê) = Tr(ê − p_{0}) ∈ Ω^{0}(ℐ ℛ̂)  (65)

are the components of a (b + B)-cycle of even degree over ℐ ℛ̂, i.e. fulfill the relations Bch_{2n}(ê) + bch_{2n+2}(ê) = 0 for any n. Here Tr is the trace over 𝒦 . In the odd case, any invertible element ĝ ∈ (𝒦 ˆ⊗ℐ ℛ̂)^{+} such that ĝ−1 ∈ 𝒦 ˆ⊗ℐ ℛ̂ gives rise to a (b + B)-cycle of odd degree with components

ch_{2n+1}(ĝ) = \frac{(−)^{n}}{\sqrt{2πi}} n! Tr(ĝ^{−1}dĝ(dĝ^{−1}dĝ)^{n}) ∈ Ω^{2n+1}(ℐ ℛ̂) .  (66)

The homology classes of these cycles are of course homotopy invariant. If ê ∈ M_{2}(𝒦 ˆ⊗ℐ ℛ̂[0, 1])^{+} is a smooth path of idempotents, we define the components of the associated Chern-Simons form as

cs_{2n+1}(ê) = (−)^{n} \frac{(2n)!}{n!} ∫_{0}^{1} Tr((−2ê + 1) ∑_{i=0}^{2n}(dê)^{i}sê(dê)^{2n+1−i})  (67)

in Ω^{2n+1}(ℐ ℛ̂). Similarly if ĝ ∈ (𝒦 ˆ⊗ℐ ℛ̂[0, 1])^{+} is a smooth path of invertibles, the components of the Chern-Simons form are for n ≥ 1

cs_{2n}(ĝ) = \frac{(−)^{n}}{\sqrt{2πi}} (n − 1)! ∫_{0}^{1} Tr(ĝ^{−1}dĝ ∑_{i=0}^{n−1}(dĝ^{−1}dĝ)^{i}dω(dĝ^{−1}dĝ)^{n−1−i}) (68)

in Ω^{2n}(ℐ ℛ̂), where ω = ĝ^{−1}sĝ, and for n = 0 we set as before cs_{0}(ĝ) = \frac{1}{\sqrt{2πi}} ∫_{0}^{1} Tr(ω). Simple algebraic manipulations show that the (b + B)- boundaries of the Chern-Simons forms yield the difference of evaluations of the Chern characters at the endpoints:

Bcs_{2n−1}(ê) + bcs_{2n+1}(ê) = ch_{2n}(ê_{1}) − ch_{2n}(ê_{0}) ,  (69)
Bcs_{2n}(ĝ) + bcs_{2n+2}(ĝ) = ch_{2n+1}(ĝ_{1}) − ch_{2n+1}(ĝ_{0}) .

The higher Chern characters (64) and their associated Chern-Simons forms are obtained by evaluation of these (b + B)-chains on the inclusion homomorphism ι_{∗} : ℐ ℛ̂ ↪ ( \begin{matrix}ℐ ℛ̂ & 0 \\ 0 & 0 \end{matrix} ) ⊂ ℳ̂_{+}^{s} followed by the chain map χ^{2n} whenever 2n+1 ≥ p.

Lemma 4.5 Let ℐ be p-summable and 2n + 1 ≥ p. For any idempotent ê ∈ M_{2}(𝒦 ˆ⊗ℐ ℛ̂)^{+} such that ê − p_{0} ∈ M_{2}(𝒦 ˆ⊗ℐ ℛ̂), and any invertible ĝ ∈ (𝒦 ˆ⊗ℐ ℛ̂)^{+} such that ĝ − 1 ∈ 𝒦 ˆ⊗ℐ ℛ̂, we define the higher Chern characters by the explicit formulas

ch_{0}^{2n}(ê) = Tr (ê − p_{0})^{2n+1} ,  (70)
ch_{1}^{2n}(ĝ) = \frac{1}{\sqrt{2πi}} \frac{(n!)^{2}}{(2n)!} Tr♮ ĝ^{−1}[(ĝ − 1)(ĝ^{−1} − 1)]^{n}dĝ ,

where we take concatenation products over ℐ and Tr is the trace over the p-th power of 𝒦 ˆ⊗ℐ. Then one has ch_{0}^{2n}(ê) = χ_{0}^{2n}ι_{∗}ch_{2n}(ê) in ℛ̂ and ch_{1}^{2n}(ĝ) = χ_{1}^{2n}ι_{∗}ch_{2n+1}(ĝ) in Ω^{1}ℛ̂_{♮} . Similarly, for any idempotent ê ∈ M_{2}(𝒦 ˆ⊗ℐ ℛ̂[0, 1])^{+} and any invertible ĝ ∈ (𝒦 ˆ⊗ℐ ℛ̂[0, 1])^{+}, we define the higher Chern-Simons forms by the explicit formulas

cs_{1}^{2n}(ê) = ∫_{0}^{1} Tr♮(−2ê + 1) ∑_{i=0}^{n}(ê − p_{0})^{2i}sê(ê − p_{0})^{2(n−i)}dê , (71)

cs_{0}^{2n}(ĝ) = \frac{1}{\sqrt{2πi}} \frac{(n!)^{2}}{(2n)!} ∫_{0}^{1} Tr ĝ^{−1}[(ĝ − 1)(ĝ^{−1} − 1)]^{n}sĝ

Then cs_{1}^{2n}(ê) = χ_{1}^{2n}ι_{∗}cs_{2n+1}(ê) in Ω^{1}ℛ̂_{♮} and cs_{0}^{2n}(ĝ) ≡ χ_{0}^{2n}ι_{∗}cs_{2n}(ĝ) mod \overline{b} in ℛ̂. Moreover the transgression relations hold:

\overline{b}cs_{1}^{2n}(ê) = ch_{0}^{2n}(ê_{1}) − ch_{0}^{2n}(ê_{0}) , ♮dcs_{0}^{2n}(ĝ) = ch_{1}^{2n}(ĝ_{1}) − ch_{1}^{2n}(ĝ_{0}) .

Proof: Let us brielfy explain the computation of the cycles χ_{0}^{2n}ι_{∗}ch_{2n}(ê) and χ_{1}^{2n}ι_{∗}ch_{2n+1}(ĝ) associated to idempotents ê ∈ M_{2}(𝒦 ˆ⊗ℐ ℛ̂)^{+} and invertibles ĝ ∈ (𝒦 ˆ⊗ℐ ℛ̂)^{+}. The upper left corner inclusion ι_{∗} : ℐ ℛ̂ ↪ ℳ̂_{+}^{s} canonically extends to a unital homomorphism (𝒦 ˆ⊗ℐ ℛ̂)^{+} → (𝒦 ˆ⊗ℳ̂_{+}^{s})^{+}, and in matrix form we can write

ι_{∗}ê = (\begin{matrix}ê & 0 \\ 0 & p_{0} \end{matrix}) , ι_{∗}ĝ = (\begin{matrix}ĝ & 0 \\ 0 & 1 \end{matrix}) .

Consequently the commutators with the odd multiplier F = ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) read

[F, ι_{∗}ê] = (\begin{matrix}0 & p_{0} − ê \\ ê − p_{0} & 0 \end{matrix}) , [F, ι_{∗}ĝ] = (\begin{matrix}0 & 1 − ĝ \\ ĝ − 1 & 0 \end{matrix}) .

It is therefore straightforward to evaluate the differential forms ch_{2n}(ê) and ch_{2n+1}(ĝ) on the chain map χ^{2n} given by (34). One finds χ_{0}^{2n}ι_{∗}ch_{2n}(ê) = ch_{0}^{2n}(ê) and χ_{1}^{2n}ι_{∗}ch_{2n+1}(ĝ) = ch_{1}^{2n}(ĝ). Similarly with the Chern-Simons forms one finds χ_{1}^{2n}ι_{∗}cs_{2n+1}(ê) = cs_{1}^{2n}(ê), whereas by setting ω = ĝ^{−1}sĝ

χ_{0}^{2n}ι_{∗}cs_{2n}(ĝ) = \frac{1}{\sqrt{2πi}} \frac{(n!)^{2}}{(2n + 1)!} ∫_{0}^{1} Tr(ω[(ĝ^{−1} − 1)(ĝ − 1)]^{n}+

(ĝ − 1)ω[(ĝ^{−1} − 1)(ĝ − 1)]^{n−1}(ĝ^{−1} − 1) + … + [(ĝ^{−1} − 1)(ĝ − 1)]^{n}ω)

coincides with cs_{0}^{2n}(ĝ) only modulo commutators. Finally, the transgression relations are an immediate consequence of Eqs. (69) and the fact that χ^{2n} is a chain map from the (b + B)-complex over ℳ̂_{+}^{s} to the complex X(ℛ̂).

In any degree 2n + 1 ≥ p the Chern characters ch_{0}^{2n} : K_{0}^{top}(ℐ ˆ⊗𝒜) → HP_{0}(𝒜) and ch_{1}^{2n} : K_{1}^{top}(ℐ ˆ⊗𝒜) → HP_{1}(𝒜) are thus obtained by first lifting idempotents e ∈ M_{2}(𝒦 ˆ⊗ℐ 𝒜)^{+} and invertibles g ∈ (𝒦 ˆ⊗ℐ 𝒜)^{+} to some ê ∈ M_{2}(𝒦 ˆ⊗ℐ ℛ̂)^{+} and ĝ ∈ (𝒦 ˆ⊗ℐ ℛ̂)^{+}, and then taking the cyclic homology classes of ch_{0}^{2n}(ê) ∈ ℛ̂ and ch_{1}^{2n}(ĝ) ∈ Ω^{1}ℛ̂_{♮} . Although ê and ĝ are only defined up to homotopy, the above lemma shows these higher Chern characters are well-defined, and moreover independent of the degree 2n because the cocycles χ^{2n} are all related by the transgression relations

χ^{2n} − χ^{2n+2} = [∂, η^{2n+1}] ∈ Hom(Ω̂ℳ̂_{+}^{s}, X(ℛ̂)) .

Passing to suspensions, lifting any idempotent e ∈ M_{2}(𝒦 ˆ⊗ℐ S𝒜)^{+} or invertible g ∈ (𝒦 ˆ⊗ℐ S𝒜)^{+} gives rise to an odd cycle cs_{1}^{2n}(ê) ∈ Ω^{1}ℛ̂_{♮} or an even cycle cs_{0}^{2n}(ĝ) ∈ ℛ̂. As expected, this is well-defined at the K-theory level and compatible with Bott periodicity:

Lemma 4.6 Let ℐ be p-summable. In any degre 2n + 1 ≥ p, the Chern-Simons forms define additive maps cs_{1}^{2n} : K_{0}^{top}(ℐ ˆ⊗S𝒜) → HP_{1}(𝒜) and cs_{0}^{2n} : K_{1}^{top}(ℐ ˆ⊗S𝒜) → HP_{0}(𝒜), independent of n, and compatible with the Bott isomorphisms:

cs_{1}^{2n} ∘ α ≡ \sqrt{2πi} ch_{1}^{2n} : K_{1}^{top}(ℐ ˆ⊗𝒜) → HP_{1}(𝒜) ,
cs_{0}^{2n} ∘ β ≡ \sqrt{2πi} ch_{0}^{2n} : K_{0}^{top}(ℐ ˆ⊗𝒜) → HP_{0}(𝒜) .

Proof: Consider an idempotent ê ∈ M_{2}(𝒦 ˆ⊗ℐ Sℛ̂)^{+}. We have to show the homotopy invariance of the cyclic homology class cs_{1}^{2n}(ê) with respect to ê. This can be shown by direct computation from Formulas (71). Define the matrix idempotent f̂ = ( \begin{matrix}ê & 0 \\ 0 & p_{0} \end{matrix} ). Then one has sf̂ = ( \begin{matrix}sê & 0 \\ 0 & 0 \end{matrix} ) and df̂ = ( \begin{matrix}dê & 0 \\ 0 & 0 \end{matrix} ), and cs_{1}^{2n}(ê) can be rewritten by means of the operator F = ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) and the supertrace τ:

cs_{1}^{2n}(ê) = (−)^{n} ∫_{0}^{1} τ♮(−2f̂+ 1) ∑_{i=0}^{n}[F, f̂]^{2i}sf̂[F, f̂]^{2(n−i)}df̂ .

Now suppose that ê depends smoothly on an additional parameter t. The above integrand may be expressed in terms of the odd differential δ = s + d + dt\frac{∂}{∂t} + [F, ] as

τ♮(−2f̂+ 1) ∑_{i=0}^{n}[F, f̂]^{2i}sf̂[F, f̂]^{2(n−i)}df̂ = \frac{−1}{n + 1} τ♮ f̂(δf̂)^{2n+1}|_{s,d}

where |_{s,d} means that we select the terms containing only s, d and not dt. Because τ♮ is a supertrace, the cocycle property (s + d + dt\frac{∂}{∂t})τ♮ f̂(δf̂)^{2n+1} = τ♮ (δf̂)^{2n+2} = 0 holds, and projecting this relation on s, d, dt yields

s τ♮ f̂(δf̂)^{2n+1}|_{d,dt} + τ♮ d(f̂(δf̂)^{2n+1})|_{s,dt} + dt\frac{∂}{∂t}τ♮ f̂(δf̂)^{2n+1}|_{s,d} = 0 .

This may be rephrased as

\frac{∂}{∂t}(Tr♮(−2ê + 1) ∑_{i=0}^{n}(ê − p_{0})^{2i}sê(ê − p_{0})^{2(n−i)}dê)

≡ s(Tr♮(−2ê + 1) ∑_{i=0}^{n}(ê − p_{0})^{2i} \frac{∂ê}{∂t}(ê − p_{0})^{2(n−i)}dê) mod ♮d ,

and integration over the current ∫_{0}^{1} shows the homotopy invariance of the class cs_{1}^{2n}(ê). Hence the map cs_{1}^{2n} : K_{0}^{top}(ℐ ˆ⊗S𝒜) → HP_{1}(𝒜) is welldefined. Its compatibility with Bott periodicity can be established, without computation, as follows. Let g ∈ (𝒦 ˆ⊗ℐ 𝒜)^{+} be an invertible element and e = α(g) ∈ M_{2}(𝒦 ˆ⊗ℐ S𝒜)^{+} its idempotent image under the Bott isomorphism. Choose an invertible lift g̃ ∈ (𝒦 ˆ⊗ST̂(ℐ 𝒜))^{+} of g and an idempotent lift ẽ ∈ M_{2}(𝒦 ˆ⊗ST̂(ℐ 𝒜))^{+} of e. The differential forms ch_{2n+1}(g̃) and cs_{2n+1}(ẽ) in Ω^{2n+1}T̂(ℐ 𝒜) defined by (66) and (67) are the components of two (b + B)-cycles ch_{∗}(g̃) and cs_{∗}(ẽ), whose projections on the odd part of the complex X(T̂(ℐ 𝒜)) are

♮ch_{1}(g̃) = \frac{1}{\sqrt{2πi}}Tr♮g^{−1}dg̃ , ♮cs_{1}(ẽ) = ∫_{0}^{1} Tr♮(−2ẽ+ 1)sẽdẽ) .

By Lemma 4.4, the cycles \sqrt{2πi}♮ch_{1}(g̃) and ♮cs_{1}(ẽ) are homologous. But we know that the projection Ω̂T̂(ℐ 𝒜) → X(T̂(ℐ 𝒜)) is a homotopy equivalence, with inverse the Goodwillie map γ. Hence \sqrt{2πi} ch_{∗}(g̃) and cs_{∗}(ẽ) are (b + B)- homologous in Ω̂T̂(ℐ 𝒜). Finally, it remains to observe that under the homomorphism ρ_{∗} : T̂𝒜 → ℳ_{+}^{s}, the invertible ρ_{∗}g̃ = ( \begin{matrix}ĝ & 0 \\ 0 & 1 \end{matrix} ) gives a choice of lifting ĝ ∈ (𝒦 ˆ⊗ℐ ℛ̂)^{+} and the idempotent ρ_{∗}(ẽ) = ( \begin{matrix}ê & 0 \\ 0 & p_{0} \end{matrix} ) gives a choice of lifting ê ∈ M_{2}(𝒦 ˆ⊗ℐ ℛ̂)^{+}. Because the cycles \sqrt{2πi} χ^{2n}ρ_{∗}ch_{∗}(g̃) and χ^{2n}ρ_{∗}cs_{∗}(ẽ) are homologous in X(ℛ̂), we have \sqrt{2πi} ch_{1}^{2n}(ĝ) ≡ cs_{1}^{2n}(ê) in HP_{1}(𝒜). We proceed similarly with the map cs_{0}^{2n}. Let ĝ ∈ (𝒦 ˆ⊗ℐ Sℛ̂)^{+} be an invertible. We have to show the homotopy invariance of the cyclic homology class cs_{0}^{2n}(ĝ). Define the invertible matrix û = ( \begin{matrix}ĝ & 0 \\ 0 & 1 \end{matrix} ). Then sû = ( \begin{matrix}sĝ & 0 \\ 0 & 0 \end{matrix} ) and one has

cs_{0}^{2n}(ĝ) = \frac{(−)^{n}}{\sqrt{2πi}} \frac{(n!)^{2}}{(2n)!} ∫_{0}^{1} τ(û^{−1}([F, û][F, û^{−1}])^{n}sû) .

Now suppose that ĝ is a smooth family of invertibles depending on an additional parameter t. The above integrand may be expressed in terms of the odd derivation δ = s + dt\frac{∂}{∂t} + [F, ] as

τ û^{−1}([F, û][F, û^{−1}])^{n}sû ≡ \frac{(−)^{n}}{2n + 1} τ(û^{−1}δû)^{2n+1}|_{s} mod \overline{b} .

One has the relation (s + dt\frac{∂}{∂t})τ(û^{−1}δû)^{2n+1} = −τ(û^{−1}δû)^{2n+2} ≡ 0 mod \overline{b}, hence by projection on s, dt

s τ(û^{−1}δû)^{2n+1}|_{dt} + dt\frac{∂}{∂t}τ(û^{−1}δû)^{2n+1}|_{s} ≡ 0 mod \overline{b} .

This may be rephrased as

\frac{∂}{∂t}(Tr ĝ^{−1}[(ĝ − 1)(ĝ^{−1} − 1)]^{n}sĝ) ≡ s(Tr ĝ^{−1}[(ĝ − 1)(ĝ^{−1} − 1)]^{n} \frac{∂ĝ}{∂t}) mod \overline{b} ,

and integration over the current ∫_{0}^{1} shows the homotopy invariance of the class cs_{0}^{2n}(ĝ). Hence the map cs_{0}^{2n} : K_{1}^{top}(ℐ ˆ⊗S𝒜) → HP_{0}(𝒜) is well-defined. Its compatibility with Bott periodicity is be established as before, replacing invertibles by idempotents and conversely.

5 Multiplicative K-theory

Let 𝒜 and ℐ be Fréchet m-algebras, ℐ being p-summable. We shall define the multiplicative K-theory groups MK_{n}^{ℐ} (𝒜) in any degree n ∈ ℤ. They are intermediate between the topological K-theory K_{n}^{top}(ℐ ˆ⊗𝒜) and the nonperiodic cyclic homology HC_{n}(𝒜). Recall from section 2 that if 0 → 𝒥 → ℛ → 𝒜 → 0 is any quasi-free extension, HC_{n}(𝒜) is computed by the quotient complex X_{n}(ℛ, 𝒥) = X(ℛ)/F_{𝒥}^{n} X(ℛ) induced by the 𝒥 -adic filtration:

HC_{n}(𝒜) = H_{n+2ℤ}(X_{n}(ℛ, 𝒥)) , ∀n ∈ ℤ .

Of course HC_{n}(𝒜) vanishes whenever n < 0. Multiplicative K-theory classes are represented by idempotents or invertibles whose higher Chern characters (Lemma 4.5) can be transgressed up to a certain order. As before we adopt the notation ℐ 𝒜 = ℐ ˆ⊗𝒜 and ℐ ℛ̂ = ℐ ˆ⊗ℛ̂, where ℛ̂ is the ℐ -adic completion of ℛ.

Definition 5.1 Let 0 → 𝒥 → ℛ → 𝒜 → 0 be any quasi-free extension of Fréchet m-algebras, and let ℐ be a p-summable Fréchet m-algebra. Choose an integer q such that 2q + 1 ≥ p. We define the multiplicative Ktheory MK_{n}^{ℐ} (𝒜), in any even degree n = 2k ∈ ℤ, as the set of equivalence classes of pairs (ê, θ) such that ê ∈ M_{2}(𝒦 ˆ⊗ℐ ℛ̂)^{+} is an idempotent and θ ∈ X_{n−1}(ℛ, 𝒥) is a chain of odd degree related by the transgression formula

ch_{0}^{2q}(ê) = \overline{b}θ ∈ X_{n−1}(ℛ, 𝒥) .

Two pairs (ê_{0}, θ_{0}) and (ê_{1}, θ_{1}) are equivalent if and only if there exists an idempotent ê ∈ M_{2}(𝒦 ˆ⊗ℐ ℛ̂[0, 1])^{+} whose evaluation yields ê_{0} and ê_{1} at the endpoints, and a chain λ ∈ X_{n−1}(ℛ, 𝒥) of even degree such that

θ_{1} − θ_{0} = cs_{1}^{2q}(ê) + ♮dλ ∈ X_{n−1}(ℛ, 𝒥) .

In the same way, we define the multiplicative K-theory MK_{n}^{ℐ} (𝒜), in any odd degree n = 2k + 1 ∈ ℤ, as the set of equivalence classes of pairs (ĝ, θ) such that ĝ ∈ (𝒦 ˆ⊗ℐ ℛ̂)^{+} is an invertible and θ ∈ X_{n−1}(ℛ, 𝒥) is a chain of even degree related by the transgression formula

ch_{1}^{2q}(ĝ) = ♮dθ ∈ X_{n−1}(ℛ, 𝒥) .

Two pairs (ĝ_{0}, θ_{0}) and (ĝ_{1}, θ_{1}) are equivalent if and only if there exists an invertible ĝ ∈ (𝒦 ˆ⊗ℐ ℛ̂[0, 1])^{+} whose evaluation yields ĝ_{0} and ĝ_{1} at the endpoints, and a chain λ ∈ X_{n−1}(ℛ, 𝒥) of odd degree such that

θ_{1} − θ_{0} = cs_{0}^{2q}(ĝ) + \overline{b}λ ∈ X_{n−1}(ℛ, 𝒥) .

We will prove in Proposition 5.4 that for any n ∈ ℤ the set MK_{n}^{ℐ} (𝒜) depends neither on the degree 2q + 1 ≥ p chosen to represent the Chern characters nor on the quasi-free extension ℛ.

Recall from section 2 that the homology H_{n−1+2ℤ}(X_{n}(ℛ, 𝒥)) is the noncommutative de Rham homology HD_{n−1}(𝒜). Hence the transgression relation ch_{0}^{2q}(ê) = \overline{b}θ ∈ X_{n−1}(ℛ, 𝒥) exactly means that the class of ch_{0}^{2q}(ê) vanishes in HD_{n−2}(𝒜). Similarly in the odd case, ch_{1}^{2q}(ĝ) = ♮dθ ∈ X_{n−1}(ℛ, 𝒥) is equivalent to ch_{1}^{2q}(ĝ) ≡ 0 in HD_{n−2}(𝒜). Since the complexes X_{n−1}(ℛ, 𝒥) vanish for n ≤ 0, we immediately deduce that MK_{n}^{ℐ} (𝒜) coincides with the topological K-theory K_{n}^{top}(ℐ ˆ⊗𝒜) whenever n ≤ 0. As in the case of topological K-theory, define an addition on MK_{n}^{ℐ} (𝒜) by direct sum of idempotents and invertibles as follows (c is the permutation matrix (52)):

even case:  (ê, θ) + (ê′, θ′) = (c(ê ⊕ ê′)c, θ + θ′) ,
odd case:  (ĝ, θ) + (ĝ′, θ′) = (ĝ ⊕ ĝ′, θ + θ′) .

This turns MK_{n}^{ℐ} (𝒜) into a semigroup, the unit being represented by (p_{0}, 0) in the even case and (1, 0) in the odd case.

Lemma 5.2 MK_{n}^{ℐ} (𝒜) is an abelian group for any n ∈ ℤ.

Proof: We first need to recall the proof that K_{0}^{top}(ℐ ˆ⊗𝒜) is a group. Let e ∈ M_{2}(𝒦 ˆ⊗ℐ 𝒜)^{+} be an idempotent, with e − p_{0} ∈ M_{2}(𝒦 ˆ⊗ℐ 𝒜). The idempotent 1 − e is orthogonal to e, as e(1 − e) = (1 − e)e = 0. If X ∈ M_{2}(ℂ) is the permutation matrix ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ), we claim that the idempotent

X(1 − e)X ∈ M_{2}(𝒦 ˆ⊗ℐ 𝒜)^{+} , with X(1 − e)X − p_{0} ∈ M_{2}(𝒦 ˆ⊗ℐ 𝒜) ,

represents the inverse class of e. Indeed, we shall construct a homotopy between the direct sum c(e ⊕ X(1 − e)X)c ∈ M_{4}(𝒦 ˆ⊗ℐ 𝒜)^{+} and the unit

p̃_{0} = (\begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix}) ∈ M_{4}(𝒦 ˆ⊗ℐ 𝒜)^{+} .

Choose a smooth real-valued function ξ ∈ C^{∞}[0, 1] ranging from ξ(0) = 0 to ξ(1) = π/2, and consider the paths of invertible matrices

R_{23} = (\begin{matrix}1 & 0 & 0 & 0 \\ 0 & cos ξ & sin ξ & 0 \\ 0 & sin ξ & − cos ξ & 0 \\ 0 & 0 & 0 & 1 \end{matrix}) , R_{14} = (\begin{matrix}cos ξ & 0 & 0 & sin ξ \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ sin ξ & 0 & 0 & − cos ξ \end{matrix}) .

A direct computation shows that the idempotents R_{23}(t)^{−1}( \begin{matrix}e & 0 \\ 0 & 0 \end{matrix} )R_{23}(t) and R_{14}(t)^{−1}( \begin{matrix}1−e & 0 \\ 0 & 0 \end{matrix} )R_{14}(t) are orthogonal for any t ∈ [0, 1]. Hence the sum

f = R_{23}^{−1} (\begin{matrix}e & 0 \\ 0 & 0 \end{matrix}) R_{23} + R_{14}^{−1} (\begin{matrix}1 − e & 0 \\ 0 & 0 \end{matrix}) R_{14} ∈ M_{4}(𝒦 ˆ⊗ℐ 𝒜 [0, 1])^{+}

is an idempotent path such that f − p̃_{0} ∈ M_{4}(𝒦 ˆ⊗ℐ 𝒜 [0, 1]), and interpolates f_{0} = p̃_{0} and f_{1} = c(e ⊕ X(1 − e)X)c. This shows that K_{0}^{top}(ℐ ˆ⊗𝒜) is a group. It is abelian because a direct sum e ⊕ e′ can be connected via a smooth path (by conjugation with respect to rotation matrices) to e′ ⊕ e. Now fix an integer 2q + 1 ≥ p and let (ê, θ) represent an element of MK_{n}^{ℐ} (𝒜) of even degree n = 2k. Hence ê ∈ M_{2}(𝒦 ˆ⊗ℐ ℛ̂)^{+} is an idempotent such that ch_{0}^{2q}(ê) = \overline{b}θ in the quotient complex X_{n−1}(ℛ, 𝒥). Consider the smooth idempotent path f̂ ∈ M_{4}(𝒦 ˆ⊗ℐ ℛ̂[0, 1])^{+} constructed as above replacing 𝒜 by its extension ℛ̂ and e by ê. It provides an interpolation between f̂_{0} = p̃_{0} and f̂_{1} = c(ê ⊕ X(1 − ê)X)c. We guess that the inverse of (ê, θ) is represented by the pair (X(1 − ê)X, cs_{1}^{2q}(f̂) − θ). Indeed, one has

\overline{b}cs_{1}^{2q}(f̂) = ch_{0}^{2q}(f̂_{1}) − ch_{0}^{2q}(f̂_{0}) = ch_{0}^{2q}(ê) + ch_{0}^{2q}(X(1 − ê)X) ,

so that ch_{0}^{2q}(X(1 − ê)X) = \overline{b}(cs_{1}^{2q}(f̂) − θ) in the complex X_{n−1}(ℛ, 𝒥) and the pair (X(1 − ê)X, cs_{1}^{2q}(f̂) − θ) represents a class in MK_{n}^{ℐ} (𝒜). Moreover, the sum

(ê, θ) + (X(1 − ê)X, cs_{1}^{2q}(f̂) − θ) = (c(ê ⊕ X(1 − ê)X)c, cs_{1}^{2q}(f̂))

is equivalent to the unit (p̃_{0}, 0) because f̂provides the interpolating idempotent. Hence MK_{n}^{ℐ} (𝒜), n = 2k is a group. Abelianity is shown as for topological K-theory, by means of another interpolation between the idempotents c(ê⊕ê′)c and c(ê′ ⊕ ê)c with the property that its Chern-Simons form cs_{1} vanishes. One proceeds similarly in the odd case. Let (ĝ, θ) represent an element of MK_{n}^{ℐ} (𝒜) of odd degree n = 2k + 1. Hence ch_{1}^{2q}(ĝ) = ♮dθ in X_{n−1}(ℛ, 𝒥). Define an invertible path û ∈ M_{2}(𝒦 ˆ⊗ℐ ℛ̂[0, 1])^{+} by means of the rotation matrix R = ( \begin{matrix}cos ξ & sin ξ \\ − sin ξ & cos ξ \end{matrix} ):

û = (\begin{matrix}ĝ & 0 \\ 0 & 1 \end{matrix}) R^{−1} (\begin{matrix}ĝ^{−1} & 0 \\ 0 & 1 \end{matrix}) R .

Then û − 1 ∈ M_{2}(𝒦 ˆ⊗ℐ ℛ̂[0, 1]), and û provides a smooth homotopy between the invertibles û_{0} = 1 and û_{1} = ( \begin{matrix}ĝ & 0 \\ 0 & ĝ^{−1} \end{matrix} ) (the same argument shows that K_{1}^{top}(ℐ ˆ⊗𝒜) is an abelian group). We guess that the inverse class of (ĝ, θ) is represented by the pair (ĝ^{−1}, cs_{0}^{2q}(û) − θ). Indeed, one has

♮dcs_{0}^{2q}(û) = ch_{1}^{2q}(û_{1}) − ch_{1}^{2q}(û_{0}) = ch_{1}^{2q}(ĝ) + ch_{1}^{2q}(ĝ^{−1})

so that ch_{1}^{2q}(ĝ^{−1}) = ♮d(cs_{0}^{2q}(û) − θ) in X_{n−1}(ℛ, 𝒥) and (ĝ^{−1}, cs_{0}^{2q}(û) − θ) represents a class in MK_{n}^{ℐ} (𝒜). Moreover, the sum

(ĝ, θ) + (ĝ^{−1}, cs_{0}^{2q}(û) − θ) = (ĝ ⊕ ĝ^{−1}, cs_{0}^{2q}(û))

is equivalent to the unit (1, 0) through the interpolating invertible û. Hence MK_{n}^{ℐ} (𝒜), n = 2k + 1 is a group as claimed. Abelianity is shown once again by means of rotation matrices.

Remark 5.3 We know that two different liftings of a given idempotent e ∈ M_{2}(𝒦 ˆ⊗ℐ 𝒜)^{+} are always homotopic in M_{2}(𝒦 ˆ⊗ℐ ℛ̂)^{+}. Hence choosing the universal free extension ℛ = T 𝒜 allows to represent any multiplicative Ktheory class of even degree by a pair (ê, θ) where ê ∈ M_{2}(𝒦 ˆ⊗ℐ T̂𝒜)^{+} is the canonical lift of some idempotent e. Moreover, the transgression formula established in the proof of Lemma 4.6 shows that two such pairs (ê_{0}, θ_{0}) and (ê_{1}, θ_{1}) are equivalent if and only if e_{0} and e_{1} can be joined by an idempotent e ∈ M_{2}(𝒦 ˆ⊗ℐ 𝒜 [0, 1])^{+} such that θ_{1} − θ_{0} ≡ cs_{1}^{2q}(ê) mod ♮d, where ê ∈ M_{2}(𝒦 ˆ⊗ℐ T̂𝒜 [0, 1])^{+} is the canonical lift of e. The same is true with invertibles: any multiplicative K-theory class of odd degree may be represented by a pair (ĝ, θ) where ĝ ∈ (𝒦 ˆ⊗ℐ T̂𝒜)^{+} is the canonical lift of some invertible g ∈ (𝒦 ˆ⊗ℐ 𝒜)^{+}. Two such pairs (ĝ_{0}, θ_{0}) and (ĝ_{1}, θ_{1}) are equivalent if and only if g_{0} and g_{1} can be joined by an invertible g ∈ (𝒦 ˆ⊗ℐ 𝒜 [0, 1])^{+} such that θ_{1} − θ_{0} ≡ cs_{0}^{2q}(ĝ) mod \overline{b}, where ĝ ∈ (𝒦 ˆ⊗ℐ T̂𝒜 [0, 1])^{+} is the canonical lift of g.

The particular case ℐ = ℂ is essentially equivalent Karoubi’s definition of multiplicative K-theory [16, 17]. The groups MK_{n}^{ℐ} (𝒜) are designed to fit in a long exact sequence

K_{n+1}^{top}(ℐ ˆ⊗𝒜) → HC_{n−1}(𝒜) →^{δ} MK_{n}^{ℐ} (𝒜) → K_{n}^{top}(ℐ ˆ⊗𝒜) → HC_{n−2}(𝒜)

(72)

The map K_{n}^{top}(ℐ ˆ⊗𝒜) → HC_{n−2}(𝒜) corresponds to the composition of the Chern character K_{n}^{top}(ℐ ˆ⊗𝒜) → HP_{n}(𝒜) with the natural map HP_{n}(𝒜) → HC_{n−2}(𝒜) induced by the projection X̂(ℛ, 𝒥) → X_{n−2}(ℛ, 𝒥). The map MK_{n}^{ℐ} (𝒜) → K_{n}^{top}(𝒜) is the forgetful map, which sends a pair (ê, θ) or (ĝ, θ) respectively on its image e or g under the projection homomorphism ℛ̂ → 𝒜. The connecting map δ : HC_{n−1}(𝒜) → MK_{n}^{ℐ} (𝒜) sends a cycle θ ∈ X_{n−1}(ℛ, 𝒥) to

δ(θ) = \{ \begin{matrix}(p_{0}, \sqrt{2πi} θ) & n even, \\ (1, \sqrt{2πi} θ) & n odd. \end{matrix}  (73)

There is also an additive Chern character map ch_{n} : MK_{n}^{ℐ} (𝒜) → HN_{n}(𝒜) defined in all degrees n ∈ ℤ, with values in negative cyclic homology. Recall that the latter is the homology in degree n mod 2 of the subcomplex F^{n−1}X̂(ℛ, 𝒥) = Ker(X̂(ℛ, 𝒥) → X_{n−1}(ℛ, 𝒥)):

HN_{n}(𝒜) = H_{n+2ℤ}(F^{n−1}X̂(ℛ, 𝒥)) .

Hence in particular HN_{n}(𝒜) = HP_{n}(𝒜) whenever n ≤ 0, and HP_{∗}, HC_{∗} HN_{∗} are related by the SBI long exact sequence (section 2). To define the Chern character ch_{n} : MK_{n}^{ℐ} (𝒜) → HN_{n}(𝒜) in even degree n = 2k, we first have to choose an integer 2q + 1 ≥ p. Then, a multiplicative K-theory class of degree n is represented by a pair (ê, θ), such that the transgression formula ch_{0}^{2q}(ê) = \overline{b}θ holds in X_{n−1}(ℛ, 𝒥). Choose an arbitrary lifting θ̃ ∈ X̂(ℛ, 𝒥) of θ, and define the negative Chern character as

ch_{n}(ê, θ) = ch_{0}^{2q}(ê) − \overline{b}θ̃ ∈ F^{n−1}X̂(ℛ, 𝒥) .  (74)

It is clearly closed, and its negative cyclic homology class does not depend on the choice of lifting θ̃, since the difference of two such liftings lies in the subcomplex F^{n−1}X̂(ℛ, 𝒥). We will show in the proposition below that it does not depend on the representative (ê, θ) of the K-theory class, nor on the integer 2q + 1 ≥ p. In odd degree n = 2k + 1, the Chern character ch_{n} : MK_{n}^{ℐ} (𝒜) → HN_{n}(𝒜) is defined exactly in the same way: take a representative (ĝ, θ) of a multiplicative K-theory class, with ch_{1}^{2q}(ĝ) = ♮dθ in X_{n−1}(ℛ, 𝒥). Then if θ̃ ∈ X̂(ℛ, 𝒥) denotes an arbitrary lifting of θ, the cycle

ch_{n}(ĝ, θ) = ch_{1}^{2q}(ĝ) − ♮dθ̃ ∈ F^{n−1}X̂(ℛ, 𝒥)  (75)

defines a negative cyclic homology class. The following proposition shows the compatibility between the K-theory exact sequence (72) and the SBI exact sequence (8), through the various Chern character maps.

Proposition 5.4 Let 𝒜 and ℐ be Fréchet m-algebras, such that ℐ is psummable. Then one has a commutative diagram with long exact rows

[figure: K_{n+1}^{top}(ℐ ˆ⊗𝒜); HC_{n−1}(𝒜); δ; MK_{n}^{ℐ} (𝒜); K_{n}^{top}(ℐ ˆ⊗𝒜); HP_{n+1}(𝒜); S; HC_{n−1}(𝒜); B̃; HN_{n}(𝒜); I; HP_{n}(𝒜)] (76)

where B̃ is the connecting map of the SBI sequence rescaled by a factor −\sqrt{2πi}.

Proof: We show the exactness of the sequence (72) in the case of even degree n = 2k (the odd case is completely similar):

K_{1}^{top}(ℐ ˆ⊗𝒜) ⟶^{ch_{1}} HC_{n−1}(𝒜) →^{δ} MK_{n}^{ℐ} (𝒜) →^{ι} K_{0}^{top}(ℐ ˆ⊗𝒜) ⟶^{ch_{0}} HC_{n−2}(𝒜) .

Fix once and for all an integer 2q + 1 ≥ p to represent the Chern characters. We first have to check that the maps δ and ι are well-defined. Let θ_{0} and θ_{1} = θ_{0} + ♮dλ be two homologous odd cycles in X_{n−1}(ℛ, 𝒥) representing the same cyclic homology class [θ] ∈ HC_{n−1}(𝒜). Their images by δ are respectively (p_{0}, \sqrt{2πi} θ_{0}) and (p_{0}, \sqrt{2πi} θ_{1}), which obviously represent the same class in MK_{n}^{ℐ} (𝒜) by virtue of the equivalence relation θ_{1} − θ_{0} = ♮dλ (take the constant idempotent p_{0} ∈ M_{2}(𝒦 ˆ⊗ℐ ℛ̂[0, 1])^{+} as interpolation, with cs_{1}^{2q}(p_{0}) = 0). Hence δ is well-defined. Now take two equivalent pairs (ê_{0}, θ_{0}) and (ê_{1}, θ_{1}) representing the same element in MK_{n}^{ℐ} (𝒜). In particular, the idempotents ê_{0} and ê_{1} are smoothly homotopic and their projections e_{0}, e_{1} ∈ M_{2}(𝒦 ˆ⊗ℐ 𝒜)^{+} define the same class in K_{0}^{top}(ℐ ˆ⊗𝒜). Since ι(ê_{0}, θ_{0}) = e_{0} and ι(ê_{1}, θ_{1}) = e_{1}, the map ι is well-defined.

Exactness at HC_{n−1}(𝒜): Let [g] ∈ K_{1}^{top}(ℐ ˆ⊗𝒜) be a class represented by an invertible element g ∈ (𝒦 ˆ⊗ℐ 𝒜)^{+}, and consider its idempotent image α(g) ∈ M_{2}(𝒦 ˆ⊗ℐ S𝒜)^{+} under the Bott isomorphism α : K_{1}^{top}(ℐ ˆ⊗𝒜) → K_{0}^{top}(ℐ ˆ⊗S𝒜). Choose any invertible lift ĝ ∈ (𝒦 ˆ⊗ℐ ℛ̂)^{+} and any idempotent lift α̂(g) ∈ M_{2}(𝒦 ˆ⊗ℐ Sℛ̂)^{+}. By Lemma 4.6, we have the equality of periodic cyclic homology classes

ch_{1}^{2q}(ĝ) ≡ \frac{1}{\sqrt{2πi}}cs_{1}^{2q}(α̂(g)) ∈ HP_{1}(𝒜) ,

hence this equality also holds in HC_{n−1}(𝒜). It follows that δ(ch_{1}(g)) is represented by

δ(\frac{1}{\sqrt{2πi}}cs_{1}^{2q}(α̂(g))) = (p_{0}, cs_{1}^{2q}(α̂(g))) .

But the idempotent path α̂(g) evaluated at the endpoints is p_{0}, so that the pairs (p_{0}, cs_{1}^{2q}(α̂(g))) and (p_{0}, 0) are equivalent in MK_{n}^{ℐ} (𝒜). Hence δ∘ch_{1} = 0. Now let a class [θ] ∈ HC_{n−1}(𝒜) be in the kernel of δ. It means that the pair δ(θ) = (p_{0}, \sqrt{2πi} θ) is equivalent to (p_{0}, 0). Hence there exists an idempotent ê ∈ M_{2}(𝒦 ˆ⊗ℐ Sℛ̂)^{+} and a chain λ such that \sqrt{2πi} θ = cs_{1}^{2q}(ê) + ♮dλ in X_{n−1}(ℛ, 𝒥). By Bott periodicity, there exists an element [g] ∈ K_{1}^{top}(ℐ ˆ⊗𝒜) such that cs_{1}^{2q}(ê) ≡ \sqrt{2πi} ch_{1}^{2q}(ĝ) in HC_{n−1}(𝒜), where ĝ ∈ (𝒦 ˆ⊗ℐ ℛ̂)^{+} is any invertible lift of g. Whence the equality of cylic homology classes [θ] ≡ ch_{1}^{2q}(ĝ). It follows that Ker δ = Im ch_{1}.

Exactness at MK_{n}^{ℐ} (𝒜): Let [θ] ∈ HC_{n−1}(𝒜) be any cyclic homology class. Then δ(θ) = (p_{0}, \sqrt{2πi} θ), and ι(δ(θ)) = p_{0} is the zero-class in topological Ktheory. Therefore ι ∘ δ = 0. Now let (ê, θ) ∈ MK_{n}^{ℐ} (𝒜) be in the kernel of ι: it means that ê is smoothly homotopic to p_{0}. Hence there exists an idempotent f̂ ∈ M_{2}(𝒦 ˆ⊗ℐ ℛ̂[0, 1])^{+}, with evaluations f̂_{0} = ê and f̂_{1} = p_{0}, and the pair (ê, θ) is equivalent to (p_{0}, θ + cs_{1}^{2q}(f̂)). Remark that the odd chain θ + cs_{1}^{2q}(f̂) ∈ X_{n−1}(ℛ, 𝒥) is closed (indeed, \overline{b}θ = ch_{0}^{2q}(ê) and \overline{b}cs_{1}^{2q}(f̂) = −ch_{0}^{2q}(ê)), and we can write

(p_{0}, θ + cs_{1}^{2q}(f̂)) = δ(\frac{1}{\sqrt{2πi}}(θ + cs_{1}^{2q}(f̂))) .

It follows that Ker ι = Im δ.

Exactness at K_{0}^{top}(ℐ ˆ⊗𝒜): Let (ê, θ) ∈ MK_{n}^{ℐ} (𝒜) represent any multiplicative K-theory class. Then ch_{0}^{2q}(ê) ≡ 0 in non-commutative de Rham homology HD_{n−2}(𝒜), and therefore also in HC_{n−2}(𝒜). Thus, the Chern character of ι(ê, θ) = e vanishes and ch_{0} ∘ ι = 0.

Now let [e] ∈ K_{0}^{top}(ℐ ˆ⊗𝒜) be in the kernel of ch_{0}. We know from section 2 that the natural map HD_{n−2}(𝒜) → HC_{n−2}(𝒜) is injective, so that ch_{0}^{2q}(ê) ≡ 0 in HD_{n−2}(𝒜) for any idempotent lift ê ∈ M_{2}(𝒦 ˆ⊗ℐ ℛ̂)^{+}. Hence there exists an odd chain θ ∈ X_{n−1}(ℛ, 𝒥) such that ch_{0}^{2q}(ê) = \overline{b}θ, and e = ι(ê, θ). This shows that Ker ch_{0} = Im ι.

Let us now show the independence of multiplicative K-theory upon the choice of degree 2q+1 ≥ p. To this end, write (ê, θ)^{q} ∈ MK_{n}^{ℐ} (𝒜)^{q} for a representative of a class obtained using the higher Chern character ch_{0}^{2q}(ê) = \overline{b}θ of degree 2q. We shall construct a map MK_{n}^{ℐ} (𝒜)^{q} → MK_{n}^{ℐ} (𝒜)^{q+1} which turns out to be an isomorphism. Let ρ : ℐ 𝒜 → ℰ^{s} ▷ ℐ^{s}𝒜 be the canonical p-summable quasihomomorphism of even degree considered in section 4, for the construction of the higher Chern characters. Recall that ℰ = ℐ^{+} ˆ⊗𝒜 with extension ℳ = ℐ^{+} ˆ⊗ℛ. From Proposition 3.10, we know that the chain maps χ^{2q} : Ω̂(ℳ̂_{+}^{s}) → X(ℛ̂) associated to ρ are related in successive degrees by the transgression formula involving the eta-cochain χ^{2q} − χ^{2q+2} = [∂, η^{2q+1}]. More precisely:

χ_{0}^{2q} − χ_{0}^{2q+2} = \overline{b}η_{1}^{2q+1} + η_{0}^{2q+1}(b + B) ,
χ_{1}^{2q} − χ_{1}^{2q+2} = ♮dη_{0}^{2q+1} + η_{1}^{2q+1}(b + B) .

The evaluation of the first equation on the (b + B)-cycle ch_{∗}(ê) ∈ Ω̂^{+}(ℐ ℛ̂) yields (see section 4; we also omit reference to the inclusion homomorphism

ι_{∗} : ℐ ℛ̂ ↪ ( \begin{matrix}ℐ ℛ̂ & 0 \\ 0 & 0 \end{matrix} ) ⊂ ℳ̂_{+}^{s})

ch_{0}^{2q}(ê) − ch_{0}^{2q+2}(ê) = \overline{b}(η_{1}^{2q+1}ch_{2q+2}(ê)) ,

with ch_{0}^{2q}(ê) = χ_{0}^{2q}ch_{2q}(ê) by Lemma 4.5. Therefore, we guess that the map MK_{n}^{ℐ} (𝒜)^{q} → MK_{n}^{ℐ} (𝒜)^{q+1} should send a pair (ê, θ)^{q} to the pair (ê, θ′)^{q+1} with θ′ = θ−η_{1}^{2q+1}ch_{2q+2}(ê). Indeed, one has the correct transgression relation

ch_{0}^{2q+2}(ê) = ch_{0}^{2q}(ê) − \overline{b}(η_{1}^{2q+1}ch_{2q+2}(ê)) = \overline{b}θ′

in the complex X_{n−1}(ℛ, 𝒥). Moreover, this map is well-defined at the level of equivalence classes: let (ê_{0}, θ_{0})^{q} and (ê_{1}, θ_{1})^{q} be two equivalent pairs. Then there exists an interpolating idempotent ê ∈ M_{2}(𝒦 ˆ⊗ℐ ℛ̂[0, 1])^{+} and a chain λ such that θ_{1} − θ_{0} = cs_{1}^{2q}(ê) + ♮dλ. Hence the respective images (ê_{0}, θ′_{0})^{q+1} and (ê_{1}, θ′_{1})^{q+1} verify

θ′_{1} − θ′_{0} = θ_{1} − θ_{0} − η_{1}^{2q+1} (ch_{2q+2}(ê_{1}) − ch_{2q+2}(ê_{0})) .

But we know the transgression relation (69)

ch_{2q+2}(ê_{1}) − ch_{2q+2}(ê_{0}) = Bcs_{2q+1}(ê) + bcs_{2q+3}(ê) ,

so that, using the identities χ_{1}^{2q}cs_{2q+1}(ê) = (♮dη_{0}^{2q+1} + η_{1}^{2q+1}B)cs_{2q+1}(ê) and −χ_{1}^{2q+2}cs_{2q+3}(ê) = η_{1}^{2q+1}bcs_{2q+3}(ê) one gets (recall cs_{1}^{2q}(ê) = χ_{1}^{2q}cs_{2q+1}(ê))

θ′_{1} − θ′_{0} = cs_{1}^{2q}(ê) + ♮dλ
−χ_{1}^{2q}cs_{2q+1}(ê) + χ_{1}^{2q+2}cs_{2q+3}(ê) + ♮d(η_{0}^{2q+1}cs_{2q+1}(ê))
= cs_{1}^{2q+2}(ê) + ♮d(λ + η_{0}^{2q+1}cs_{2q+1}(ê)) .

Hence (ê_{0}, θ′_{0})^{q+1} and (ê_{1}, θ′_{1})^{q+1} are equivalent in MK_{n}^{ℐ} (𝒜)^{q+1}. It remains to show that the map MK_{n}^{ℐ} (𝒜)^{q} → MK_{n}^{ℐ} (𝒜)^{q+1} is an isomorphism. Consider the following diagram:

[figure: HC_{n−1}(𝒜); δ; MK_{n}^{ℐ} (𝒜)^{q}; K_{0}^{top}(ℐ ˆ⊗𝒜); HC_{n−1}(𝒜); δ; MK_{n}^{ℐ} (𝒜)^{q+1}; K_{0}^{top}(ℐ ˆ⊗𝒜)]

For any cyclic homology class [θ] ∈ HC_{n−1}(𝒜) represented by a closed chain θ, one has δ(θ) = (p_{0}, \sqrt{2πi} θ)^{q} in MK_{n}^{ℐ} (𝒜)^{q}. But observe that η_{1}^{2q+1}ch_{2q+2}(p_{0}) = 0, so that (p_{0}, \sqrt{2πi} θ)^{q} is mapped to (p_{0}, \sqrt{2πi} θ)^{q+1} in MK_{n}^{ℐ} (𝒜)^{q+1}. Hence the left square is commutative. Moreover the right square is obviously commutative. The isomorphism MK_{n}^{ℐ} (𝒜)^{q} ≅ MK_{n}^{ℐ} (𝒜)^{q+1} then follows from the five-lemma.

The negative Chern character ch_{n} : MK_{n}^{ℐ} (𝒜)^{q} → HN_{n}(𝒜) is also independent of q, and compatible with the SBI exact sequence. Indeed if (ê, θ)^{q} is a representative of a class in MK_{n}^{ℐ} (𝒜)^{q}, one has by definition

ch_{n}(ê, θ)^{q} = ch_{0}^{2q}(ê) − \overline{b}θ̃ ,

where θ̃ ∈ X̂(ℛ, 𝒥) is an arbitrary lifting of θ. First remark that ch_{n} is well-defined at the level of equivalence classes: if (ê_{0}, θ_{0})^{q} and (ê_{1}, θ_{1})^{q} are equivalent, there exists an interpolation ê ∈ M_{2}(𝒦 ˆ⊗ℐ ℛ̂[0, 1])^{+} and a chain λ such that θ_{1} −θ_{0} = cs_{1}^{2q}(ê)+♮dλ in X_{n−1}(ℛ, 𝒥). Let θ̃_{0}, θ̃_{1} and λ̃ be arbitrary liftings; then there exists a chain μ ∈ F^{n−1}X̂(ℛ, 𝒥) such that θ̃_{1} − θ̃_{0} = cs_{1}^{2q}(ê) + ♮dλ̃ + μ in X̂(ℛ, 𝒥). Hence the difference

ch_{n}(ê_{1}, θ_{1})^{q} − ch_{n}(ê_{0}, θ_{0})^{q} = ch_{0}^{2q}(ê_{1}) − ch_{0}^{2q}(ê_{0}) − \overline{b}(θ̃_{1} − θ̃_{0})
= \overline{b}cs_{1}^{2q}(ê) − \overline{b}(cs_{1}^{2q}(ê) + ♮dλ̃ + μ) = −\overline{b}μ

is a coboundary of the subcomplex F^{n−1}X̂(ℛ, 𝒥), and the Chern character ch_{n} : MK_{n}^{ℐ} (𝒜)^{q} → HN_{n}(𝒜) is well-defined. Now if (ê, θ)^{q} ∈ MK_{n}^{ℐ} (𝒜)^{q} is any class, its image in MK_{n}^{ℐ} (𝒜)^{q+1} is represented by (ê, θ′)^{q+1} with θ′ = θ − η_{1}^{2q+1}ch_{2q+2}(ê). One has

ch_{n}(ê, θ′)^{q+1} = ch_{0}^{2q+2}(ê) − \overline{b}(θ̃ − η_{1}^{2q+1}ch_{2q+2}(ê))
= ch_{0}^{2q+2}(ê) − \overline{b}(θ̃) + χ_{0}^{2q}ch_{2q}(ê) − χ_{0}^{2q+2}ch_{2q+2}(ê)
= ch_{0}^{2q}(ê) − \overline{b}(θ̃) = ch_{n}(ê, θ)^{q} ,

and the negative Chern character does not depend on the degree q. Finally, for any cyclic homology class [θ] ∈ HC_{n−1}(𝒜), one has

ch_{n}(δ(θ)) = ch_{n}(p_{0}, \sqrt{2πi} θ) = −\sqrt{2πi} \overline{b}(θ̃) ,

which shows the commutativity of the middle square (76). The compatibility between the negative Chern character and the periodic Chern character on topological K-theory is obvious, whence the commutativity of the right square (76).

Concerning the independence of MK_{n}^{ℐ} (𝒜) with respect to the choice of quasi-free extension ℛ, it suffices to consider the universal extension 0 → J𝒜 → T 𝒜 → 𝒜 → 0 together with the classifying homomorphisms T 𝒜 → ℛ and J𝒜 → 𝒥. The various Chern characters and Chern-Simons forms constructed in X̂(ℛ, 𝒥) are obtained from the universal ones in X̂(T 𝒜 , J𝒜) by applying the chain map X̂(T 𝒜 , J𝒜) → X̂(ℛ, 𝒥), which we know is a homotopy equivalence compatible with the adic filtrations. Once again the conclusion follows from the five-lemma. The case of odd degree n = 2k +1 is established along the same lines, replacing idempotents by invertibles.

Before ending this section we need to establish the invariance of topological and multiplicative K-theory with respect to adjoint actions of multipliers on the p-summable Fréchet m-algebra ℐ. We say that U is a multiplier if it defines continuous linear maps (left and right multiplications) x ↦ Ux and x ↦ xU on ℐ, which commute and fulfill

i) U(xy) = (Ux)y , (xU)y = x(Uy) , (xy)U = x(yU)  ∀x, y ∈ ℐ ,
ii) Tr([U, ℐ^{p}]) = 0 .

U is invertible if there exists a multiplier U^{−1} such that the compositions UU^{−1} and U^{−1}U induce the identity on ℐ, while left and right multiplications by U and U^{−1} commute. In this case the adjoint action of U defined by Ad U(x) = U^{−1}xU is a continuous automorphism of ℐ preserving the trace on ℐ^{p}. If 𝒜 is any Fréchet m-algebra, the adjoint action of U extends to the tensor product 𝒦 ˆ⊗ℐ 𝒜 by acting trivially on the factors 𝒦 and 𝒜, thus defines an automorphism of K_{n}^{top}(ℐ ˆ⊗𝒜). Similarly if 0 → 𝒥 → ℛ → 𝒜 → 0 is a quasi-free extension, and (ê, θ) (resp. (ĝ, θ)) represents a multiplicative K-theory class of even (resp. odd) degree, the adjoint action of U extends to an automorphism of the pro-algebra 𝒦 ˆ⊗ℐ ℛ̂ and define maps

Ad U : (ê, θ) ↦ (U^{−1}êU, θ) , (ĝ, θ) ↦ (U^{−1}ĝU, θ) .  (77)

The images represent multiplicative K-theory classes because the invariance of the trace implies ch_{0}^{2q}(U^{−1}êU) = ch_{0}^{2q}(ê) = \overline{b}θ and ch_{0}^{2q}(U^{−1}ĝU) = ch_{0}^{2q}(ĝ) = ♮dθ. The adjoint action is actually well-defined at the level of K-theory:

Lemma 5.5 Let U be an invertible multiplier of ℐ. Then the adjoint action Ad U induces the identity on K_{n}^{top}(ℐ ˆ⊗𝒜) and MK_{n}^{ℐ} (𝒜).

Proof: First we show that an idempotent e ∈ M_{2}(𝒦 ˆ⊗ℐ 𝒜)^{+}, with e − p_{0} ∈ M_{2}(𝒦 ˆ⊗ℐ 𝒜), is smoothly homotopic to its adjoint U^{−1}eU. Introduce the idempotent f_{0} = ( \begin{matrix}e & 0 \\ 0 & p_{0} \end{matrix} ) ∈ M_{4}(𝒦 ˆ⊗ℐ 𝒜)^{+}, and choose a smooth real-valued function ξ ∈ C^{∞}[0, 1] such that ξ(0) = 0 and ξ(1) = π/2. We define a path W of invertible multipliers of M_{4}(𝒦 ˆ⊗ℐ 𝒜) by means of the formula

W = R^{−1} (\begin{matrix}1 & 0 \\ 0 & U \end{matrix}) R , R = (\begin{matrix}cos ξ & sin ξ \\ − sin ξ & cos ξ \end{matrix}) ,

where each entry should be viewed as a 2×2 block matrix. Hence, W commutes with the matrix p̃_{0} = ( \begin{matrix}p_{0} & 0 \\ 0 & p_{0} \end{matrix} ). The smooth path of idempotents f = W^{−1}f_{0}W thus provides an interpolation between f_{0} and f_{1} = ( \begin{matrix}U^{−1}eU & 0 \\ 0 & p_{0} \end{matrix} ). Put in another way, cfc interpolates the K-theoretic sums e + p_{0} and U^{−1}eU + p_{0}. This shows that e and U^{−1}eU define the same topological K-theory class. Now suppose that (ê, θ) ∈ MK_{n}^{ℐ} (𝒜) represents a multiplicative K-theory class, with ê ∈ M_{2}(𝒦 ˆ⊗ℐ ℛ̂)^{+}, θ ∈ X_{n−1}(ℛ, 𝒥) and ch_{0}^{2q}(ê) = \overline{b}θ. We define as before f̂_{0} = ( \begin{matrix}ê & 0 \\ 0 & p_{0} \end{matrix} ), and f̂ = W^{−1}f̂_{0}W provides an interpolation between f̂_{0} and f̂_{1} = ( \begin{matrix}U^{−1}êU & 0 \\ 0 & p_{0} \end{matrix} ). If s : C^{∞}[0, 1] → Ω^{1}[0, 1] denotes the differential over [0, 1] and d : ℛ̂ → Ω^{1}ℛ̂ the noncommutative differential, the Chern-Simons form (71) associated to cf̂c reads

cs_{1}^{2q}(cf̂c) = ∫_{0}^{1} Tr♮(−2f̂+ 1) ∑_{i=0}^{q}(f̂− p̃_{0})^{2i}sf̂(f̂− p̃_{0})^{2(q−i)}df̂ .

One has df̂ = W^{−1}df̂_{0}W and sf̂ = W^{−1}(−sW W^{−1}f̂_{0} + f̂_{0}sW W^{−1})W, hence

Tr♮(−2f̂+ 1) ∑_{i=0}^{q}(f̂− p̃_{0})^{2i}sf̂(f̂− p̃_{0})^{2(q−i)}df̂

= −Tr♮(−2f̂_{0} + 1) ∑_{i=0}^{q}(f̂_{0} − p̃_{0})^{2i}sW W^{−1}f̂_{0}(f̂_{0} − p̃_{0})^{2(q−i)}df̂_{0}

+Tr♮(−2f̂_{0} + 1) ∑_{i=0}^{q}(f̂_{0} − p̃_{0})^{2i}f̂_{0}sW W^{−1}(f̂_{0} − p̃_{0})^{2(q−i)}df̂_{0}

Observe that Tr♮ is a trace. In the first term of the r.h.s., we can use the identity f̂_{0}(f̂_{0} − p̃_{0})^{2(q−i)} = (f̂_{0} − p̃_{0})^{2(q−i)}f̂_{0} which holds for any two idempotents f̂_{0} and p̃_{0}, and then f̂_{0}df̂_{0}(−2f̂_{0} +1) = f̂_{0}df̂_{0}. In the second term of the r.h.s., we simply write (−2f̂_{0} +1)(f̂_{0} −p̃_{0})^{2i}f̂_{0} = (−2f̂_{0} +1)f̂_{0}(f̂_{0} −p̃_{0})^{2i} = −f̂_{0}(f̂_{0} −p̃_{0})^{2i}. Hence we arrive at

Tr♮(−2f̂+ 1) ∑_{i=0}^{q}(f̂− p̃_{0})^{2i}sf̂(f̂− p̃_{0})^{2(q−i)}df̂

= − ∑_{i=0}^{q} Tr♮ (f̂_{0} − p̃_{0})^{2i}sW W^{−1}(f̂_{0} − p̃_{0})^{2(q−i)}f̂_{0}df̂_{0}

− ∑_{i=0}^{q} Tr♮ (f̂_{0} − p̃_{0})^{2i}sW W^{−1}(f̂_{0} − p̃_{0})^{2(q−i)}df̂_{0} f̂_{0}

= − ∑_{i=0}^{q} Tr♮ sW W^{−1}(f̂_{0} − p̃_{0})^{2i}df̂_{0}(f̂_{0} − p̃_{0})^{2(q−i)}

by the idempotent property f̂_{0}df̂_{0} + df̂_{0} f̂_{0} = df̂_{0}. It remains to show that the latter sum is a boundary:

− ∑_{i=0}^{q} Tr♮ sW W^{−1}(f̂_{0} −p̃_{0})^{2i}df̂_{0}(f̂_{0} −p̃_{0})^{2(q−i)} = ♮d(Tr sW W^{−1}(f̂_{0} −p̃_{0})^{2q+1}) .

Indeed d anticommutes with sW W^{−1}, and d(f̂_{0} − p̃_{0}) = df̂_{0}. Hence if we can show that the terms Tr♮ sW W^{−1}(f̂_{0} − p̃_{0})^{2i+1}df̂_{0}(f̂_{0} − p̃_{0})^{2j+1} vanish, the conclusion follows. Since f̂_{0}df̂_{0}f̂_{0} = 0, we can write

Tr♮ sW W^{−1}(f̂_{0} − p̃_{0})^{2i+1}df̂_{0}(f̂_{0} − p̃_{0})^{2j+1}

= Tr♮ sW W^{−1}(f̂_{0} − p̃_{0})^{2i}(−f̂_{0}df̂_{0}p̃_{0} − p̃_{0}df̂_{0}f̂_{0} + p̃_{0}df̂_{0}p̃_{0})(f̂_{0} − p̃_{0})^{2j}
= Tr♮ sW W^{−1}(f̂_{0} − p̃_{0})^{2i}(−f̂_{0}df̂_{0}p̃_{0} − df̂_{0}f̂_{0}p̃_{0} + df̂_{0}p̃_{0})(f̂_{0} − p̃_{0})^{2j}
= 0 ,

where we used the fact that p̃_{0} commutes with sW W^{−1} and the even powers of f̂_{0} − p̃_{0}. Hence cs_{1}^{2q}(cf̂c) ≡ 0 mod ♮d, which shows that the pairs (ê, θ) and (Û^{−1}êU, θ) are equivalent. The adjoint action of U on multiplicative K-theory groups in even degrees is thus the identity. One proceeds in the same fashion with odd groups. Let g ∈ (𝒦 ˆ⊗ℐ 𝒜)^{+} be an invertible such that g − 1 ∈ 𝒦 ˆ⊗ℐ 𝒜. Introduce u_{0} = ( \begin{matrix}g & 0 \\ 0 & 1 \end{matrix} ) and the invertible path u = W^{−1}u_{0}W, where W = R^{−1}( \begin{matrix}1 & 0 \\ 0 & U \end{matrix} )R is now viewed as a path of invertible multipliers of M_{2}(𝒦 ˆ⊗ℐ 𝒜). Hence u interpolates between u_{0} and u_{1} = ( \begin{matrix}U^{−1}gU & 0 \\ 0 & 1 \end{matrix} ). This shows that g and U^{−1}gU define the same topological K-theory class. Now suppose that (ĝ, θ) ∈ MK_{n}^{ℐ} (𝒜) represents a multiplicative K-theory class, with ĝ ∈ (𝒦 ˆ⊗ℐ ℛ̂)^{+}, θ ∈ X_{n−1}(ℛ, 𝒥) and ch_{1}^{2q}(ĝ) = ♮dθ. We define û_{0} = ( \begin{matrix}ĝ & 0 \\ 0 & 1 \end{matrix} ), and û = W^{−1}û_{0}W provides an interpolation between û_{0} and û_{1} = ( \begin{matrix}U^{−1}ĝU & 0 \\ 0 & 1 \end{matrix} ). The Chern-Simons form (71) associated to û reads

cs_{0}^{2q}(û) = \frac{1}{\sqrt{2πi}} \frac{(q!)^{2}}{(2q)!} ∫_{0}^{1} Tr û^{−1}[(û − 1)(û^{−1} − 1)]^{q}sû .

Using sû = W^{−1}(−sW W^{−1}û_{0} + û_{0}sW W^{−1})W, one gets

Tr û^{−1}[(û − 1)(û^{−1} − 1)]^{q}sû = −Tr û_{0}^{−1}[(û_{0} − 1)(û_{0}^{−1} − 1)]^{q}sW W^{−1}û_{0}

+Tr û_{0}^{−1}[(û_{0} − 1)(û_{0}^{−1} − 1)]^{q}û_{0}sW W^{−1}

≡ 0 mod \overline{b}

Hence cs_{0}^{2q}(û) ≡ 0 mod \overline{b} and the pair (Û^{−1}ĝU, θ) is equivalent to (ĝ, θ). The adjoint action of U on the odd multiplicative K-theory groups is therefore the identity.

Example 5.6 Take 𝒜 = ℂ and ℐ = ℒ^{p}(H) a Schatten ideal. It is known that K_{0}^{top}(ℐ) = ℤ and K_{1}^{top}(ℐ) = 0. Furthermore HC_{n}(ℂ) = ℂ for n = 2k ≥ 0 and vanishes otherwise. Hence the exact sequence yields

MK_{n}^{ℐ} (ℂ) = \{ \begin{matrix}ℤ & n ≤ 0 even \\ ℂ^{×} & n > 0 odd \\ 0 & otherwise \end{matrix}

The multiplicative K-theory of ℂ is the natural target for index maps in even degree, and for regulator maps in odd degree (see [6] and Example 6.4).

Multiplicative K-theory has close connections with higher algebraic K-theory [16, 29]. In fact there exists a morphism K_{n}^{alg}(𝒜) → MK_{n}^{ℐ} (𝒜) in all degrees, and composition with the negative Chern character coincides with the Jones-Goodwillie map K_{n}^{alg}(𝒜) → HN_{n}(𝒜) [15]. See [7] for an exact sequence relating topological and algebraic K-theories of locally convex algebras stabilized by operator ideals.

6 Riemann-Roch-Grothendieck theorem

In this section we construct direct images of topological and multiplicative Ktheory under quasihomomorphisms and show their compatibility with the Ktheory and cyclic homology exact sequences. This provides a noncommutative version of the Riemann-Roch-Grothendieck theorem. If ℐ is a p-summable Fréchet m-algebra, with trace Tr : ℐ^{p} → ℂ, the tensor product ℐ ˆ⊗ℐ is in a natural way a p-sumable algebra with trace Tr ˆ⊗Tr. We demand that ℐ is provided with an external product as follows.

Definition 6.1 A p-summable Fréchet m-algebra ℐ is multiplicative if there exists a continuous algebra homomorphism (external product)

⊠ : ℐ ˆ⊗ℐ → ℐ

such that the composition Tr ∘ ⊠ coincides with the trace Tr ˆ⊗Tr on (ℐ ˆ⊗ℐ)^{p}. Two external products ⊠ and ⊠′ are equivalent if there exists an invertible multiplier U of ℐ such that ⊠′ = Ad U ∘ ⊠ on ℐ ˆ⊗ℐ.

Hence if ℐ is multiplicative the homomorphism ⊠ induces additive maps K_{n}^{top}(ℐ ˆ⊗ℐ ˆ⊗𝒜) → K_{n}^{top}(ℐ ˆ⊗𝒜) and MK_{n}^{ℐ ˆ⊗ℐ} (𝒜) → MK_{n}^{ℐ} (𝒜), clearly compatible with the commutative diagram of Proposition 5.4. Moreover two equivalent products induce the same maps, the adjoint action Ad U being trivial on K-theory by Lemma 5.5. In practice the algebra ℐ often arises with external products defined only modulo equivalence:

Example 6.2 Let ℐ = ℒ^{p}(H) be the Schatten ideal of p-summable operators on a separable infinite-dimensional Hilbert space H, provided with the operator trace. The tensor product ℒ^{p}(H) ˆ⊗ℒ^{p}(H) is naturally mapped to the algebra ℒ^{p}(H ⊗H), and choosing an isomorphism of Hilbert spaces H ⊗H ≅ H allows to identify ℒ^{p}(H ⊗ H) with ℒ^{p}(H) modulo the adjoint action of unitary operators U ∈ ℒ (H). The product ⊠ : ℒ^{p}(H) ˆ⊗ℒ^{p}(H) → ℒ^{p}(H) is thus compatible with the traces, and canonically defined modulo the adjoint action of unitary operators.

Let 𝒜 and ℬ be any Fréchet m-algebras. Let ρ : 𝒜 → ℰ^{s} ▷ ℐ^{s} ˆ⊗ℬ be a quasihomomorphism of parity p mod 2, and suppose that ℐ is finitely summable (the exact summability degree is irrelevant for the moment). We want to show that ρ induces an additive map

ρ_{!} : K_{n}^{top}(ℐ ˆ⊗𝒜) → K_{n−p}^{top}(ℐ ˆ⊗ℬ)  ∀n ∈ ℤ  (78)

provided ℐ is multiplicative. This is has nothing to do with cyclic homology and we don’t need to assume ℰ admissible. Thanks to Bott periodicity, it is sufficient to define ρ_{!} on K_{1}^{top}(ℐ ˆ⊗𝒜), where it is given by very explicit formulas. Suppose first that p is even. Then ρ is described by a pair of homomorphisms (ρ_{+}, ρ_{−}) : 𝒜 ⇉ ℰ which coincide modulo ℐ ˆ⊗ℬ. For any invertible element g ∈ (𝒦 ˆ⊗ℐ ˆ⊗𝒜)^{+} with g − 1 ∈ 𝒦 ˆ⊗ℐ ˆ⊗𝒜 representing a K-theory class in K_{1}^{top}(ℐ ˆ⊗𝒜), one has ρ_{±}(g) ∈ (𝒦 ˆ⊗ℐ ˆ⊗ℰ)^{+} and ρ_{+}(g) − ρ_{−}(g) ∈ 𝒦 ˆ⊗ℐ ˆ⊗ℐ ˆ⊗ℬ, where the homomorphisms ρ_{+} and ρ_{−} are extended to the unitalized algebra (𝒦 ˆ⊗ℐ ˆ⊗𝒜)^{+} by acting trivially on the factor 𝒦 ˆ⊗ℐ and preserving the unit. set

ρ_{!}(g) = ρ_{+}(g)ρ_{−}(g)^{−1} ∈ (𝒦 ˆ⊗ℐ ˆ⊗ℐ ˆ⊗ℬ)^{+} .  (79)

Using the homomorphism ⊠ : ℐ ˆ⊗ℐ → ℐ, we may therefore consider ρ_{!}(g) as an invertible element of (𝒦 ˆ⊗ℐ ˆ⊗ℬ)^{+} such that ρ_{!}(g) − 1 ∈ 𝒦 ˆ⊗ℐ ˆ⊗ℬ. It is clear that the homotopy class of ρ_{!}(g) only depends on the homotopy class of g, hence the map ρ_{!} : K_{1}^{top}(ℐ ˆ⊗𝒜) → K_{1}^{top}(ℐ ˆ⊗ℬ) is well-defined. When p is odd, ρ is a homomorphism 𝒜 → M_{2}(ℰ) such that the off-diagonal terms lie in ℐ ˆ⊗ℬ. For any invertible element g ∈ (𝒦 ˆ⊗ℐ ˆ⊗𝒜)^{+} as above, one has ρ(g) ∈ M_{2}(𝒦 ˆ⊗ℐ ˆ⊗ℰ)^{+} with off-diagonal elements in 𝒦 ˆ⊗ℐ ˆ⊗ℐ ˆ⊗ℬ. Set

ρ_{!}(g) = ρ(g)^{−1}p_{0}ρ(g) ∈ M_{2}(𝒦 ˆ⊗ℐ ˆ⊗ℐ ˆ⊗ℬ)^{+} ,  (80)

where p_{0} = ( \begin{matrix}1 & 0 \\ 0 & 0 \end{matrix} ) is the trivial matrix idempotent. Again applying the external product ⊠ we may consider ρ_{!}(g) as an idempotent of M_{2}(𝒦 ˆ⊗ℐ ˆ⊗ℬ)^{+} such that ρ_{!}(g) − p_{0} ∈ M_{2}(𝒦 ˆ⊗ℐ ˆ⊗ℬ). The homotopy class only depends on the homotopy class of g and we thus obtain ρ_{!} : K_{1}^{top}(ℐ ˆ⊗𝒜) → K_{0}^{top}(ℐ ˆ⊗ℬ). To define ρ_{!} on K_{0}^{top}(ℐ ˆ⊗𝒜) it suffices to pass to the suspensions S𝒜 = 𝒜 (0, 1) and Sℬ = ℬ(0, 1), then apply the pushforward map constructed above ρ_{!} : K_{1}^{top}(ℐ ˆ⊗S𝒜) → K_{1−p}^{top}(ℐ ˆ⊗Sℬ) with trivial action on the factor C^{∞}(0, 1). The Bott isomorphisms K_{n}^{top}(ℐ ˆ⊗·) ≅ K_{n+1}^{top}(ℐ ˆ⊗S·) allow to define ρ_{!} for the original algebras through a graded-commutative diagram

[figure: K_{0}^{top}(ℐ ˆ⊗𝒜); ∼; ρ_{!}; K_{1}^{top}(ℐ ˆ⊗S𝒜); ρ_{!}; K_{−p}^{top}(ℐ ˆ⊗ℬ); ∼; K_{1−p}^{top}(ℐ ˆ⊗Sℬ)]  (81)

Note the following subtlety concerning graduations: since K_{n}^{top} has parity n mod 2 by definition, the Bott isomorphisms are odd. As a consequence, when p is also odd, the above diagram must be anti-commutative. These conventions are necessary if we want to avoid sign problems with the theorem below.

Now choose a quasi-free extension 0 → 𝒥 → ℛ → ℬ → 0 for ℬ, and suppose that the algebra ℰ ▷ ℐ ˆ⊗ℬ is ℛ-admissible. We impose the following compatibility between the parity of the quasihomomorphism ρ and the summability degree of ℐ: in the even case ℐ is (p + 1)-summable with p even, and in the odd case ℐ is p-summable with p odd (this complicated choice is dictated by the theorem below). In both cases the bivariant Chern character ch^{p}(ρ) ∈ HC^{p}(𝒜 , ℬ) constructed in section 3 induces a map

ch^{p}(ρ) : HC_{n}(𝒜) → HC_{n−p}(ℬ)  ∀n ∈ ℤ .  (82)

Combining ρ_{!} with the bivariant Chern character yields a transformation in multiplicative K-theory, compatible with the diagram (76) of Proposition 5.4. This will be detailed in the proof of the following noncommutative version of the Riemann-Roch-Grothendieck theorem:

Theorem 6.3 Let 𝒜, ℬ be Fréchet m-algebras, and choose a quasi-free extension 0 → 𝒥 → ℛ → ℬ → 0. Let ρ : 𝒜 → ℰ^{s} ▷ ℐ^{s} ˆ⊗ℬ be a quasihomomorphism of parity p mod 2, where ℐ is multiplicative and (p + 1)-summable in the even case, p-summable in the odd case. Suppose that ℰ ▷ ℐ ˆ⊗ℬ is ℛ-admissible. Then ρ defines a transformation in multiplicative K-theory ρ_{!} : MK_{n}^{ℐ} (𝒜) → MK_{n−p}^{ℐ}(ℬ) compatible with the K-theory exact sequences for 𝒜 and ℬ, whence a graded-commutative diagram

[figure: K_{n+1}^{top}(ℐ ˆ⊗𝒜); ρ_{!}; HC_{n−1}(𝒜); ch^{p}(ρ); MK_{n}^{ℐ} (𝒜); ρ_{!}; K_{n}^{top}(ℐ ˆ⊗𝒜); ρ_{!}; K_{n+1−p}^{top}(ℐ ˆ⊗ℬ); HC_{n−1−p}(ℬ); MK_{n−p}^{ℐ}(ℬ); K_{n−p}^{top}(ℐ ˆ⊗ℬ)]

(83)

The vertical arrows are invariant under conjugation of quasihomomorphisms; the arrow in topological K-theory K_{n}^{top}(ℐ ˆ⊗𝒜) → K_{n−p}^{top}(ℐ ˆ⊗ℬ) is also invariant under homotopy of quasihomomorphisms. Moreover (83) is compatible with the commutative diagram of Theorem 3.13 (with connecting map B rescaled by a factor −\sqrt{2πi}) after taking the Chern characters MK_{n}^{ℐ} → HN_{n} and K_{n}^{top}(ℐ ˆ⊗ . ) → HP_{n}.

Proof: As a general rule, the bivariant cyclic cohomomology HC^{p}(𝒜 , ℬ) is described as the cohomology of the complex Hom^{p}(X̂(T 𝒜 , J𝒜), X̂(ℛ, 𝒥)) of linear maps of order ≤ p, where we choose the universal free extension 0 → J𝒜 → T 𝒜 → 𝒜 → 0 for 𝒜 and the quasi-free extension 0 → 𝒥 → ℛ → ℬ → 0 for ℬ. By hypothesis, the algebra ℰ ▷ ℐ ˆ⊗ℬ is ℛ-admissible (Definition 3.2), i.e. one has a commutative diagram

[figure: 0; 𝒩; ℳ; ℰ; 0; 0; ℐ ˆ⊗𝒥; ℐ ˆ⊗ℛ; ℐ ˆ⊗ℬ; 0]

verifying adequate properties with respect to the trace over ℐ. The detailed construction of the pushforward map in multiplicative K-theory ρ_{!} : MK_{n}^{ℐ} (𝒜) → MK_{n−p}^{ℐ}(ℬ) depends on the respective parities of n and p.

i) n = 2k + 1 is odd and p = 2q is even. Our first task is to understand the composition of the topological Chern character ch_{1}^{p} : K_{1}^{top}(ℐ ˆ⊗𝒜) → HP_{1}(𝒜) with the bivariant Chern character ch^{p}(ρ) ∈ HC^{p}(𝒜 , ℬ). For notational simplicity, we write as usual ℐ 𝒜 for the tensor product ℐ ˆ⊗𝒜. Without loss of generality, we may suppose that an element of K_{1}^{top}(ℐ 𝒜) is represented by an invertible g ∈ (ℐ 𝒜)^{+} such that g − 1 ∈ ℐ 𝒜 (indeed the algebra 𝒦 of smooth compact operators plays a trivial role in what follows). Since the universal free extension T 𝒜 is chosen, we can take the canonical lift ĝ ∈ (ℐ T̂𝒜)^{+} which corresponds to the image of g under the canonical linear inclusion 𝒜 ↪ Ω̂^{+}𝒜 ≅ T̂𝒜 as the the subspace of zero-forms. Its inverse is given by the series (57), with 𝒦 replaced by ℐ. The p-th higher Chern character of ĝ is then represented by the cycle (70)

ch_{1}^{p}(ĝ) = \frac{1}{\sqrt{2πi}} \frac{(q!)^{2}}{p!} Tr♮ĝ^{−1}[(ĝ − 1)(ĝ^{−1} − 1)]^{q}dĝ ∈ Ω^{1}T̂𝒜_{♮} ,

Observe that (p + 1) powers of ℐ appear in the products of (ĝ − 1), (ĝ^{−1} − 1) and dĝ hence the trace Tr : ℐ^{p+1} → ℂ is well-defined. On the other hand, the bivariant Chern character of the quasihomomorphism ρ (section 3) is represented by the composition of chain maps ch^{p}(ρ) = χ^{p}ρ_{∗}γ : X(T̂𝒜) → Ω̂T̂𝒜 → Ω̂ℳ̂_{+}^{s} → X(ℛ̂), hence the composite ch^{p}(ρ) · ch_{1}^{p}(ĝ) requires to compute first the image of ch_{1}^{p}(ĝ) under the Goodwillie equivalence γ : X(T̂𝒜) → Ω̂T̂𝒜. This tricky computation can be simplified as follows. We use the isomorphism ℐ 𝒜 ≅ ℂ ˆ⊗ℐ 𝒜 to identify ĝ with the invertible element

û = 1 + e ⊗ (ĝ − 1) ∈ (ℂ ˆ⊗ℐ 𝒜)^{+} ↪ (T̂ℂ ˆ⊗ℐ T̂𝒜)^{+} ,

where e is the unit of ℂ. As usual we regard ℂ ˆ⊗ℐ 𝒜 as the subspace of zeroforms of the algebra T̂ℂ ˆ⊗ℐ T̂𝒜 ≅ Ω̂^{+}ℂ ˆ⊗ℐ Ω̂^{+}𝒜. It is not hard to calculate that the inverse of û is given by the series

û^{−1} = ∑_{i≥0} ((dede)^{i} ⊗[(ĝ^{−1} −1)(ĝ−1)]^{i} +e(dede)^{i} ⊗[(ĝ^{−1} −1)(ĝ−1)]^{i}(ĝ^{−1} −1))

with the convention (dede)^{0} = 1. Observe that the power of ℐ is equal to the power of e in each term of this series. Also, recall that the canonical lift of e is the idempotent

ê = e + ∑_{i≥1} \frac{(2i)!}{(i!)^{2}}(e − \frac{1}{2})(dede)^{i} ∈ T̂ℂ .

We define the fundamental class of degree p = 2q as the trace [2q] : T̂ℂ → ℂ vanishing on all the differential forms e(dede)^{i} and (dede)^{i} except e(dede)^{q}, and normalized so that [2q] ê = 1. One thus have

[2q] e(dede)^{q} = \frac{(q!)^{2}}{p!} , [2q] (anything else) = 0 .

Of course, [2q] is the generator in degree p of the cyclic cohomology of ℂ. The fact that it is a trace over T̂ℂ is crucial. Indeed, one finds the identity

Tr♮[2q] û^{−1}dû = \frac{(q!)^{2}}{p!} Tr♮ĝ^{−1}[(ĝ − 1)(ĝ^{−1} − 1)]^{q}dĝ ∈ Ω^{1}T̂𝒜_{♮} ,

so that the Chern character ch_{1}^{p}(ĝ) is exactly the cycle \frac{1}{\sqrt{2πi}} Tr♮[2q] û^{−1}dû. This simplifies drastically the computation of γch_{1}^{p}(ĝ). The Goodwillie equivalence γ is explicitly constructed in section 2; it is based on the linear map ϕ : T̂𝒜 → Ω^{2}T̂𝒜 verifying the properties ϕ(xy) = ϕ(x)y+xϕ(y)+dxdy for all x, y ∈ T̂𝒜, and ϕ(a) = 0 whenever a ∈ 𝒜. We extend ϕ to a linear map

ϕ : (T̂ℂ ˆ⊗ℐ T̂𝒜)^{+} → T̂ℂ ˆ⊗ℐ Ω^{2}T̂𝒜

acting by the identity on the factor T̂ℂ ˆ⊗ℐ and setting ϕ(1) = 0. This implies ϕ(ûû^{−1}) = 0 = ϕ(û)û^{−1} + ûϕ(û^{−1}) + dûdû^{−1}. Moreover û lies in (ℂ ˆ⊗ℐ 𝒜)^{+} so that ϕ(û) = 0, and one gets

ϕ(û^{−1}) = −û^{−1}dûdû^{−1} .

Now, extending ϕ in all degrees as in section 2 one gets a linear map ϕ : T̂ℂ ˆ⊗ℐ Ω^{i}T̂𝒜 → T̂ℂ ˆ⊗ℐ Ω^{i+2}T̂𝒜 for any i ∈ ℕ. The following computation is then straightforward (remark that Tr[2q] is a trace hence cyclic permutations are allowed; moreover the fundamental class [2q] selects (p + 1) powers of e, hence of ℐ, so that Tr is well-defined):

γ(Tr♮[2q] û^{−1}dû) = ∑_{i≥0} Tr[2q] ϕ^{i}(û^{−1}dû) = ∑_{i≥0}(−)^{i}i! Tr[2q] û^{−1}dû(dû^{−1}dû)^{i} .

Hence γch_{1}^{p}(ĝ) is equal to this (b+B)-cycle over T̂𝒜, divided by a factor \sqrt{2πi}. It remains to apply the chain map χ^{p}ρ_{∗} : Ω̂T̂𝒜 → X(ℛ̂) associated to the quasihomomorphism ρ : 𝒜 → ℰ^{s} ▷ ℐ^{s}ℬ. In 2 × 2 matrix notation, the image of any x ∈ T̂𝒜 under the lifted quasihomomorphism ρ_{∗} : T̂𝒜 → ℳ̂^{s} ▷ ℐ^{s}ℛ̂ and the odd multiplier F read

ρ_{∗}x = (\begin{matrix}x_{+} & 0 \\ 0 & x_{−} \end{matrix}) ∈ ℳ̂_{+}^{s} , F = (\begin{matrix}0 & 1 \\ 1 & 0 \end{matrix}) ,

and the difference x_{+} − x_{−} is therefore an element of the pro-algebra ℐ ℛ̂. On the other hand, the odd component of the chain map χ^{p}ρ_{∗} evaluated on a (p + 1)-form x_{0}dx_{1} … dx_{p+1} is given by Eqs. (34):

\frac{q!}{(p + 1)!} ∑_{i=1}^{p+1} τ′♮(ρ_{∗}x_{0}[F, ρ_{∗}x_{1}] … d(ρ_{∗}x_{i}) … [F, ρ_{∗}x_{p+1}])

where τ′ = \frac{1}{2}τ(F [F, ]) is the modified supertrace of even degree. Then we extend canonically ρ_{∗} to a unital homomorphism (T̂ℂ ˆ⊗ℐ T̂𝒜)^{+} → (T̂ℂ ˆ⊗ℐ ℳ̂_{+}^{s})^{+} by taking the identity on the factor T̂ℂ ˆ⊗ℐ. One thus has ρ_{∗}û = ( \begin{matrix}û_{+} & 0 \\ 0 & û_{−} \end{matrix} ) with û_{+} − û_{−} ∈ T̂ℂ ˆ⊗ℐ ℐ ℛ̂. A direct computation gives

ch^{p}(ρ) (Tr♮[2q] û^{−1}dû) = (−)^{q}q! χ_{1}^{p}ρ_{∗} Tr[2q](û^{−1}dû(dû^{−1}dû)^{q})
= \frac{(q!)^{2}}{p!} Tr♮[2q] u^{−1}[(ũ − 1)(u^{−1} − 1)]^{q}dũ ,

where ũ = û_{+}û_{−}^{−1} ∈ (T̂ℂ ˆ⊗ℐ ℐ ℛ̂)^{+} may be considered as an invertible element of the pro-algebra (T̂ℂ ˆ⊗ℐ ℛ̂)^{+} after applying the homomorphism ⊠ : ℐ ˆ⊗ℐ → ℐ. Dividing by a factor \sqrt{2πi}, the right-hand-side should be defined as the Chern character ch_{1}^{p}(ũ), cf. (70). One thus gets the identity

ch^{p}(ρ) · ch_{1}^{p}(ĝ) = ch_{1}^{p}(ũ)

at the level of cycles in X(ℛ̂). Now, observe that the projection of ũ onto the quotient algebra (ℂ ˆ⊗ℐ ℬ)^{+} is u_{+}u_{−}^{−1} = 1 + e ˆ⊗(ρ_{+}(g)ρ_{−}(g)^{−1} − 1). It corresponds to the direct image ρ_{+}(g)ρ_{−}(g)^{−1} = ρ_{!}(g) by virtue of the isomorphism (ℂ ˆ⊗ℐ ℬ)^{+} ≅ (ℐ ℬ)^{+}. Hence we expect that ch_{1}^{p}(ũ) is homologous to the Chern character of any invertible lift { \atop ρ_{!}(g)} ∈ (ℐ ℛ̂)^{+}. To see this, consider an invertible path v̂ ∈ (T̂ℂ ˆ⊗ℐ ℛ̂[0, 1])^{+} connecting homotopically v̂_{0} = ũ to v̂_{1} = 1+ê⊗({ \atop ρ_{!}(g)}−1), and such that its projection onto (ℂ ˆ⊗ℐ ℬ[0, 1])^{+} is the constant invertible function 1 + e ⊗ (ρ_{!}(g) − 1) over [0, 1]. Such a path always exists, for example the linear interpolation

v̂_{t} = t(1 + ê ⊗ ({ \atop ρ_{!}(g)} − 1)) + (1 − t)ũ ,  t ∈ [0, 1]  (84)

works. Since the evaluation of the fundamental class [2q] on the canonical idempotent lift ê is the unit, a little computation shows the equality

ch_{1}^{p}(v̂_{1}) = ch_{1}^{p}({ \atop ρ_{!}(g)}) ∈ Ω^{1}ℛ̂_{♮}

at the level of cycles. Moreover, the Chern-Simons form associated to v̂, defined in analogy with formulas (71)

cs_{0}^{p}(v̂) = \frac{1}{\sqrt{2πi}} \frac{(q!)^{2}}{p!} ∫_{0}^{1} dt Tr[2q] v̂^{−1}[(v̂ − 1)(v̂^{−1} − 1)]^{q} \frac{∂v̂}{∂t}

fulfills the transgression relation in the complex X(ℛ̂)

♮dcs_{0}^{p}(v̂) = ch_{1}^{p}(v̂_{1}) − ch_{1}^{p}(v̂_{0}) = ch_{1}^{p}({ \atop ρ_{!}(g)}) − ch_{1}^{p}(ũ)

as wanted. We are now in a position to define the map ρ_{!} on multiplicative Ktheory. Let a pair (ĝ, θ) represent a class in MK_{n}^{ℐ} (𝒜) of odd degree n = 2k+1. From Remark 5.3, we know that ĝ ∈ (ℐ T̂𝒜)^{+} can be taken as the canonical lift of some invertible element g ∈ (ℐ 𝒜)^{+}. Then, the transgression θ is a chain of even degree in the quotient complex X_{n−1}(T 𝒜 , J𝒜), and the relation ch_{1}^{p}(ĝ) = ♮dθ holds in X_{n−1}(T 𝒜 , J𝒜). We set

ρ_{!}(ĝ, θ) = ({ \atop ρ_{!}(g)} , ch^{p}(ρ) · θ + cs_{0}^{p}(v̂)) ∈ MK_{n−p}^{ℐ}(ℬ)  (85)

where { \atop ρ_{!}(g)} ∈ (ℐ ℛ̂)^{+} is any invertible lift of ρ_{!}(g) and v̂ is an invertible path constructed as above. Let us explain why this defines a multiplicative Ktheory class. First, the bivariant Chern character ch^{p}(ρ) ∈ HC^{p}(𝒜 , ℬ) induces a morphism of quotient complexes X_{n−1}(T 𝒜 , J𝒜) → X_{n−p−1}(ℛ, 𝒥), hence ch^{p}(ρ) · θ is a well-defined chain of even degree in X_{n−p−1}(ℛ, 𝒥). Regarding also cs_{0}^{p}(v̂) as an element of X_{n−p−1}(ℛ, 𝒥), we see that the relation

♮d(ch^{p}(ρ) · θ + cs_{0}^{p}(v̂)) = ch^{p}(ρ) · ch_{1}^{p}(ĝ) + ch_{1}^{p}({ \atop ρ_{!}(g)}) − ch_{1}^{p}(ũ) = ch_{1}^{p}({ \atop ρ_{!}(g)})

holds in this quotient complex, hence ρ_{!}(ĝ, θ) represents a class in MK_{n−p}^{ℐ}(ℬ).

In fact, the latter does not depend on the choice of lifting { \atop ρ_{!}(g)}, nor on the invertible path v̂. This can be proved simultaneously with the fact that the equivalence class of ρ_{!}(ĝ, θ) depends only on the equivalence class of (ĝ, θ). To show this, consider two equivalent pairs (ĝ_{0}, θ_{0}) and (ĝ_{1}, θ_{1}) representing the same element of MK_{n}^{ℐ} (𝒜). It means there exists a homotopy ĝ ∈ (ℐ T̂𝒜 [0, 1]_{x})^{+} between ĝ_{0} and ĝ_{1} (we denote by x the variable of this homotopy, which should not be confused with the variable t used in the definition of the interpolation (84)), and a chain λ ∈ X_{n−1}(T 𝒜 , J𝒜) such that θ_{1} − θ_{0} = cs_{0}^{p}(ĝ) + \overline{b}λ. From Remark 5.3, we may suppose that ĝ_{0}, ĝ_{1} and ĝ are respectively the canonical lifts of invertibles g_{0}, g_{1} ∈ (ℐ 𝒜)^{+} and g ∈ (ℐ 𝒜 [0, 1]_{x})^{+}. By definition one has

ρ_{!}(ĝ_{i}, θ_{i}) = ({ \atop ρ_{!}(g_{i})} , ch^{p}(ρ) · θ_{i} + cs_{0}^{p}(v̂(g_{i}))) , i = 0, 1 ,

where v̂(g_{i}) ∈ (T̂ℂ ˆ⊗ℐ ℛ̂[0, 1]_{t})^{+} is a choice of invertible path associated to g_{i}, for example by Eq. (84). Choose an invertible path { \atop ρ_{!}(g)} ∈ (ℐ ℛ̂[0, 1]_{x})^{+} interpolating { \atop ρ_{!}(g_{0})} and { \atop ρ_{!}(g_{1})}: it can be chosen as a lift of the path ρ_{!}(g) = ρ_{+}(g)ρ_{−}(g)^{−1}. Our goal is to show that the relation

ch^{p}(ρ) · (θ_{1} − θ_{0}) + cs_{0}^{p}(v̂(g_{1})) − cs_{0}^{p}(v̂(g_{0})) ≡ cs_{0}^{p}({ \atop ρ_{!}(g)}) mod \overline{b}

holds in X_{n−p−1}(ℛ, 𝒥). As before we identify the canonical lift ĝ of g with the invertible element

û = 1 + e ⊗ (ĝ − 1) ∈ (ℂ ˆ⊗ℐ 𝒜 [0, 1]_{x})^{+} ↪ (T̂ℂ ˆ⊗ℐ T̂𝒜 [0, 1]_{x})^{+} ,

and we remark that the higher Chern-Simons form cs_{0}^{p}(ĝ) given by Lemma 4.5, Eqs. (71), can be written as

cs_{0}^{p}(ĝ) = \frac{1}{\sqrt{2πi}} ∫_{0}^{1} Tr[2q] û^{−1}sû

where s = dx\frac{∂}{∂x} is the de Rham coboundary acting on the space of differential forms Ω[0, 1]_{x}. It follows that the computation of ch^{p}(ρ) · cs_{0}^{p}(ĝ) requires first to evaluate the Goodwillie equivalence γ on the one-form ω̂ = û^{−1}sû. To this end, we extend ϕ to a linear map

ϕ : T̂ℂ ˆ⊗ℐ Ω^{i}T̂𝒜 ˆ⊗Ω[0, 1]_{x} → T̂ℂ ˆ⊗ℐ Ω^{i+2}T̂𝒜 ˆ⊗Ω[0, 1]_{x}

acting by the identity on the factors T̂ℂ ˆ⊗ℐ and Ω[0, 1]_{x}. The algebraic property of ϕ implies ϕ(û^{−1}sû) = ϕ(û^{−1})sû + û^{−1}sϕ(û) + dû^{−1}d(sû). From ϕ(û) = 0, ϕ(û^{−1}) = −û^{−1}dûdû^{−1} and dû^{−1} = −û^{−1}dûû^{−1} we deduce

ϕ(ω̂) = −û^{−1}dûdω̂ .

Then the image of [2q] ω̂ under the Goodwillie equivalence is a straightforward computation, taking into account the tracial property of the fundamental class Tr[2q] and the fact that Ω[0, 1]_{x} is a commutative algebra:

γ(Tr[2q] ω̂) = ∑_{i≥0} Tr[2q]ϕ^{i}(ω̂) =

Tr[2q] ω̂ + ∑_{i≥1}(−)^{i}(i − 1)! ∑_{j=0}^{i−1} Tr[2q] û^{−1}dû(dû^{−1}dû)^{j}dω̂(dû^{−1}dû)^{i−j−1} .

This is a chain in the bicomplex Ω̂T̂𝒜 ˆ⊗Ω[0, 1]_{x} endowed with the boundary maps (b + B) and s. Its is related to the (b + B)-cocycle γ(Tr♮[2q]û^{−1}dû) via the descent equation

(b + B)γ(Tr[2q]ω̂) + sγ(Tr♮[2q]û^{−1}dû) = 0 ,

which can be shown either by direct computation, or simply by observing that ♮d(Tr[2q]ω)+s(Tr♮[2q]û^{−1}dû) = 0 and γ♮d = (b+B)γ, γs = sγ. Next we have to evaluate the image of γ(Tr[2q] ω̂) by the chain map χ^{p}ρ_{∗} : Ω̂T̂𝒜 → X(ℛ̂), whose even component evaluated on a p-form x_{0}dx_{1} … dx_{p} over T̂𝒜 reads

\frac{q!}{(p + 1)!} ∑_{λ∈S_{p+1}} ε(λ)τ′(ρ_{∗}x_{λ(0)}[F, ρ_{∗}x_{λ(1)}] … [F, ρ_{∗}x_{λ(p)}]) .

Denote as before ũ = û_{+}û_{−}^{−1} ∈ (T̂ℂ ˆ⊗ℐ ℛ̂[0, 1]_{x})^{+}, and ω̃ = u^{−1}sũ. One finds:

ch^{p}(ρ) (Tr[2q] ω̂) = \frac{(q!)^{2}}{(p + 1)!} Tr[2q] (û_{−}^{−1}ω̃[(u^{−1} − 1)(ũ − 1)]^{q}û_{−}+

(ũ − 1)ω̃[(u^{−1} − 1)(ũ − 1)]^{q−1}(u^{−1} − 1) + … + û_{−}^{−1}[(u^{−1} − 1)(ũ − 1)]^{q}ω̃û_{−})

After evaluation on the current \frac{1}{\sqrt{2πi}} ∫_{x=0}^{1}, the right-hand-side may be identified, modulo commutators, with the Chern-Simons form cs_{0}^{p}(ũ) defined in analogy with (71). One thus gets

ch^{p}(ρ) · cs_{0}^{p}(ĝ) ≡ cs_{0}^{p}(ũ) mod \overline{b} .

Now, introduce a parameter t ∈ [0, 1] and choose an invertible interpolation

v̂ ∈ (T̂ℂ ˆ⊗ℐ ℛ̂[0, 1]_{x}[0, 1]_{t})^{+} between v̂_{t=0} = ũ and v̂_{t=1} = 1 + ê ⊗ ({ \atop ρ_{!}(g)} − 1),

with the property that it restricts to v̂(g_{0}) for x = 0 and to v̂(g_{1}) for x = 1. The projection of v̂ on the algebra (ℂ ˆ⊗ℐ ℬ[0, 1]_{x}[0, 1]_{t})^{+} may be chosen constant with respect to t. In the proof of Lemma 4.6 we established at any point (x, t) ∈ [0, 1]^{2} the identity

\frac{∂}{∂t}(Tr[2q] v̂^{−1}[(v̂−1)(v̂^{−1}−1)]^{q}sv̂) ≡ s(Tr[2q] v̂^{−1}[(v̂−1)(v̂^{−1}−1)]^{q} \frac{∂v̂}{∂t}) mod \overline{b} ,

and integrating over [0, 1]^{2} this implies

cs_{0}^{p}(v̂_{t=1}) − cs_{0}^{p}(v̂_{t=0}) ≡ cs_{0}^{p}(v̂_{x=1}) − cs_{0}^{p}(v̂_{x=0}) mod \overline{b} .

Taking into account that v̂_{x=0} = v̂(g_{0}) and v̂_{x=1} = v̂(g_{1}), we calculate mod \overline{b} in the complex X_{n−p−1}(ℛ, 𝒥)

ch^{p}(ρ) · (θ_{1} − θ_{0}) + cs_{0}^{p}(v̂(g_{1})) − cs_{0}^{p}(v̂(g_{0}))

≡ ch^{p}(ρ) · cs_{0}^{p}(ĝ) + cs_{0}^{p}(v̂_{t=1}) − cs_{0}^{p}(v̂_{t=0}) mod \overline{b}

≡ cs_{0}^{p}(ũ) + cs_{0}^{p}(1 + ê ⊗ ({ \atop ρ_{!}(g)} − 1)) − cs_{0}^{p}(ũ) mod \overline{b}

≡ cs_{0}^{p}({ \atop ρ_{!}(g)}) mod \overline{b} .

Hence the direct images ρ_{!}(ĝ_{0}, θ_{0}) and ρ_{!}(ĝ_{1}, θ_{1}) are equivalent and the map ρ_{!} : MK_{n}^{ℐ} (𝒜) → MK_{n−p}^{ℐ}(ℬ) for n = 2k + 1 and p = 2q is well-defined. It is obviously compatible with the push-forward map in topological K-theory ρ_{!} : K_{1}^{top}(ℐ 𝒜) → K_{1}^{top}(ℐ ℬ). The compatibility with the push-forward map in cyclic homology ch^{p}(ρ) : HC_{n−1}(𝒜) → HC_{n−p−1}(ℬ) is clear once we remark that the Chern-Simons form cs_{0}^{2q}(v̂) vanish whenever v̂ = 1. Hence the diagram (83) is commutative. We have to check the invariance of ρ_{!} with respect to conjugation of quasihomomorphisms. Let ρ_{0} and ρ_{1} be two conjugate quasihomomorphisms 𝒜 → ℰ^{s} ▷ ℐ^{s}ℬ. Hence there exists an invertible element U ∈ (ℰ_{+}^{s})^{+} such that ρ_{1} = U^{−1}ρ_{0}U. We follow the proof of Proposition 3.12 and remark that the lifting homomorphisms ρ_{0∗}, ρ_{1∗} : T̂𝒜 → ℳ_{+}^{s} factor through homomorphisms φ_{0}, φ_{1} : T̂𝒜 → T̂ℰ_{+}^{s}. The maps ρ_{0!}, ρ_{1!} : MK_{n}^{ℐ} (𝒜) → MK_{n−p}^{ℐ}(ℬ) are obtained by composition of the pushforward maps φ_{0!}, φ_{1!} : MK_{n}^{ℐ} (𝒜) → MK_{n}^{ℐ} (ℰ_{+}^{s}) induced by the homomorphisms

φ_{i!}(ĝ, θ) = (φ_{i}(ĝ), φ_{i}(θ))

with the map MK_{n}^{ℐ} (ℰ_{+}^{s}) → MK_{n−p}^{ℐ}(ℬ) associated with the natural (p + 1)- summable quasihomomorphism of even degree ℰ_{+}^{s} → ℰ^{s} ▷ ℐ^{s}ℬ. Hence it is sufficient to check that the maps φ_{0!} and φ_{1!} : MK_{n}^{ℐ} (𝒜) → MK_{n}^{ℐ} (ℰ_{+}^{s}) coincide. From the proof of 3.12 we know that φ_{1} is smoothly homotopic to Û^{−1}φ_{0}Û, where Û ∈ (T̂ℰ_{+}^{s})^{+} is a lifting of U, and the interpolating homomorphism φ : T̂𝒜 → T̂ℰ_{+}^{s}[0, 1] is constant modulo the ideal Ĵℰ_{+}^{s}. Consequently the morphisms X(φ_{1}) and X(Û^{−1}φ_{0}Û) : X(T̂𝒜) → X(T̂ℰ_{+}^{s}) are homotopic,

X(φ_{1}) − X(Û^{−1}φ_{0}Û) = [∂, H]

with H ∈ Hom^{0}(X(T 𝒜 , J𝒜), X(T ℰ_{+}^{s}, Jℰ_{+}^{s})) a cochain of order zero. Let us now compare the images of (ĝ, θ) ∈ MK_{n}^{ℐ} (𝒜) under the pushworwards φ_{1!} and (Û^{−1}φ_{0}Û)_{!}. The images φ_{1}(ĝ) and Û^{−1}φ_{0}(ĝ)Û are smoothly homotopic, with interpolation φ(ĝ) ∈ (ℐ T̂ℰ_{+}^{s}[0, 1])^{+}. If moreover ĝ is the canonical lift of an invertible element g ∈ (ℐ 𝒜)^{+}, Chern-Simons form associated to φ(ĝ) can be written as

cs_{0}^{p}(φ(ĝ)) = \frac{1}{\sqrt{2πi}} \frac{(q!)^{2}}{p!} ∫_{0}^{1} Tr φ(ĝ^{−1})φ[(ĝ − 1)(ĝ^{−1} − 1)]^{q}sφ(ĝ)
= \frac{1}{\sqrt{2πi}} ∫_{0}^{1} Tr[2q] φ(û^{−1})sφ(û) ,

where as usual û = 1 + e ⊗ (ĝ − 1) is the invertible associated to ĝ, with φ(û) = 1 + e ⊗ (φ(ĝ) − 1) ∈ (T̂ℂ ˆ⊗ℐ T̂ℰ_{+}^{s}[0, 1])^{+}. By construction (Proposition 3.12 i)), the r.h.s. coincides with the evaluation of H on the Chern character ch_{1}^{p}(ĝ) = \frac{1}{\sqrt{2πi}}Tr♮[2q] û^{−1}dû, and because H is of order zero one has

cs_{0}^{p}(φ(ĝ)) = Hch_{1}^{p}(ĝ) = H(♮dθ) ≡ φ_{1}(θ) − (Û^{−1}φ_{0}Û)(θ) mod \overline{b}

in the complex X_{n−1}(T ℰ_{+}^{s}, Jℰ_{+}^{s}). This proves that φ_{1!}(ĝ, θ) is equivalent to (Û^{−1}φ_{0}Û)_{!}(ĝ, θ). Now it remains to show that the image (Û^{−1}φ_{0}Û)_{!}(ĝ, θ) = (Û^{−1}φ_{0}(ĝ)Û, Û^{−1}φ_{0}(θ)Û) is equivalent to φ_{0!}(ĝ, θ) = (φ_{0}(ĝ), φ_{0}(θ)). Here we mimic the proof of Lemma 5.5 and construct a homotopy between the invertible matrices ( \begin{matrix}φ_{0}(ĝ) & 0 \\ 0 & 1 \end{matrix} ) and ( \begin{matrix}Û^{−1}φ_{0}(ĝ)Û & 0 \\ 0 & 1 \end{matrix} ), whose associated Chern-Simons form is a \overline{b}-boundary. Since Û^{−1}φ_{0}(θ)Û ≡ φ_{0}(θ) mod \overline{b}, we conclude that ρ_{0!} and ρ_{1!} agree on topological and multiplicative K-theories. Finally we have to check the compatibility with the negative Chern character MK_{∗}^{ℐ} → HN_{∗}. For any pair (ĝ, θ) ∈ MK_{n}^{ℐ} (𝒜), one has

ch_{n}(ĝ, θ) = ch_{1}^{p}(ĝ) − ♮dθ̃ ∈ F^{n−1}X̂(T 𝒜 , J𝒜),

where θ̃ is any lift of θ in X̂(T 𝒜 , J𝒜). On the other hand, if ĝ is the canonical lift of some g ∈ (ℐ 𝒜)^{+}, its image ρ_{!}(ĝ, θ) ∈ MK_{n−p}^{ℐ}(ℬ) is represented by the

pair ({ \atop ρ_{!}(g)}, ch^{p}(ρ) · θ + cs_{0}^{p}(v̂)) constructed above, so that

ch_{n−p}(ρ_{!}(ĝ, θ)) = ch_{1}^{p}({ \atop ρ_{!}(g)}) − ♮d(ch^{p}(ρ) · θ̃ + cs_{0}^{p}(v̂)) ∈ F^{n−p−1}X̂(ℛ, 𝒥) .

But we know that the relation ch_{1}^{p}({ \atop ρ_{!}(g)}) − ♮dcs_{0}^{p}(v̂) = ch^{p}(ρ) · ch_{1}^{p}(ĝ) actually holds in the complex X̂(ℛ, 𝒥) = X(ℛ̂). Therefore

ch_{n−p}(ρ_{!}(ĝ, θ)) = ch^{p}(ρ) · (ch_{1}^{p}(ĝ) − ♮dθ̃) = ch^{p}(ρ) · ch_{n}(ĝ, θ) ,

and (83) is compatible with the diagram of Theorem 3.13. ii) n = 2k is even and p = 2q is even. As in the case of topological K-theory we pass to the suspensions of 𝒜 and ℬ. We shall only sketch the procedure. The multiplicative K-theory group of even degree MK_{n}^{ℐ} (𝒜) has an alternative description in terms of the set MK_{n}^{′ℐ} (𝒜) of equivalence classes of pairs (ĝ, θ), where ĝ ∈ (𝒦 ˆ⊗ℐ ST̂𝒜)^{+} is an invertible and θ ∈ X_{n−1}(T 𝒜 , J𝒜) is a chain of odd degree such that cs_{0}^{p}(ĝ) = \overline{b}θ. The equivalence relation is based on a higher transgression of the Chern-Simons form: (ĝ_{0}, θ_{0}) is equivalent to (ĝ_{1}, θ_{1}) iff there exists an invertible interpolation ĝ ∈ (𝒦 ˆ⊗ℐ ST̂𝒜 [0, 1])^{+} and a chain of even degree λ such that

θ_{1} − θ_{0} = cs_{1}^{′p}(ĝ) + ♮dλ ∈ X_{n−1}(T 𝒜 , J𝒜) ,

where the odd chain cs_{1}^{′p}(ĝ) ∈ X(T̂𝒜) is defined modulo ♮d by the higher transgression formula (see the proof of Lemma 4.6)

\overline{b}cs_{1}^{′p}(ĝ) = cs_{0}^{p}(ĝ_{1}) − cs_{0}^{p}(ĝ_{0}) .

Like MK, one can show that MK_{n}^{′ℐ} (𝒜) is an abelian group inserted between HC_{∗}(𝒜) and K_{∗}^{top}(ℐ S𝒜) in an exact sequence. More precisely there is a commutative diagram with exact rows:

\xymatrix{K_{1}^{top}(ℐ 𝒜) \ar[r]^{^{ch^{1}}} \ar[d]^{α} & HC_{n−1}(𝒜) \ar[r]^{δ} \ar[d]^{×\sqrt{2πi}} & MK_{n}^{ℐ} (𝒜) \ar[r] \ar[d] & K_{0}^{top}(ℐ 𝒜) \ar[r]^{^{ch^{0}}} \ar[d]_{β} & HC_{n−2}(𝒜) \ar[d]_{×\sqrt{2πi}} \\ K_{0}^{top}(ℐ S𝒜) \ar[r]^{_{cs_{1}}} & HC_{n−1}(𝒜) \ar[r]^{δ} & MK_{n}^{′ℐ} (𝒜) \ar[r] & K_{1}^{top}(ℐ S𝒜) \ar[r]^{_{cs_{0}}} & HC_{n−2}(𝒜)}

Because for even n the group MK′_{n} is constructed from invertibles, it has odd parity by convention. The (odd) map MK_{n}^{ℐ} (𝒜) → MK_{n}^{′ℐ} (𝒜) sends a pair (ê, θ) to (β(ê), \sqrt{2πi}θ + l_{1}^{p}(ê)), where β(ê) = (1 + (z − 1)ê)(1 + (z − 1)p_{0})^{−1} is the invertible image of ê under the Bott map, and l_{1}^{p}(ê) is the transgressed cochain defined modulo ♮d by \overline{b}(l_{1}^{p}(ê)) = cs_{0}^{p}(β(ê)) − \sqrt{2πi} ch_{0}^{p}(ê). The map MK_{n}^{′ℐ} (𝒜) → K_{1}^{top}(ℐ S𝒜) is the forgetful map, and HC_{n−1}(𝒜) → MK_{n}^{′ℐ} (𝒜) sends a cycle θ to (1, \sqrt{2πi} θ). By the five lemma, MK_{n}^{′ℐ} (𝒜) is thus isomorphic to MK_{n}^{ℐ} (𝒜). One easily checks that the negative Chern character ch_{n} : MK_{n}^{′ℐ} (𝒜) → HN_{n}(𝒜) given by ch_{n}(ĝ, θ) = cs_{0}^{p}(ĝ) − \overline{b}θ̃ coincides with the negative Chern character on MK_{n}^{ℐ} (𝒜) up to a factor \sqrt{2πi}. Hence it suffices to constru ct the pushforward morphism ρ_{!} for the groups MK′_{n}, whose elements are represented by invertibles of the suspended algebras:

[figure: K_{0}^{top}(ℐ S𝒜); ρ_{!}; HC_{n−1}(𝒜); ch^{p}(ρ); MK_{n}^{′ℐ} (𝒜); ρ_{!}; K_{1}^{top}(ℐ S𝒜); ρ_{!}; K_{−p}^{top}(ℐ Sℬ); HC_{n−1−p}(ℬ); MK_{n−p}^{′ℐ}(ℬ); K_{1−p}^{top}(ℐ Sℬ)]

This can be done explicitly as in case i), with the only difference that the Chern character ch_{1}^{p} and Chern-Simons transgression cs_{0}^{p} are now replaced respectively by cs_{0}^{p} and the higher transgression cs_{1}^{′p}. The needed formulas were already established in i): let g ∈ (ℐ S𝒜)^{+} be any invertible with canonical lift ĝ ∈ (ℐ ST̂𝒜)^{+}. One can write

ch^{p}(ρ) · cs_{0}^{p}(ĝ) = cs_{0}^{p}(ũ) − \overline{b}k_{1}^{p}(û)

with the invertibles û = 1 + e ⊗ (ĝ − 1) ∈ (T̂ℂ ˆ⊗ℐ ST̂𝒜)^{+} and ũ = û_{+}û_{−}^{−1} ∈ (T̂ℂ ˆ⊗ℐ Sℛ̂)^{+}, and k_{1}^{p}(û) is a chain defined mod ♮d. Let { \atop ρ_{!}(g)} ∈ (ℐ Sℛ̂)^{+} be any invertible lift of ρ_{!}(g) = ρ_{+}(g)ρ_{−}(g)^{−1}, and v̂ ∈ (T̂ℂ ˆ⊗ℐ Sℛ̂[0, 1])^{+} be an invertible interpolation between v̂_{0} = ũ and v̂_{1} = 1 + ê⊗ ({ \atop ρ_{!}(g)} − 1). Then one has

\overline{b}cs_{1}^{′p}(v̂) = cs_{0}^{p}({ \atop ρ_{!}(g)}) − cs_{0}^{p}(ũ) .

Therefore if (ĝ, θ) represents a class in MK_{n}^{′ℐ} (𝒜) we define its pushforward as the multiplicative K-theory class over ℬ

ρ_{!}(ĝ, θ) = ({ \atop ρ_{!}(g)} , ch^{p}(ρ) · θ + k_{1}^{p}(û) + cs_{1}^{′p}(v̂)) ∈ MK_{n−p}^{′ℐ}(ℬ)  (86)

with the odd chain ch^{p}(ρ) · θ + k_{1}^{p}(û) + cs_{1}^{′p}(v̂) sitting in X_{n−p−1}(ℛ, 𝒥). One shows the consistency of ρ_{!} with the various equivalence relations using the properties of higher Chern-Simons transgressions. Details are left to the reader.

iii) n = 2k + 1 is odd and p = 2q + 1 is odd. We first establish an explicit formula for the composition of the topological Chern character ch_{1}^{2q} : K_{1}^{top}(ℐ 𝒜) → HP_{1}(𝒜) with the bivariant Chern character ch^{p}(ρ) ∈ HC^{p}(𝒜 , ℬ). Remark that ℐ is (2q + 1)-summable by hypothesis hence ch_{1}^{2q} is well-defined. As in case i) let g ∈ (ℐ 𝒜)^{+} be an invertible, ĝ ∈ (ℐ T̂𝒜)^{+} its canonical lift and û = 1+e⊗(ĝ−1) ∈ (T̂ℂ ˆ⊗ℐ T̂𝒜)^{+} the associated invertible. Recall that

ch_{1}^{2q}(ĝ) = \frac{1}{\sqrt{2πi}} Tr♮[2q] û^{−1}dû ∈ Ω^{1}T̂𝒜_{♮} ,

and the image of Tr♮[2q] û^{−1}dû under the Goodwillie equivalence is the (b+B)- cycle over T̂𝒜

γ(Tr♮[2q] û^{−1}dû) = ∑_{i≥0}(−)^{i}i! Tr[2q] û^{−1}dû(dû^{−1}dû)^{i} .

Now the quasihomomorphism ρ : 𝒜 → ℰ^{s} ▷ ℐ^{s}ℬ is of odd degree. Hence, the image of an element x ∈ T̂𝒜 under the lifted quasihomomorphism ρ_{∗} : T̂𝒜 → ℳ̂^{s} ▷ ℐ^{s}ℛ̂ is a 2 × 2 matrix over ℳ̂whose off-diagonal entries lie in ℐ ℛ̂. Moreover the multiplier F is given by the matrix

F = ε (\begin{matrix}1 & 0 \\ 0 & −1 \end{matrix}) = ε(2p_{0} − 1)

where ε is the odd generator of the Clifford algebra C_{1}. Thus the commutator [p_{0}, ρ_{∗}x] lies in the matrix algebra M_{2}(ℐ ℛ̂) for any x ∈ T̂𝒜. On the other hand, the component of the chain map χ^{p}ρ_{∗} : Ω̂T̂𝒜 → X(ℛ̂) evaluated on a p-form x_{0}dx_{1} … dx_{p} reads

−\frac{Γ(q + \frac{3}{2})}{(p + 1)!} ∑_{λ∈S_{p+1}} ε(λ)τ(ρ_{∗}x_{λ(0)}[F, ρ_{∗}x_{λ(1)}] … [F, ρ_{∗}x_{λ(p)}]) ,

where τ(ε·) = −\sqrt{2i} Tr(·) is the odd supertrace (see section 3). As in case i), let us extend ρ_{∗} to a unital homomorphism (T̂ℂ ˆ⊗ℐ T̂𝒜)^{+} → (T̂ℂ ˆ⊗ℐ ℳ̂_{+}^{s})^{+}. Using Γ(q + \frac{3}{2}) = \sqrt{π} p!/(2^{p}q!) with p = 2q + 1, one gets by direct computation

ch^{p}(ρ) · ch_{1}^{2q}(ĝ) = \frac{1}{2}Tr[2q] (u^{−1}[p_{0}, ũ])^{p} + \frac{1}{2}Tr[2q] ([p_{0}, ũ]u^{−1})^{p}
= Tr[2q] (u^{−1}[p_{0}, ũ])^{p} − \overline{b}\frac{1}{2}Tr♮[2q] (u^{−1}[p_{0}, ũ])^{p}u^{−1}dũ

where ũ = ρ_{∗}û is an invertible element of the algebra (T̂ℂ ˆ⊗ℐ ℳ̂_{+}^{s})^{+}, and the commutator [p_{0}, ũ] ∈ M_{2}(T̂ℂ ˆ⊗ℐ ℐ ℛ̂)^{+} may be considered as an element of M_{2}(T̂ℂ ˆ⊗ℐ ℛ̂)^{+} after applying the homomorphism ⊠ : ℐ ˆ⊗ℐ → ℐ. The first term of the r.h.s. is recognized as the higher Chern character ch_{0}^{2q}(f̃) = Tr[2q](f̃ − p_{0})^{p} given by (70) for the idempotent f̃ = u^{−1}p_{0}ũ ∈ M_{2}(T̂ℂ ˆ⊗ℐ ℐ ℛ̂)^{+} (or M_{2}(T̂ℂ ˆ⊗ℐ ℛ̂)^{+}), whence the equality

ch^{p}(ρ) · ch_{1}^{2q}(ĝ) = ch_{0}^{2q}(f̃) − \overline{b} \frac{1}{2}Tr♮[2q] (f̃− p_{0})^{p}u^{−1}dũ

of cycles in X(ℛ̂). Then, observe that the projection of f̃ to the algebra M_{2}(ℂ ˆ⊗ℐ ℬ)^{+} is the idempotent p_{0} + e ⊗ (ρ(g)^{−1}p_{0}ρ(g) − p_{0}). Using the isomorphism (ℂ ˆ⊗ℐ ℬ)^{+} ≅ (ℐ ℬ)^{+}, this idempotent may be identified with the direct image ρ_{!}(g) = ρ(g)^{−1}p_{0}ρ(g). Hence, it is possible to relate ch_{0}^{2q}(f̃) with the Chern character of a given idempotent lift { \atop ρ_{!}(g)} ∈ M_{2}(ℐ ℛ̂)^{+}, via a homotopy with parameter t ∈ [0, 1]. Let f̂ ∈ M_{2}(T̂ℂ ˆ⊗ℐ ℛ̂[0, 1])^{+} be an idempotent path lifting the constant family p_{0} + e ⊗ (ρ_{!}(g) − p_{0}) and connecting the two endpoints

f̂_{0} = f̃ ,  f̂_{1} = p_{0} + ê ⊗ ({ \atop ρ_{!}(g)} − p_{0}) .

ê ∈ T̂ℂ is the canonical idempotent lift of the unit e ∈ ℂ as in case i). The lifting f̂ is thus defined up to homotopy (at least after stabilization by the matrix algebra 𝒦 ). The property [2q] ê = 1 implies the equality

ch_{0}^{2q}(f̂_{1}) = ch_{0}^{2q}({ \atop ρ_{!}(g)}) ∈ ℛ̂

at the level of cycles. Furthermore, in analogy with Eqs. (71) the Chern-Simons form associated to the idempotent f̂ is defined by

cs_{1}^{2q}(f̂) = ∫_{0}^{1} dt Tr♮[2q] (−2f̂+ 1) ∑_{i=0}^{q}(f̂− p_{0})^{2i} \frac{∂f̂}{∂t}(f̂− p_{0})^{2(q−i)}df̂ ,

and fulfills the transgression relation in X(ℛ̂)

\overline{b}cs_{1}^{2q}(f̂) = ch_{0}^{2q}(f̂_{1}) − ch_{0}^{2q}(f̂_{0}) = ch_{0}^{2q}({ \atop ρ_{!}(g)}) − ch_{0}^{2q}(f̃) .

This leads to the definition of the map ρ_{!} on multiplicative K-theory. Let (ĝ, θ) represent a class in MK_{n}^{ℐ} (𝒜) of odd degree n = 2k + 1. By Remark 5.3 we may suppose that ĝ is the canonical lift of some invertible g ∈ (ℐ 𝒜)^{+}, and θ ∈ X_{n−1}(T 𝒜 , J𝒜) is a transgression of the Chern character ch_{1}^{2q}(ĝ) = ♮dθ. We set

ρ_{!}(ĝ, θ) = ({ \atop ρ_{!}(g)} , −ch^{p}(ρ) · θ + h_{1}^{2q}(ũ) + cs_{1}^{2q}(f̂)) ∈ MK_{n−p}^{ℐ}(ℬ)  (87)

where { \atop ρ_{!}(g)} ∈ M_{2}(ℐ ℛ̂)^{+} is any idempotent lift of ρ_{!}(g), h_{1}^{2q}(ũ) is the chain \frac{1}{2}Tr♮[2q] (f̃− p_{0})^{p}u^{−1}dũ, and f̂ is an idempotent path constructed as above. The minus sign in front of ch^{p}(ρ) · θ is necessary because the bivariant Chern character ch^{p}(ρ) is of odd degree p = 2q + 1. This ensures the correct transgression relation

\overline{b}(−ch^{p}(ρ) · θ + h_{1}^{2q}(ũ) + cs_{1}^{2q}(f̂))

= ch^{p}(ρ) · ch_{1}^{2q}(ĝ) + \overline{b} \frac{1}{2}Tr♮[2q] (f̃− p_{0})^{p}u^{−1}dũ + ch_{0}^{2q}({ \atop ρ_{!}(g)}) − ch_{0}^{2q}(f̃)
= ch_{0}^{2q}({ \atop ρ_{!}(g)})

in the quotient complex X_{n−p−1}(ℛ, 𝒥), which shows that ρ_{!}(ĝ, θ) indeed defines an element of MK_{n−p}^{ℐ}(ℬ). Its class does not dependent on the chosen

idempotent lift { \atop ρ_{!}(g)} nor on the path f̂, and moreover ρ_{!} is compatible with the equivalence relation on multiplicative K-theory. We proceed as in case i) and let (ĝ_{0}, θ_{0}) and (ĝ_{1}, θ_{1}) be two equivalent representatives of a class in MK_{n}^{ℐ} (𝒜), provided with an interpolation ĝ ∈ (ℐ T̂𝒜 [0, 1]_{x})^{+} and a chain λ ∈ X_{n−1}(T 𝒜 , J𝒜) such that θ_{1} − θ_{0} = cs_{0}^{2q}(ĝ) + \overline{b}λ. From Remark 5.3 the elements ĝ_{0}, ĝ_{1} and ĝ can be taken as the canonical lifts of g_{0}, g_{1} ∈ (ℐ 𝒜)^{+} and g ∈ (ℐ 𝒜 [0, 1]_{x})^{+}. Denoting by ρ_{∗}û(g_{i}) = ũ(g_{i}) ∈ (T̂ℂ ˆ⊗ℐ ℳ̂_{+}^{s})^{+} the invertible and f̂(g_{i}) ∈ M_{2}(T̂ℂ ˆ⊗ℐ ℛ̂[0, 1]_{t})^{+} the idempotent path associated to g_{i}, we have to establish the relation

−ch^{p}(ρ) · (θ_{1} − θ_{0}) + h_{1}^{2q}(ũ(g_{1})) − h_{1}^{2q}(ũ(g_{0})) + cs_{1}^{2q}(f̂(g_{1})) − cs_{1}^{2q}(f̂(g_{0}))

≡ cs_{1}^{2q}({ \atop ρ_{!}(g)}) mod ♮d

in the complex X_{n−p−1}(ℛ, 𝒥), where { \atop ρ_{!}(g)} ∈ M_{2}(ℐ ℛ̂[0, 1]_{x})^{+} is a choice of idempotent interpolation between the liftings { \atop ρ_{!}(g_{i})}’s. As usual, let û = 1 + e ⊗ (ĝ − 1) ∈ (ℂ ˆ⊗ℐ 𝒜 [0, 1]_{x})^{+} ↪ (T̂ℂ ˆ⊗ℐ T̂𝒜 [0, 1]_{x})^{+} be the invertible identification with ĝ. We know the equality

cs_{0}^{2q}(ĝ) = \frac{1}{\sqrt{2πi}} ∫_{0}^{1} Tr[2q] û^{−1}sû ,

where s is the de Rham differential on Ω[0, 1]_{x}. Set ω̂ = û^{−1}sû. The computation of ch^{p}(ρ) · cs_{0}^{2q}(ĝ) involves the formula

γ(Tr[2q] ω̂) =

Tr[2q] ω̂ + ∑_{i≥1}(−)^{i}(i − 1)! ∑_{j=0}^{i−1} Tr[2q] û^{−1}dû(dû^{−1}dû)^{j}dω̂(dû^{−1}dû)^{i−j−1} ,

as well as the component of the chain map χ^{p}ρ_{∗} evaluated on a (p + 1)-form x_{0}dx_{1} … dx_{p+1} over T̂𝒜:

−\frac{Γ(q + \frac{3}{2})}{(p + 1)!} ∑_{i=1}^{p+1} τ♮(ρ_{∗}x_{0}[F, ρ_{∗}x_{1}] … d(ρ_{∗}x_{i}) … [F, ρ_{∗}x_{p+1}])

Denote as before ũ = ρ_{∗}û the invertible image in (T̂ℂ ˆ⊗ℐ ℳ̂_{+}^{s}[0, 1]_{x})^{+}, the associated idempotent f̃ = u^{−1}p_{0}ũ ∈ M_{2}(T̂ℂ ˆ⊗ℐ ℛ̂[0, 1]_{x})^{+}, and the Maurer-Cartan form ω̃ = u^{−1}sũ. One gets

ch^{p}(ρ) (Tr[2q] ω̂) =

−\frac{\sqrt{2πi}}{2}Tr♮[2q] ( ∑_{i=0}^{2q}(f̃− p_{0})^{i}[p_{0}, ω̃](f̃− p_{0})^{2q−i}u^{−1}dũ + (f̃− p_{0})^{p}dω̃) .

Now observe that ũ_{x=0} = ũ(g_{0}) and ũ_{x=1} = ũ(g_{1}), so that after integration over the current \frac{1}{\sqrt{2πi}} ∫_{x=0}^{1} we get the identity (recall ch^{p}(ρ) is odd)

−ch^{p}(ρ) · cs_{0}^{2q}(ĝ) + h_{1}^{2q}(ũ(g_{1})) − h_{1}^{2q}(ũ(g_{0}))

= \frac{1}{\sqrt{2πi}} ∫_{0}^{1} ch^{p}(ρ) (Tr[2q] ω̂) + \frac{1}{2} ∫_{0}^{1} s Tr♮[2q] (f̃− p_{0})^{p}u^{−1}dũ

= ∫_{0}^{1} Tr♮[2q] ( ∑_{i=1}^{q}(f̃− p_{0})^{2i−1}[p_{0}, ω̃](f̃− p_{0})^{2(q−i)+1}u^{−1}dũ − (f̃− p_{0})^{p}dω̃)

On the other hand, let us calculate the Chern-Simons form associated to the idempotent f̃,

cs_{1}^{2q}(f̃) = ∫_{0}^{1} Tr♮[2q] (−2f̃+ 1) ∑_{i=0}^{q}(f̃− p_{0})^{2i}sf̃(f̃− p_{0})^{2(q−i)}df̃ ,

in terms of ω̃. Since by definition f̃ = u^{−1}p_{0}ũ, the structure equation sf̃ = [f̃, ω̃] follows and one finds

cs_{1}^{2q}(f̃) = −♮d ∫_{0}^{1} Tr[2q] (f̃− p_{0})^{p}ω̃

+ ∫_{0}^{1} Tr♮[2q] ( ∑_{i=1}^{q}(f̃− p_{0})^{2i−1}[p_{0}, ω̃](f̃− p_{0})^{2(q−i)+1}u^{−1}dũ − (f̃− p_{0})^{p}dω̃)

Thus holds the fundamental relation

−ch^{p}(ρ) · cs_{0}^{2q}(ĝ) + h_{1}^{2q}(ũ(g_{1})) − h_{1}^{2q}(ũ(g_{0})) ≡ cs_{1}^{2q}(f̃) mod ♮d .

Now let f̂ ∈ M_{2}(T̂ℂ ˆ⊗ℐ ℛ̂[0, 1]_{x}[0, 1]_{t})^{+} be an idempotent interpolation between f̂_{t=0} = f̃ and f̂_{t=1} = p_{0} + ê ⊗ ({ \atop ρ_{!}(g)} − p_{0}), with the property that it restricts to f̂(g_{0}) for x = 0 and to f̂(g_{1}) for x = 1. The projection of f̂ to the algebra M_{2}(ℂ ˆ⊗ℐ ℬ[0, 1]_{x}[0, 1]_{t})^{+} may be chosen constant with respect to t. In the proof of Lemma 4.6 we established the following identity at any point (x, t) ∈ [0, 1]^{2}:

\frac{∂}{∂t}(Tr♮[2q] (−2f̂+ 1) ∑_{i=0}^{q}(f̂− p_{0})^{2i}sf̂(f̂− p_{0})^{2(q−i)}df̂)

≡ s(Tr♮[2q] (−2f̂+ 1) ∑_{i=0}^{q}(f̂− p_{0})^{2i} \frac{∂f̂}{∂t}(f̂− p_{0})^{2(q−i)}df̂) mod ♮d ,

and integration over the square [0, 1]^{2} yields

cs_{1}^{2q}(f̂_{t=1}) − cs_{1}^{2q}(f̂_{t=0}) ≡ cs_{1}^{2q}(f̂_{x=1}) − cs_{1}^{2q}(f̂_{x=0}) mod ♮d .

Since f̂_{x=0} = f̂(g_{0}) and f̂_{x=1} = f̂(g_{1}) we calculate, modulo ♮d in the complex X_{n−p−1}(ℛ, 𝒥)

−ch^{p}(ρ) · (θ_{1} − θ_{0}) + h_{1}^{2q}(ũ(g_{1})) − h_{1}^{2q}(ũ(g_{0})) + cs_{1}^{2q}(f̂(g_{1})) − cs_{1}^{2q}(f̂(g_{0}))

≡ −ch^{p}(ρ) · cs_{0}^{2q}(ĝ) + h_{1}^{2q}(ũ(g_{1})) − h_{1}^{2q}(ũ(g_{0})) + cs_{1}^{2q}(f̂_{t=1}) − cs_{1}^{2q}(f̂_{t=0})
≡ cs_{1}^{2q}(f̃) + cs_{1}^{2q}(p_{0} + ê ⊗ ({ \atop ρ_{!}(g)} − p_{0})) − cs_{1}^{2q}(f̃)
≡ cs_{1}^{2q}({ \atop ρ_{!}(g)}) mod ♮d

as wanted. Hence ρ_{!}(ĝ_{0}, θ_{0}) and ρ_{!}(ĝ_{1}, θ_{1}) are equivalent and the map ρ_{!} : MK_{n}^{ℐ} (𝒜) → MK_{n−p}^{ℐ}(ℬ) for n = 2k + 1 and p = 2q + 1 is well-defined. Its compatibility with the map ρ_{!} on topological K-theory is obvious. Concerning its compatibility with the map ch^{p}(ρ) : HC_{n−1}(𝒜) → HC_{n−p−1}(ℬ), we should take care of a minus sign which shows that the middle square of (83) is actually anticommutative; this has to be so because all the maps involved in this square are of odd degree. Hence the diagram (83) is graded commutative. The invariance of ρ_{!} with respect to conjugation of quasihomomorphisms is proved exactly as in case i), by decomposing ρ_{!} as the pushforward map φ_{!} : MK_{n}^{ℐ} (𝒜) → MK_{n}^{ℐ} (ℰ_{+}^{s}) induced by the homomorphism φ : T̂𝒜 → T̂ℰ_{+}^{s}, followed by the map MK_{n}^{ℐ} (ℰ_{+}^{s}) → MK_{n−p}^{ℐ}(ℬ) associated with the natural p-summable quasihomomorphism of odd degree ℰ_{+}^{s} → ℰ^{s} ▷ ℐ^{s}ℬ. Also the compatibility with the negative Chern character is easily established.

iv) n = 2k is even and p = 2q + 1 is odd. As in case ii) we pass to the suspensions of 𝒜 and ℬ and work with the group MK_{n}^{′ℐ} (𝒜). Hence a multiplicative K-theory class of degree n over 𝒜 is represented by a pair (ĝ, θ) of an invertible ĝ ∈ (𝒦 ˆ⊗ℐ ST̂𝒜)^{+} and an odd chain θ ∈ X_{n−1}(T 𝒜 , J𝒜) such that cs_{0}^{2q}(ĝ) = \overline{b}θ. We are thus led to build a morphism

ρ_{!} : MK_{n}^{′ℐ} (𝒜) → MK_{n−p}^{′ℐ}(ℬ) ,

where the group MK_{n−p}^{′ℐ}(ℬ), for n − p odd, is represented by pairs (ê, θ) of idempotent ê ∈ M_{2}(𝒦 ˆ⊗ℐ Sℛ̂)^{+} and chain of even degree θ ∈ X_{n−p−1}(ℛ, 𝒥) such that cs_{1}^{2q}(ê) = ♮dθ. Note that the parity of MK_{n−p}^{′ℐ}(ℬ) is even. We already established the needed formulas in case iii): let g ∈ (ℐ S𝒜)^{+} be any invertible with canonical lift ĝ ∈ (ℐ ST̂𝒜)^{+}. One can write

−ch^{p}(ρ) · cs_{0}^{2q}(ĝ) = cs_{1}^{2q}(f̃) − ♮dk_{0}^{2q}(û)

with the invertible û = 1 + e ⊗ (ĝ − 1) ∈ (T̂ℂ ˆ⊗ℐ ST̂𝒜)^{+}, the idempotent f̃ = u^{−1}p_{0}ũ ∈ M_{2}(T̂ℂ ˆ⊗ℐ Sℛ̂)^{+} where ũ = ρ_{∗}û, and the chain k_{0}^{2q}(û) =

− ∫_{0}^{1} Tr[2q] (f̃ − p_{0})^{p}ω̃ where ω̃ = u^{−1}sũ. Let { \atop ρ_{!}(g)} ∈ M_{2}(ℐ Sℛ̂)^{+} be any

idempotent lift of ρ_{!}(g) = ρ(g)^{−1}p_{0}ρ(g), and f̂ ∈ M_{2}(T̂ℂ ˆ⊗ℐ Sℛ̂[0, 1])^{+} be an idempotent interpolation between f̂_{0} = f̃ and f̂_{1} = p_{0} + ê⊗ ({ \atop ρ_{!}(g)} − p_{0}). Then one has by means of the higher transgressions (see the proof of Lemma 4.6)

♮dcs_{0}^{′2q}(f̂) = cs_{1}^{2q}({ \atop ρ_{!}(g)}) − cs_{1}^{2q}(f̃) ,

with cs_{0}^{′2q}(f̂) defined modulo \overline{b}. Therefore if (ĝ, θ) represents a class in MK_{n}^{′ℐ} (𝒜) we define its pushforward as the multiplicative K-theory class over ℬ

ρ_{!}(ĝ, θ) = ({ \atop ρ_{!}(g)} , ch^{p}(ρ) · θ + k_{0}^{2q}(û) + cs_{0}^{′2q}(f̂)) ∈ MK_{n−p}^{′ℐ}(ℬ)  (88)

with the chain ch^{p}(ρ)·θ+k_{0}^{2q}(û)+cs_{0}^{′2q}(f̂) of even degree sitting in the quotient complex X_{n−p−1}(ℛ, 𝒥).

Example 6.4 For ℬ = ℂ a quasihomomorphism 𝒜 → ℰ^{s} ▷ℐ^{s} induces a map MK_{n}^{ℐ} (𝒜) → MK_{n−p}^{ℐ}(ℂ). Thus if ℐ is a Schatten ideal on a Hilbert space, Example 5.6 yields index maps or regulators, depending on the degrees:

MK_{n}^{ℐ} (𝒜) → ℤ  if n ≤ p , n ≡ p mod 2 ,
MK_{n}^{ℐ} (𝒜) → ℂ^{×}  if n > p , n ≡ p + 1 mod 2 .

7 Assembly maps and crossed products

In this section we illustrate the general theory of secondary characteristic classes with the specific example of crossed product algebras, and build an ”assembly map” for multiplicative K-theory modelled on the Baum-Connes construction [1].

Let 𝒜 be a unital Fréchet m-algebra and Γ a countable discrete group acting on 𝒜 from the right by automorphisms. The action of an element γ ∈ Γ on a ∈ 𝒜 reads a^{γ}. We impose the action to be almost isometric in the following sense: for each submultiplicative seminorm ∥ · ∥_{α} on 𝒜 there exists a constant C_{α} such that

∥a^{γ}∥_{α} ≤ C_{α}∥a∥_{α}  ∀ a ∈ 𝒜 , γ ∈ Γ .  (89)

The algebraic tensor product 𝒜 ⊗ ℂΓ is identified with the space of 𝒜 -valued functions with finite support over Γ. Thus any element of 𝒜 ⊗ ℂΓ is a finite linear combination of symbols aγ^{∗} with a_{γ} ∈ 𝒜 and γ ∈ Γ. The star refers to a contravariant notation. We endow 𝒜 ⊗ ℂΓ with the crossed product defined in terms of symbols by

(a_{1}γ_{1}^{∗})(a_{2}γ_{2}^{∗}) = a_{1}(a_{2})^{γ_{1}}(γ_{2}γ_{1})^{∗} ∀ a_{i} ∈ 𝒜 , γ_{i} ∈ Γ .

The crossed product algebra 𝒜 ⋊Γ is an adequate completion of the above space consisting of 𝒜 -valued functions with “rapid decay” over Γ. This requires to fix once and for all a right-invariant distance function d : Γ × Γ → ℝ_{+}. Endow the space 𝒜 ⊗ ℂΓ with the seminorms

∥b∥_{α,β} = C_{α} ∑_{γ∈Γ} σ_{β}(γ)∥b(γ)∥_{α} ∀ b ∈ 𝒜 ⊗ ℂΓ ,

where the ℝ_{+}-valued function σ_{β}(γ) := (1 + d(γ, 1))^{β}, for β ≥ 0, fulfills the property σ_{β}(γ_{1}γ_{2}) ≤ σ_{β}(γ_{1})σ_{β}(γ_{2}). One easily checks that ∥ · ∥_{α,β} is submultiplicative with respect to the crossed product, hence the completion of 𝒜 ⊗ ℂΓ for the family of seminorms (∥ · ∥_{α,β}) yields a Fréchet m-algebra 𝒜 ⋊ Γ. Multiplicative K-theory classes of 𝒜 ⋊ Γ may be obtained by adapting the assembly map construction of [1]. The idea is to replace the noncommutative space 𝒜 ⋊ Γ by a more classical space, for which the secondary invariants are presumably easier to describe. Consider a compact Riemannian manifold M without boundary, and let P ⟶^{Γ} M be a Γ-covering. Γ acts on P from the left by deck transformations. Denote by

𝒜_{P} := C^{∞}(P; 𝒜)^{Γ}

the algebra of Γ-invariant smooth 𝒜 -valued functions over P: any function a ∈ 𝒜_{P} verifies a(γ^{−1} · x) = (a(x))^{γ}, ∀x ∈ P, γ ∈ Γ. Thus 𝒜_{P} is the algebra of smooth sections of a non-trivial bundle with fibre 𝒜 over M. It can be represented as a subalgebra of matrices over C^{∞}(M) ˆ⊗(𝒜 ⋊Γ) = C^{∞}(M; 𝒜 ⋊Γ) as follows. Let (U_{i}), i = 1, …, m be a finite open covering of M trivializing the bundle P, via a set of sections s_{i} : U_{i} → P and locally constant transition functions γ_{ij} : U_{i} ∩ U_{j} → Γ:

γ_{ij}γ_{jk} = γ_{ik} over U_{i} ∩ U_{j} ∩ U_{k} , s_{i}(x) = γ_{ij} · s_{j}(x) ∀x ∈ U_{i} ∩ U_{j} .

Choose a partition of unity c_{i} ∈ C^{∞}(M) relative to this covering: supp c_{i} ⊂ U_{i} and ∑_{i=1}^{m} c_{i}(x)^{2} = 1. From these data consider the homomorphism ρ : 𝒜_{P} → M_{m}(C^{∞}(M) ˆ⊗(𝒜 ⋊ Γ)) sending an element a ∈ 𝒜_{P} to the m × m matrix ρ(a) whose components, as (𝒜 ⋊ Γ)-valued functions over M, read

ρ(a)_{ij}(x) := c_{i}(x)c_{j}(x)a(s_{i}(x))γ_{ji}^{∗} ∀ i, j = 1, …, m , ∀x ∈ M .

Of course ρ depends on the choice of trivialization (U_{i}, s_{i}) and partition of unity (c_{i}), but different choices are related by conjugation in a suitably large matrix algebra. Indeed, if (U′_{α}, s′_{α}), α = 1, …, μ denotes another trivialization with transition functions γ′_{αβ} and partition of unity (c′_{α}), we get a corresponding homomorphism ρ′ : 𝒜_{P} → M_{μ}(C^{∞}(M) ˆ⊗(𝒜 ⋊ Γ)). Introduce the rectangular matrices u, v over C^{∞}(M) ˆ⊗(𝒜 ⋊ Γ) with components

u_{iα}(x) = c_{i}(x)c′_{α}(x)γ_{αi}^{∗} , v_{αi}(x) = c′_{α}(x)c_{i}(x)γ_{iα}^{∗} ,

(recall 𝒜 is unital by hypothesis hence ℂΓ ⊂ 𝒜 ⋊ Γ), where the mixed transition functions γ_{iα}, γ_{αi} are defined by s_{i}(x) = γ_{iα} · s′_{α}(x) and s′_{α}(x) = γ_{αi} · s_{i}(x) for any x ∈ U_{i} ∩ U′_{α}. Then uv and vu are idempotent square matrices, and for any element a ∈ 𝒜_{P} one has ρ(a) = uρ′(a)v and ρ′(a) = vρ(a)u. Moreover, the invertible square matrix of size m + μ

W = (\begin{matrix}1 − uv & −u \\ v & 1 − vu \end{matrix}) , W^{−1} = (\begin{matrix}1 − uv & u \\ −v & 1 − vu \end{matrix})

verifies W^{−1}( \begin{matrix}ρ(a) & 0 \\ 0 & 0 \end{matrix} )W = ( \begin{matrix}0 & 0 \\ 0 & ρ′(a) \end{matrix} ), which shows that the homomorphisms ρ and ρ′ are stably conjugate. In order to get a quasihomomorphism from 𝒜_{P} to 𝒜 ⋊ Γ, we need a K-cycle for the Fréchet algebra C^{∞}(M) (see Example 3.3). By a standard procedure [4, 5], such a K-cycle D may be constructed from an elliptic pseudodifferential operator or a Toeplitz operator over M. We shall suppose that D is of parity p mod 2, and of summability degree p+1 (even case) or p (odd case). Hence (see Example 3.3) in the even case one has an infinite-dimensional separable Hilbert space H with two continuous representations C^{∞}(M) ⇉ ℒ = ℒ (H) which agree modulo the Schatten ideal ℐ = ℒ^{p+1}(H), whereas in the odd case the algebra C^{∞}(M) is represented in the matrix algebra ( \begin{matrix}ℒ & ℐ \\ ℐ & ℒ \end{matrix} ) with ℐ = ℒ^{p}(H). Therefore, upon choosing an isomorphism H ≅ H ˆ⊗ℂ^{m} the composition of ρ : 𝒜_{P} → M_{m}(C^{∞}(M) ˆ⊗(𝒜 ⋊Γ)) with the Hilbert space representation induced by the K-cycle D leads to a quasihomomorphism of parity p mod 2

ρ_{D} : 𝒜_{P} → ℰ^{s} ▷ ℐ^{s} ˆ⊗(𝒜 ⋊ Γ) ,

with intermediate algebra ℰ = ℒ ˆ⊗(𝒜 ⋊Γ) (or (ℒ ⋉ℐ) ˆ⊗(𝒜 ⋊Γ), see Example 3.4). Note that ℒ and ℐ may be replaced by other suitable operator algebras, if needed. From the discussion above we see that ρ_{D} depends only on D up to conjugation by an invertible element W ∈ ℰ_{+}^{s}. Taking into account the Chern characters in negative and periodic cyclic homology, the Riemann-Roch-Grothendieck Theorem 6.3 thus yields cube diagrams of the following kind:

Corollary 7.1 A K-cycle D over C^{∞}(M) as above yields for any integer n ∈ ℤ a commutative diagram

[figure: K_{n}^{top}(ℐ 𝒜_{P}); K_{n−p}^{top}(ℐ (𝒜 ⋊ Γ)); MK_{n}^{ℐ} (𝒜_{P}); MK_{n−p}^{ℐ}(𝒜 ⋊ Γ); HP_{n}(𝒜_{P}); HP_{n−p}(𝒜 ⋊ Γ); HN_{n}(𝒜_{P}); HN_{n−p}(𝒜 ⋊ Γ)]

where the horizontal arrows are induced by the quasihomomorphism ρ_{D} : 𝒜_{P} → ℰ^{s} ▷ ℐ^{s} ˆ⊗(𝒜 ⋊ Γ).

The background square describes the topological side of the Riemann-Roch-Grothendieck theorem, namely the compatibility between the push-forward in topological K-theory and the bivariant Chern character in periodic cyclic homology. One may choose D as a representative of the fundamental class in the K-homology of M. If moreover M is a model for the classifying space BΓ, one may choose P as the universal bundle EΓ. For torsion-free groups Γ the morphism K_{n}^{top}(ℐ 𝒜_{P}) → K_{n−p}^{top}(ℐ (𝒜 ⋊ Γ)) thus obtained is related to the Baum-Connes assembly map [1] and exhausts many (in some cases, all the) interesting topological K-theory classes of 𝒜 ⋊ Γ. The foreground square provides a lifting of the topological situation at the level of multiplicative K-theory and negative cyclic homology, i.e. secondary characteristic classes. Hence a part of MK_{∗}^{ℐ} (𝒜 ⋊ Γ) may be obtained by direct images of multiplicative K-theory over 𝒜_{P}. Note that in contrast to the topological situation, the push-forward map in multiplicative K-theory does not exhaust all the interesting classes over 𝒜 ⋊ Γ.

Let us now deal with the case 𝒜 = C^{∞}(N), for a compact smooth Riemannian manifold N, endowed with a left action of Γ by diffeomorphisms. We provide 𝒜 with its usual Fréchet topology, and condition (89) forces the Γ-action be ”almost isometric” on N. The crossed product 𝒜 ⋊ Γ is then isomorphic to a certain convolution algebra of functions over the smooth étale groupoid Γ ⋉ N, describing a highly noncommutative space when the action of Γ is not proper. The commutative algebra 𝒜_{P} is the subalgebra of smooth Γ-invariant functions a ∈ C^{∞}(P × N), a(γ · x, γ · y) = a(x, y) for any (x, y) ∈ P × N, and is thus isomorphic to the algebra of smooth functions over the (compact) quotient manifold Q = Γ\(P × N).

The problem is therefore reduced to the computation of secondary invariants for the classical space Q. The cyclic homology of 𝒜_{P} = C^{∞}(Q) has been determined by Connes [4] and is computable from the de Rham complex of differential forms over Q. We will see that the multiplicative K-theory MK_{∗}^{ℐ} (𝒜_{P}) is closely related (though not isomorphic) to Deligne cohomology. We first recall some definitions. Let Ω^{n}(Q) denote the space of complex, smooth differential n-forms over Q, d the de Rham coboundary, Z_{dR}^{n}(Q) = Ker(d : Ω^{n} → Ω^{n+1}) the space of closed n-forms and B_{dR}^{n}(Q) = Im(d : Ω^{n−1} → Ω^{n}) the space of exact n-forms. By de Rham’s theorem, the de Rham cohomology H_{dR}^{n}(Q) = Z_{dR}^{n}(Q)/B_{dR}^{n}(Q) is isomorphic to the Čech cohomology of Q with complex coefficients Ȟ^{n}(Q; ℂ). For any half-integer q we define the additive group ℤ(q) := (2πi)^{q}ℤ ⊂ ℂ (the square root of 2πi must be chosen consistentl y with the Chern character on K_{1}^{top}). Let \underline{Ω}^{k} denote the sheaf of differential k-forms over Q and consider for any n ∈ ℕ the complex of sheaves

0 ⟶ \underline{ℤ}(n/2) ⟶ \underline{Ω}^{0} ⟶^{d} \underline{Ω}^{1} ⟶^{d} … ⟶^{d} \underline{Ω}^{n−1} ⟶ 0  (90)

where the constant sheaf \underline{ℤ}(n/2) sits in degree 0 and \underline{Ω}^{k} in degree k + 1. The map \underline{ℤ}(n/2) → \underline{Ω}^{0} is induced by the natural inclusion of constant functions into complex-valued functions. By definition the (smooth) Deligne cohomology H_{𝒟}^{n}(Q; ℤ(n/2)) is the hyperhomology of (90) in degree n. The natural projection onto the constant sheaf \underline{ℤ}(n/2) yields a welldefined map from H_{𝒟}^{n}(Q; ℤ(n/2)) to the Čech cohomology with integral coefficients Ȟ^{n}(Q; ℤ(n/2)). On the other extreme, the de Rham coboundary d : \underline{Ω}^{n−1} → \underline{Ω}^{n} sends a Deligne n-cocycle to a globally defined closed n-form over Q, called its curvature, which only depends on the Deligne cohomology class. It follows from the definitions that the image of the curvature in de Rham cohomology coincides with the complexification of the Čech cohomology class of the Deligne cocycle. One thus gets a commutative diagram in any degree n

\xymatrix{H_{𝒟}^{n}(Q; ℤ(n/2)) \ar[r] \ar[d]_{d} & Ȟ^{n}(Q; ℤ(n/2)) \ar[d]^{⊗ℂ} \\ Z_{dR}^{n}(Q) \ar[r] & H_{dR}^{n}(Q)}  (91)

This has to be compared with the commutative square involving the multiplicative and topological K-theories of the algebra 𝒜_{P} = C^{∞}(Q), with their Chern characters:

\xymatrix{MK_{n}^{ℐ} (𝒜_{P}) \ar[r] \ar[d] & K_{n}^{top}(ℐ 𝒜_{P}) \ar[d] \\ HN_{n}(𝒜_{P}) \ar[r] & HP_{n}(𝒜_{P})}  (92)

In fact one can construct, at least in low degrees n, an explicit transformation from Deligne cohomology to multiplicative K-theory, and the curvature morphism captures the lowest degree part of the negative Chern character. Let us explain this with some details. Firstly, it is well-known that the bottom line of (91) is included as a direct summand in the bottom line of (92). Since we deal essentially with the X-complex description of cyclic homology (section 2), we recall how the latter is related to the de Rham cohomology of Q. Choose the universal free extension 0 → J𝒜_{P} → T 𝒜_{P} → 𝒜_{P} → 0. The cyclic homology of 𝒜_{P} is computed by the X-complex X(T̂𝒜_{P}) of the pro-algebra T̂𝒜_{P} = \underleftarrow{lim}_{n} T 𝒜_{P} /(J𝒜_{P})^{n}, together with its filtration by the subcomplexes F^{n}X̂(T 𝒜_{P} , J𝒜_{P}). As a pro-vector space, X(T̂𝒜_{P}) is isomorphic to the completed space of noncommutative differential forms Ω̂𝒜_{P}, and the J𝒜_{P} -adic filtration coincides with the Hodge filtration F^{n}Ω̂𝒜_{P}. A canonical chain map X(T̂𝒜_{P}) → Ω^{∗}(Q) is given by projecting noncommutative differential forms to commutative ones, up to a rescaling:

Ω^{n}𝒜_{P} ∋ a_{0}da_{1} … da_{n} → λ_{n}a_{0}da_{1} … da_{n} ∈ Ω^{n}(Q)

with λ_{n} = (−)^{k} \frac{k!}{(2k)!} if n = 2k and λ_{n} = (−)^{k} \frac{k!}{(2k+1)!} if n = 2k + 1. These factors are fixed in order to get exactly a chain map. Clearly it sends the Hodge filtration of Ω̂𝒜_{P} onto the natural filtration by degree on Ω^{∗}(Q). The following proposition is a reformulation of Connes’ version of the Hochschild-Kostant-Rosenberg theorem [4].

Proposition 7.2 The chain map X(T̂𝒜_{P}) → Ω^{∗}(Q) is a homotopy equivalence compatible with the filtrations. Hence follow the isomorphisms

HP_{n}(𝒜_{P}) = ⨁_{k∈ℤ} H_{dR}^{n+2k}(Q) ,

HC_{n}(𝒜_{P}) = \frac{Ω^{n}(Q)}{B_{dR}^{n}(Q)} ⊕ ⨁_{k<0} H_{dR}^{n+2k}(Q) ,  (93)
HN_{n}(𝒜_{P}) = Z_{dR}^{n}(Q) ⊕ ⨁_{k>0} H_{dR}^{n+2k}(Q) .

Of course the injections Z_{dR}^{n}(Q) → HN_{n}(𝒜_{P}) and H_{dR}^{n}(Q) → HP_{n}(𝒜_{P}) are compatible with the forgetful maps Z_{dR}^{n}(Q) → H_{dR}^{n}(Q) and HN_{n}(𝒜_{P}) → HP_{n}(𝒜_{P}). It is useful to calculate the image of the Chern character of idempotents and invertibles under the chain map X(T̂𝒜_{P}) → Ω^{∗}(Q). Let e ∈ M_{2}(𝒦 ˆ⊗𝒜_{P})^{+} be an idempotent such that e − p_{0} ∈ M_{2}(𝒦 ˆ⊗𝒜_{P}), and let ê ∈ M_{2}(𝒦 ˆ⊗T̂𝒜_{P})^{+} be its canonical lift. The image of the Chern character ch_{0}(ê) = Tr(ê − p_{0}) ∈ X(T̂𝒜_{P}) is the differential form of even degree

ch_{dR}(e) = Tr(e − p_{0}) + ∑_{k≥1} \frac{(−)^{k}}{k!} Tr((e − \frac{1}{2})(dede)^{k}) ∈ Ω^{+}(Q) .

Let g ∈ (𝒦 ˆ⊗𝒜_{P})^{+} be an invertible such that g − 1 ∈ 𝒦 ˆ⊗𝒜_{P}, and let ĝ ∈ (𝒦 ˆ⊗T̂𝒜_{P})^{+} be its canonical lift. The image of the Chern character ch_{1}(ĝ) = \frac{1}{\sqrt{2πi}}♮ĝ^{−1}dĝ ∈ X(T̂𝒜_{P}) is the differential form of odd degree

ch_{dR}(g) = \frac{1}{\sqrt{2πi}} ∑_{k≥0}(−)^{k} \frac{k!}{(2k + 1)!} Tr(g^{−1}dg(dg^{−1}dg)^{k}) ∈ Ω^{−}(Q) .

We now construct explicit morphisms H_{𝒟}^{n}(Q; ℤ(n/2)) → MK_{n}^{ℐ} (𝒜_{P}) in degrees n = 0, 1, 2. In fact the ideal ℐ is irrelevant and the previous morphisms factor through the multiplicative group MK_{n}(𝒜_{P}) := MK_{n}^{ℂ}(𝒜_{P}) associated to the 1-summable algebra ℂ. Then choosing any rank one injection ℂ → ℐ induces a unique map MK_{n}(𝒜_{P}) → MK_{n}^{ℐ} (𝒜_{P}) by virtue of Lemma 5.5.

n = 0: Then ℤ(0) = ℤ and the complex 0 → \underline{ℤ}(0) → 0 calculates the Čech cohomology of Q with coefficients in ℤ. Hence H_{𝒟}^{0} (Q; ℤ(0)) = Ȟ^{0}(Q; ℤ) is the additive group of ℤ-valued locally constant functions over Q. The map

H_{𝒟}^{0} (Q; ℤ(0)) → MK_{0}(𝒜_{P}) ≅ K_{0}^{top}(𝒜_{P})  (94)

associates to such a function f the K-theory class of the trivial complex vector bundle of rank f over Q.

n = 1: Then ℤ(1/2) = \sqrt{2πi} ℤ and H_{𝒟}^{1} (Q; ℤ(1/2)) is the hyperhomology in degree 1 of the complex 0 → \underline{ℤ}(1/2) → \underline{Ω}^{0} → 0. Choose a good covering (U_{i}) of Q. A Deligne 1-cocycle is given by a collection (f_{i}, n_{ij}) of smooth functions f_{i} : U_{i} → ℂ and locally constant functions n_{ij} : U_{i} ∩ U_{j} → ℤ related by the descent equations

f_{i} − f_{j} = \sqrt{2πi} n_{ij} over U_{i} ∩ U_{j} , n_{jk} − n_{ik} + n_{ij} = 0 over U_{i} ∩ U_{j} ∩ U_{k} .

The cocycle is trivial if the f_{i}’s are ℤ(1/2)-valued. Taking the exponentials g_{i} = exp(\sqrt{2πi}f_{i}) one gets invertible smooth functions which agree on the overlaps U_{i} ∩ U_{j}, hence define a global invertible function g over Q. The latter is equal to 1 exactly when the cocycle (f_{i}, n_{ij}) is trivial. Hence H_{𝒟}^{1} (Q; ℤ(1/2)) is the multiplicative group C^{∞}(Q)^{×} of complex-valued invertible functions over Q. On the other hand, the elements of MK_{1}(𝒜_{P}) are represented by pairs (ĝ, θ) of an invertible ĝ ∈ (𝒦 ˆ⊗T̂𝒜_{P})^{+} and a cochain θ ∈ X_{0}(T 𝒜_{P} , J𝒜_{P}) ≅ C^{∞}(Q). We get a map

H_{𝒟}^{1} (Q; ℤ(1/2)) ≅ C^{∞}(Q)^{×} → MK_{1}(𝒜_{P})  (95)

by sending an invertible g ∈ C^{∞}(Q)^{×} to the multiplicative K-theory class of (ĝ, 0), with ĝ the canonical lift of g (to be precise one should replace g by 1 + (g − 1_{𝒜_{P}} ) ∈ (𝒜_{P})^{+}). This map identifies the curvature morphism H_{𝒟}^{1} (Q; ℤ(1/2)) → Z_{dR}^{1}(Q) with the lowest degree part of the negative Chern character ch_{1} : MK_{1}(𝒜_{P}) → HN_{1}(𝒜_{P}). Indeed the curvature of an element g ∈ C^{∞}(Q)^{×} is by definition the closed one-form

df_{i} = \frac{1}{\sqrt{2πi}} g_{i}^{−1}dg_{i} = \frac{1}{\sqrt{2πi}} g^{−1}dg ∀i ,

globally defined over Q. But this coincides with the Z_{dR}^{1}(Q)-component of the negative Chern character ch_{1}(ĝ, 0).

n = 2: Then ℤ(1) = 2πi ℤ and H_{𝒟}^{2} (Q; ℤ(1)) is the hyperhomology in degree 2 of the complex 0 → \underline{ℤ}(1) → \underline{Ω}^{0} → \underline{Ω}^{1} → 0. A Deligne cocycle relative to the finite good covering (U_{i}), i = 1, …, m is a collection (A_{i}, f_{ij}, n_{ijk}) of one-forms A_{i} over U_{i}, smooth functions f_{ij} : U_{i} ∩ U_{j} → ℂ and locally constant functions n_{ijk} : U_{i} ∩ U_{j} ∩ U_{k} → ℤ, subject to the descent equations

A_{i} − A_{j} = df_{ij} , −f_{jk} + f_{ik} − f_{ij} = 2πi n_{ijk} , n_{jkl} − n_{ikl} + n_{ijl} − n_{ijk} = 0 .

Equivalently, passing to the exponentials g_{ij} = exp f_{ij} a cocycle is a collection (A_{i}, g_{ij}) such that A_{i} − A_{j} = g_{ij}^{−1}dg_{ij} and g_{ij}g_{jk} = g_{ik}. Two cocycles (A_{i}, g_{ij}) and (A′_{i}, g′_{ij}) are cohomologous iff there exists a collection of smooth invertible functions (gauge transformations) α_{i} : U_{i} → ℂ^{×} such that

A′_{i} = A_{i} + α_{i}^{−1}dα_{i} , g′_{ij} = α_{i}g_{ij}α_{j}^{−1} .

One sees that a Deligne cohomology class is nothing else but a complex line bundle over Q, described by the smooth transition functions g_{ij} : U_{i}∩U_{j} → ℂ^{×}, together with a connection given locally by the one-forms A_{i}, up to gauge transformation. Hence

H_{𝒟}^{2} (Q; ℤ(1)) ={isomorphism classes of complex line bundles with connection}

The group law is the tensor product of line bundles with connections. The curvature morphism H_{𝒟}^{2} (Q; ℤ(1)) → Z_{dR}^{2}(Q) maps a cocycle (A_{i}, g_{ij}) to the globally defined closed two-form dA_{i} ∀i, i.e. the curvature of the connection of the corresponding line bundle. The construction of the morphism from Deligne cohomology to multiplicative K-theory requires to fix a partition of unity (c_{i}) relative to the finite covering: supp c_{i} ⊂ U_{i} and ∑_{i} c_{i}(x)^{2} = 1 ∀x ∈ Q. Given a Deligne cocycle (A_{i}, g_{ij}), we construct the idempotent e_{+} ∈ M_{m}(C^{∞}(Q)) of rank 1 whose matrix elements are the functions

(e_{+})_{ij} = c_{i}g_{ij}c_{j} ∈ C_{c}^{∞}(U_{i} ∩ U_{j}) ,

and let e_{−} = 1_{𝒜_{P}} be the unit of C^{∞}(Q) (the constant function 1 over Q). Define e ∈ M_{2}(𝒜_{P})^{+} as the idempotent matrix ( \begin{matrix}1−e_{−} & 0 \\ 0 & e_{+} \end{matrix} ). The 0th degree of the Chern character ch_{dR}(e) is Tr(e − p_{0}) = Tr(e_{+}) − 1_{𝒜_{P}} = 0, so that the class of e in K_{0}^{top}(𝒜_{P}) is a virtual bundle of rank 0. To get a multiplicative K-theory class (ê, θ) ∈ MK_{2}(𝒜_{P}), we must adjoin to the canonical idempotent lift ê ∈ M_{2}(T̂𝒜_{P})^{+} an odd cochain θ ∈ X_{1}(T 𝒜_{P} , J𝒜_{P}) ≅ X(𝒜_{P}). Since 𝒜_{P} is the commutative algebra of smooth functions over a compact manifold, its X-complex reduces to the de Rham complex of Q truncated in degrees ≥ 2, X(𝒜_{P}) : C^{∞}(Q) →^{d} Ω^{1}(Q). The fact that e is of virtual rank zero insures that any choice of one-form θ ∈ Ω^{1}(Q) satisfies the correct transgression relation ch_{0}(ê) = \overline{b}θ = 0 in the complex X(𝒜_{P}). We s et

θ = − ∑_{i} c_{i}^{2}A_{i} ∈ Ω^{1}(Q) .

Lemma 7.3 The assignement (A_{i}, g_{ij}) ↦ (ê, θ) yields a well-defined morphism

H_{𝒟}^{2} (Q; ℤ(1)) → MK_{2}(𝒜_{P}) ,  (96)

and the curvature of (A_{i}, g_{ij}) corresponds to the Z_{dR}^{2}(Q) component of the negative Chern character ch_{2}(ê, θ) ∈ HN_{2}(𝒜_{P}).

Proof: We have to show that the multiplicative K-theory class of (ê, θ) only depends on the Deligne cohomology class of (A_{i}, g_{ij}). Thus let (A′_{i}, g′_{ij}) be another representative, with A′_{i} = A_{i} +α_{i}^{−1}dα_{i} and g′_{ij} = α_{i}g_{ij}α_{j}^{−1}. This yields a new pair (ê′, θ′) with (e′_{+})_{ij} = c_{i}g′_{ij}c_{j} and θ′ = − ∑_{i} c_{i}^{2}A′_{i}. We show that (ê, θ) and (ê′, θ′) represent the same class in MK_{2}(𝒜_{P}) by using the following general fact: if e and e′ = W^{−1}eW are conjugate by an invertible W, then (ê, θ) is equivalent to (ê′, θ + cs_{1}(f)), where f is the idempotent interpolation between the matrices ( \begin{matrix}e & 0 \\ 0 & p_{0} \end{matrix} ) and ( \begin{matrix}W^{−1}eW & 0 \\ 0 & p_{0} \end{matrix} ) constructed as in Lemma 5.5. One has

cs_{1}(f) ≡ Tr(W^{−1}(e − p_{0})dW) mod d in Ω^{1}(Q) ,

so that finally (ê, θ) is equivalent to (ê′, θ+Tr(W^{−1}(e−p_{0})dW)). In the present situation g′_{ij} = α_{i}g_{ij}α_{j}^{−1}, hence the idempotent e′_{+} is conjugate to e_{+} via the diagonal matrix W_{+} = diag (α_{1}^{−1} …, α_{m}^{−1}), and of course e′_{−} = e_{−} = 1_{𝒜_{P}} so one can choose W_{−} = 1. One calculates

Tr(W^{−1}(e − p_{0})dW) = Tr(W_{+}^{−1}e_{+}dW_{+}) = ∑_{i} α_{i}c_{i}^{2}d(α_{i}^{−1})
= − ∑_{i} c_{i}^{2}(A′_{i} − A_{i}) = θ′ − θ ,

hence (ê, θ) and (ê′, θ′) represent the same multiplicative K-theory class. Now we leave the cocycle (A_{i}, g_{ij}) fixed and change the partition of unity (c_{i}) to (c′_{i}), ∑_{i}(c′_{i})^{2} = 1, whence a new idempotent (e′_{+})_{ij} = c′_{i}g_{ij}c′_{j} and a new cochain θ′ = − ∑_{i}(c′_{i})^{2}A_{i}. Introduce the matrices u_{ij} = c_{i}g_{ij}c′_{j} and v_{ij} = c′_{i}g_{ij}c_{j}. Then one has e_{+} = uv, e′_{+} = vu, and the invertible matrix W_{+} = ( \begin{matrix}1−uv & −u \\ v & 1−vu \end{matrix} ) stably conjugates e_{+} and e′_{+} in the sense that W_{+}^{−1}( \begin{matrix}e_{+} & 0 \\ 0 & 0 \end{matrix} )W_{+} = ( \begin{matrix}0 & 0 \\ 0 & e′_{+} \end{matrix} ). A direct computation yields

Tr(W_{+}^{−1}( \begin{matrix}e_{+} & 0 \\ 0 & 0 \end{matrix} )dW_{+}) = ∑_{i,j}(c′_{i})^{2}c_{j}^{2}g_{ij}dg_{ji} = ∑_{i,j}(c′_{i})^{2}c_{j}^{2}(A_{j} − A_{i}) = θ′ − θ ,

hence the multiplicative K-theory class of (ê, θ) does not depend on the choice of partition of unity. A similar argument shows that it does not depend on the good covering (U_{i}). It remains to calulate the lowest component of the negative Chern character. By definition ch_{2}(ê, θ) is the cycle of even degree in X(T̂𝒜_{P}) given by Tr(ê − p_{0}) − \overline{b}θ̃, where θ̃ is an arbitrary lifting of θ ∈ X(𝒜_{P}). Thus, the image of ch_{2}(ê, θ) under the chain map X(T̂𝒜_{P}) → Ω^{∗}(Q) has a component of degree two given by

−Tr((e − \frac{1}{2})dede) − dθ = − ∑_{i} d(c_{i}^{2})A_{i} + d ∑_{i} c_{i}^{2}A_{i} = ∑_{i} c_{i}^{2}dA_{i} = dA_{i} ,

and coincides with the curvature of the line bundle (A_{i}, g_{ij}).

If one forgets the connection A_{i}, the morphism H_{𝒟}^{2} (Q; ℤ(1)) → MK_{2}(𝒜_{P}) just reduces to the elementary map Ȟ^{2}(Q; ℤ(1)) → K_{2}^{top}(𝒜_{P}) which associates to an isomorphism class of line bundles over Q its topological K-theory class.

Example 7.4 The simplest non-trivial example is provided by the celebrated non-commutative torus [3]. Here 𝒜 is the algebra of smooth functions over the circle N = S^{1} = ℤ\ℝ. Conventionnally we parametrize the points of N by the variable y. The group Γ = ℤ acts on N by rotations of angle α ∈ ℝ:

y ↦ y + nα ∀ y ∈ N , n ∈ ℤ .

When ℤ is provided with its natural distance, the crossed product 𝒜 ⋊ℤ is isomorphic to the algebra 𝒜_{α} of the non-commutative torus, presented for example in [3] by generators and relations: let V_{1} ∈ 𝒜 be the function V_{1}(y) = e^{2πiy} over N and V_{2} = 1^{∗} ∈ ℂℤ be the element corresponding to the generator 1 ∈ ℤ. Then V_{1} and V_{2} are invertible elements of 𝒜_{α} and fulfill the noncommutativity relation

V_{2}V_{1} = e^{2πiα}V_{1}V_{2} .  (97)

Moreover any element of 𝒜_{α} is a power series ∑_{(n_{1},n_{2})∈ℤ^{2}} a_{n_{1}n_{2}}V_{1}^{n_{1}}V_{2}^{n_{2}} with coefficients a_{n_{1}n_{2}} ∈ ℂ of rapid decay. For α ∈ ℚ this algebra is Morita equivalent to the smooth functions over an ordinary (commutative) 2-torus, and its multiplicative K-theory in any degree turns out to be completely determined by Deligne cohomology in this case. The situation is more interesting for α ∉ ℚ. Following the discussion above we introduce the universal principal ℤ-bundle P = Eℤ = ℝ over the classifying space M = Bℤ = ℤ\ℝ. Conventionnally we parametrize the points of P by the variable x. Thus, 𝒜_{P} = C^{∞}(P; 𝒜)^{ℤ} is the mapping torus algebra

𝒜_{P} = {a ∈ C^{∞}(P × N) | a(x + 1, y + α) = a(x, y) , ∀ x ∈ P , y ∈ N} .

Equivalently it is the algebra of smooth functions over the commutative 2-torus Q = ℤ\(P × N), quotient of ℝ^{2} by the lattice generated by the vectors (1, α) and (0, 1). Now to get a quasihomomorphism from 𝒜_{P} to 𝒜_{α} we need a K-cycle D for the circle manifold M. Let H = L^{2}(M) be the Hilbert space of squareintegrable complex-valued functions. The algebra C^{∞}(M) is represented in the algebra of bounded operators ℒ (H) by pointwise multiplication. D will be represented by the Toeplitz operator in ℒ (H) which projects H onto the Hardy space H_{+} ⊂ H:

D(e^{2πi nx}) = \{ \begin{matrix}e^{2πi nx} & if & n ≥ 0 \\ 0 & if & n < 0 \end{matrix}

One thus obtains a polarization of the Hilbert space H = H_{+} ⊕ H_{−}, with H_{−} the kernel of D. In the Fourier basis e^{2πi nx} of H, the representation C^{∞}(M) → ℒ (H) is easily seen to factor through the matrix subalgebra

(\begin{matrix}𝒯 & 𝒦 \\ 𝒦 & 𝒯 \end{matrix} ) ⊂ (\begin{matrix}ℒ (H_{+}) & ℒ (H_{−}, H_{+}) \\ ℒ (H_{+}, H_{−}) & ℒ (H_{−}) \end{matrix} ) = ℒ (H) ,

where 𝒯 is the smooth Toeplitz algebra (the elementary non-trivial extension of C^{∞}(S^{1}) by th ealgebra 𝒦 of smooth compact operators, see [9] ). The induced quasihomomorphism ρ_{D} : 𝒜_{P} → ℰ^{s} ▷ 𝒦 ^{s} ˆ⊗𝒜_{α}, with ℰ = 𝒯 ˆ⊗𝒜_{α}, is therefore 1-summable and of odd parity. Theorem 6.3 yields a graded-commutative diagram (remark that MK_{n}^{𝒦} ≅ MK_{n})

\xymatrix{K_{n+1}^{top}(𝒜_{P}) \ar[r] \ar[d] & HC_{n−1}(𝒜_{P}) \ar[r] \ar[d]^{ch^{1}(ρ_{D})} & MK_{n}(𝒜_{P}) \ar[r] \ar[d] & K_{n}^{top}(𝒜_{P}) \ar[d] \\ K_{n}^{top}(𝒜_{α}) \ar[r] & HC_{n−2}(𝒜_{α}) \ar[r] & MK_{n−1}(𝒜_{α}) \ar[r] & K_{n−1}^{top}(𝒜_{α})} (98)

in any degree n ∈ ℤ. The group K_{n}^{top}(𝒜_{P}) is isomorphic to the topological Ktheory of the 2-torus Q. Hence in even degree, K_{0}^{top}(𝒜_{P}) = ℤ ⊕ ℤ is generated by the trivial line bundle over Q together with the Bott class, whereas in odd degree K_{1}^{top}(𝒜_{P}) = ℤ ⊕ ℤ is generated by the invertible functions g_{1}(x, y) = e^{2πi x} and g_{2}(x, y) = e^{2πi(αx−y)}. The pushforward map K_{n}^{top}(𝒜_{P}) → K_{n−1}^{top}(𝒜_{α}) is known to be an isomorphism (Baum-Connes). In particular the Bott class and the trivial line bundle over Q are mapped respectively to the classes of the invertible elements V_{1} and V_{2} in K_{1}^{top}(𝒜_{α}). For multiplicative K-theory the situation is more involved. In degree n = 1 the map

MK_{1}(𝒜_{P}) ≅ H_{𝒟}^{1} (Q; ℤ(1/2)) ≅ C^{∞}(Q)^{×} → MK_{0}(𝒜_{α}) ≅ K_{0}^{top}(𝒜_{α})

simply factors through the topological K-theory group K_{1}^{top}(𝒜_{P}). In degree n = 2 one still has an isomorphism MK_{2}(𝒜_{P}) ≅ H_{𝒟}^{2} (Q; ℤ(1)), and (98) amounts to

[figure: K_{1}^{top}(𝒜_{P}); ^{ch^{1}}; \frac{Ω^{1}(Q)}{dΩ^{0}(Q)}; ch^{1}(ρ_{D}); H_{𝒟}^{2} (Q; ℤ(1)); K_{0}^{top}(𝒜_{P}); K_{0}^{top}(𝒜_{α}); HC_{0}(𝒜_{α}); MK_{1}(𝒜_{α}); K_{1}^{top}(𝒜_{α}); 0] (99)

The map Ω^{1}/dΩ^{0} → H_{𝒟}^{2} (Q; ℤ(1)) associates to a one-form \frac{1}{\sqrt{2πi}} A over Q the isomorphism class of the trivial line bundle with connection −A, while the range of H_{𝒟}^{2} (Q; ℤ(1)) → K_{0}^{top}(𝒜_{P}) is generated by the Bott class. For generic values of α ∉ ℚ the commutator subspace [𝒜_{α}, 𝒜_{α}] may not be closed in 𝒜_{α}, therefore the quotient HC_{0}(𝒜_{α}) = HH_{0}(𝒜_{α}) = 𝒜_{α}/[𝒜_{α}, 𝒜_{α}] may not be separated. However the quotient \overline{HC}_{0}(𝒜_{α}) by the closure of the commutator subspace turns out to be isomorphic to ℂ, via the canonical trace

𝒜_{α} → ℂ , V_{1}^{n_{1}}V_{2}^{n_{2}} ↦ \{ \begin{matrix}1 & if n_{1} = n_{2} = 0 , \\ 0 & otherwise . \end{matrix}

With these identifications the evaluation of the Chern character ch^{1}(ρ_{D}) : Ω^{1}/dΩ^{0} → \overline{HC}_{0}(𝒜_{α}) ≅ ℂ on a one-form A = A_{x}dx + A_{y}dy is easily performed and one finds

ch^{1}(ρ_{D}) (\frac{A}{\sqrt{2πi}}) = \frac{1}{2πi} ∫_{0}^{1} dy ∫_{0}^{1} dx A_{x}(x, y) .

In particular ch^{1}(ρ_{D}) · ch_{1}(g_{1}) = 1 and ch^{1}(ρ_{D}) · ch_{1}(g_{2}) = α, and one recovers the well-known fact ([4]) that the image of K_{0}^{top}(𝒜_{α}) in \overline{HC}_{0}(𝒜_{α}) is the subgroup ℤ + αℤ ⊂ ℂ. We may analogously define a new multiplicative K-theory group \overline{MK}_{1}(𝒜_{α}) whose elements are represented by pairs (ĝ, θ) with θ ∈ ℂ instead of θ ∈ 𝒜_{α}/[𝒜_{α}, 𝒜_{α}]. Because K_{1}^{top}(𝒜_{α}) is generated by the invertibles V_{1} and V_{2} any class in \overline{MK}_{1}(𝒜_{θ}) is represented by a pair (V_{1}^{n_{1}}V_{2}^{n_{2}}, θ) for some integers n_{1}, n_{2} and a complex number θ. Using a homotopy one shows that this pair is equivalent to (e^{−\sqrt{2πi}θ}V_{1}^{n_{1}}V_{2}^{n_{2}}, 0), and by exactness \overline{MK}_{1}(𝒜_{α}) is the quotient of the multiplicative group ℂ^{×}⟨V_{1}⟩⟨V_{2}⟩ ⊂ GL_{1}(𝒜_{α}) by its commutator subgroup ⟨e^{2πiα}⟩ ⊂ ℂ^{×}, or equivalently the abelianization

\overline{MK}_{1}(𝒜_{α}) = (ℂ^{×}⟨V_{1}⟩⟨V_{2}⟩)_{ab} .  (100)

Since the Bott class of K_{0}(𝒜_{P}) is sent to [V_{1}] ∈ K_{1}^{top}(𝒜_{α}), one sees that the range of H_{𝒟}^{2} (Q; ℤ(1)) → \overline{MK}_{1}(𝒜_{α}) coincides with the subgroup ℂ^{×}⟨V_{1}⟩.

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362 Denis Perrot

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Denis Perrot Université de Lyon Université Lyon 1 CNRS, UMR 5208 Institut Camille Jordan 43, bd du 11 novembre 1918 69622 Villeurbanne Cedex France perrot@math.univ-lyon1.fr 364

Documenta Mathematica 13 (2008)