Permanence Properties of C*-exact Groups

Eberhard Kirchberg and Simon Wassermann

Received: March 31, 1999

Communicated byJoachim Cuntz

Abstract. It is shown that the class of exact groups, as defined in a previous paper, is closed under various operations, such as passing to a closed subgroup and taking extensions. Taken together, these results imply, in particular, that all almost-connected locally compact groups are exact. The proofs of the permanence properties use a result relating the exactness of sequences of maps in which corresponding algebras are strongly Morita equivalent. The statement of this result is based on a notion of reduced twisted crossed product for covariant systems which are twisted in the sense of Green. The theory of these reduced twisted crossed products and the proof of the exactness result are given in the first part of the paper.

1991 Mathematics Subject Classification: 46L05, 46L55, 22D25

1. Introduction.

Given a locally compact group G, let C_{G}^{∗} be the category whose objects are the pairs (A, α) consisting of a C*-algebra A and a continuous action α of G on A, and whose maps are the G-equivariant *-homomorphisms between C*- algebras with continuous G-actions. Following [KW], the group G is said to be C*-exact (or just exact) if the reduced crossed product functor A → A ⋊_{α,r} G, for (A, α) ∈ C_{G}^{∗}, is short-exact. To be more precise, G is exact if and only if, whenever (I, α), (A, β) and (B, γ) are elements of C_{G}^{∗} and there is a Gequivariant short exact sequence

\xymatrix{0 \ar[r] & I \ar[r]^{ι} & A \ar[r]^{q} & B \ar[r] & 0,}

of maps, the corresponding sequence

\xymatrix{0 \ar[r] & I ⋊_{α,r} G \ar[r]^{ι_{r}} & A ⋊_{β,r} G \ar[r]^{q_{r}} & B ⋊_{γ,r} G \ar[r] & 0}

of reduced crossed products is exact. This is equivalent to saying that for (A, α) ∈ C_{G}^{∗}, if I is an α_{G}-invariant ideal of A, then the quotient (A ⋊_{α,r} G)/(I ⋊_{α|,r} G) is canonically isomorphic to (A/I) ⋊_{α̇,r} G, where α| and α̇ are the restriction and quotient actions of G on I and A/I, respectively.

We introduced group exactness in [KW], primarily as a criterion for the continuity of crossed products of continuous bundles of C*-algebras. Given a continuous bundle A = {A, X, A_{x}} over a locally compact Hausdorff space X with a continuous fibre-preserving action α of a group G on the bundle C*- algebra A, it is not in general clear that the reduced crossed product bundle A ⋊_{α,r} G = {A ⋊_{α,r} G, X, A_{x} ⋊_{α_{x},r} G} is continuous, though we know of no instance where continuity fails. One of the main results in [KW] is that, for a given G, A ⋊_{α,r} G is continuous for all pairs (A, α) if and only if G is exact. It is thus of some importance to know which groups are exact, and it is this problem which is addressed in this paper.

The most basic question is whether, in fact, all locally compact groups are exact. We have so far not been able to resolve this question even in the discrete case, and to the best of our knowledge the exactness of arbitrary discrete groups remains a significant open problem. What we are able to show is that the class of exact groups is closed under various operations such as passing to closed subgroups and taking extensions. Moreover groups possessing closed exact subgroups of finite covolume or which are cocompact are themselves exact. Using these permanence results we can show that groups from a wide class, including, in particular, all connected groups, are exact.

To prove these results we use adaptations of a number of techniques from the theory of induced representations of C*-algebras. Originally formulated by Rieffel to give an interpretation of Mackey’s theory of induced representations of groups in terms of C*-algebras, this theory has been developed by P. Green [Gr] and others to give powerful techniques for handling crossed products of C*-algebras. The main tools that we use to prove the permanence results are imprimitivity theorems asserting strong Morita equivalences between various C*-crossed products by a group G on the one hand and by a closed subgroup H of G on the other. These results all follow either from Green’s generalisation to crossed products of Rieffel’s imprimitivity theorem [Gr, §2], or from Raeburn’s symmetric generalisation of Green’s theorem [Rae, Theorem 1.1]. We shall use Green’s notion of a twisted action of a group G on a C*-algebra [Gr] to prove that exactness is preserved on taking extensions.

Let N be a closed normal subgroup of G and suppose that (A, α) ∈ C_{G}^{∗}. Using α to denote also the restriction α|N, G has a canonical continuous action γ on A ⋊_{α} N, and there is a canonical homomorphic embedding τ : N → U(A ⋊_{α} N), where U(A ⋊_{α} N) is the unitary group of the multiplier algebra M(A⋊_{α}N). The map τ, which is an example of a twisting map, satisfies various compatibility conditions relative to γ (see §2). The system {A ⋊_{α} N, G, γ, τ} is an example of a twisted covariant system in the sense of Green.

In general a twisted covariant system {A, G, α, τ} consists of a continuous action α of G on A and a continuous group monomorphism τ from a closed normal subgroup N of G into the unitary group U(A) of the multiplier algebra M(A) satisfying the aforementioned compatibility conditions. There is a natural idea of a twist-preserving covariant pair of representations of a twisted covariant system and the full twisted crossed product A ⋊_{α,τ} G is defined as the unique quotient of the usual full crossed product A ⋊_{α} G which is universal for the representations obtained as the integrated forms of the twist-preserving covariant pairs of representations of {A, G}. If (γ, τ) is the twisted action of the previous paragraph, the crossed products (A ⋊_{α|N} N) ⋊_{γ,τ} G and A ⋊_{α} G are canonically *-isomorphic, by [Gr, §1]. By a result of Echterhoff [Ech, Theorem 1], if {B, G, N, α, τ} is a twisted covariant system, there is an associated covariant system {C, G/N, β} such that the twisted crossed product B⋊_{α,τ} G is strongly Morita equivalent to the ordinary crossed product C ⋊_{β} (G/N). Combining these results, one finds that, with A, G, α and N as above, A ⋊_{α|N} N is strongly Morita equivalent to C ⋊_{β} (G/N), where C = (C_{0}(G/N) ⊗ A) ⋊_{Δ^{α}} G, Δ^{α} being the diagonal action of G on C_{0}(G/N) ⊗ A ≅ C_{0}(G/N, A), and β is a certain action of G/N on C.

An analogous result for reduced crossed products is used in §5 to show that an extension of an exact group by an exact group is exact. This requires the definition, for a given a twisted covariant system {A, G, α, τ}, of a reduced twisted crossed product A ⋊_{α,τ,r} G, which reduces to the ordinary reduced crossed product if the twisting is trivial, that is, if N = {1}. Although there are definitions of twisted reduced crossed products in the literature for twisted actions coming from cocycles, so far as we are aware none has been given hitherto for twisted actions in the sense of Green. Using the reduced twisted crossed product we show that A⋊_{α,r} G is strongly Morita equivalent to (A⋊_{α,r} N) ⋊_{γ,τ,r} G for a suitable twisted action (γ, τ) of G on A ⋊_{α,r} N. In fact our result is sharper, in that the Morita equivalence we establish is functorial in A in a certain sense.

The other permanence results are also proved using analogues for reduced crossed products of known imprimitivity theorems for full crossed products. In order to unify our techniques as much as possible, we give a general imprimitivity theorem in §2 which covers all the cases we need. This section also contains a brief review of twisted covariant systems, full twisted crossed products and other relevant background material. We define the reduced twisted crossed product in §3, and deduce imprimitivity results for reduced crossed products that parallel those for full crossed products in §2. These results are the basis of the proofs of the permanence properties mentioned above, which are established in §§4 and 5. The permanence properties are applied in §6 to show that groups of various types are exact.

During the writing of this paper we have benefited from discussions with a number of people. We should particularly like to thank Etienne Blanchard, Siegfried Echterhoff and George Skandalis for valuable comments and observations. We also have pleasure in thanking the following for support:

The EPSRC for a Visiting Fellowship for the first author to visit Glasgow. The DFG for supporting a visit by the second author to Berlin.

Gert Pedersen for invitations to visit Copenhagen, and the Danish Research Council for supporting these visits.

The Volkswagen Stiftung for a Research in Pairs Fellowship at the Mathematisches Forschungsinstitut, Oberwolfach.

The second author would like to express his gratitude to Etienne Blanchard and his colleagues at the Université d’Aix-Marseille II at Luminy for inviting him to visit Luminy.

2. Imprimitivity results for full twisted crossed products.

In this section we recall the basic ideas of Green’s theory of twisted group actions and state some of the imprimitivity results which will be used in later sections to prove the permanence results. Most of this material is a straightforward generalisation of [Gr,§§1,2], but we have found it necessary to make some aspects of the theory which are not immediately accessible in Green’s treatment more explicit. Throughout the paper all groups will be assumed locally compact. Our notation follows that of [KW], for the most part. For each locally compact group G, m_{G} will denote a particular left Haar measure on G and Δ_{G} the modular function. For (A, α) ∈ C_{G}^{∗}, the full and reduced crossed products of A by G are denoted by A ⋊_{α} G and A ⋊_{α,r} G, respectively. If (B, β) ∈ C_{G}^{∗}, and θ : A → B is a completely positive G-equivariant map, then θ_{u} and θ_{r} will denote the canonical extensions of the map f → θ(f); C_{c}(G, A) → C_{c}(G, B), where (θ(f))(s) = θ(f(s)) for s ∈ G, to completely positive maps

A ⋊_{α} G → B ⋊_{β} G

and

A ⋊_{α,r} G → B ⋊_{β,r} G,

respectively. If {π, V } is a covariant pair of representations of the covariant system {A, G, α}, then π ⋊ V will denote the corresponding integrated form representation of the full crossed product A ⋊_{α} G.

Let G be a locally compact group, let H be a closed subgroup of G and let (A, α) ∈ C_{H}^{∗}. Recall that the C*-algebra Ind(A, α) is the *-subalgebra of the C*-algebra C_{b}(G, A) of bounded continuous A-valued functions on G consisting of those functions f such that

α_{h}(f(xh)) = f(x)

for h, x ∈ H, so that the function x → ||f(x)|| is constant on left cosets of H, and such that the associated continuous function on G/H given by

xH → ||f(x)||

is in C_{0}(G/H). As is easily seen, Ind(A, α) is closed in C_{b}(G, A), and is, moreover, the bundle algebra of a continuous bundle of C*-algebras on G/H with constant fibre A. In general this bundle is nontrivial, though if the action α extends to an action of G on A, then the bundle is isomorphic to the trivial bundle on G/H with fibre A. In fact when α is defined on all of G, an automorphism ν of C_{b}(G, A) is defined by

(ν(f))(s) = α_{s}(f(s)).

For f ∈ Ind(A, α), x ∈ G and h ∈ H, (ν(f))(xh) = α_{x}((f)(x)), so that ν(f) is constant on left cosets of H. If we identify ν(f) with the corresponding function in C_{0}(G/H, A), the restriction of ν to Ind(A, α) gives an isomorphism ν_{A} of Ind(A, α) onto C_{0}(G/H, A).

A continuous action α̃ of G on Ind(A, α) is given by

(α̃_{g}(ψ))(s) = ψ(g^{−1}s).

If α is defined on all of G, for ψ ∈Ind(A, α), g, s ∈ G,

(ν(α̃_{g}(ψ)))(s) = α_{s}((α̃_{g}(ψ))(s)) = α_{s}(ψ(g^{−1}s)) = (Δ_{g}^{α}(ν(ψ)))(s),

where Δ^{α} is the diagonal action of G on C_{0}(G/H, A) given by

(Δ_{g}^{α}(f))(s) = α_{g}(f(g^{−1}s))

for f ∈ C_{0}(G/H, A). Thus ν_{A} is an equivariant isomorphism between the covariant systems {Ind(A, α), G, α̃} and {C_{0}(G/H, A), G, Δ^{α}}.

Let E_{0} and B_{0} be the *-algebras C_{c}(G,Ind(A, α)) and B_{0} = C_{c}(H, A), with the convolution products relative to the actions α̃ and α, respectively, and let X_{0} = C_{c}(G, A). The algebras E_{0} and B_{0} are taken to have the C*-norms and positive cones resulting from their canonical embeddings in Ind(A, α) ⋊α̃ G and A ⋊_{α} H, respectively. The linear space X_{0} is given an E_{0}–B_{0} bimodule structure and E_{0}- and B_{0}-valued inner products as follows. For f ∈ E_{0}, g ∈ B_{0}, and x, y ∈ X_{0}, fx, xg, ⟨x, y⟩_{E_{0}} and ⟨x, y⟩_{B_{0}} are defined by

(fx)(r) = ∫_{G} f(s, r)x(s^{−1}r)dm_{G}(s)
(xg)(r) = ∫_{H} δ(t)α_{t}(x(rt)g(t^{−1}))dm_{H}(t)
⟨x, y⟩_{E_{0}}(s, r) = ∫_{H} Δ_{G}(rs^{−1}t)α_{t}(x(rt)y(s^{−1}rt)^{∗})dm_{H}(t)
⟨x, y⟩_{B_{0}}(t) = δ(t) ∫_{G} x(s)^{∗}α_{t}(y(st))dm_{G}(s),

where δ(t) = Δ_{G}(t)^{1/2}/Δ_{H}(t)^{1/2}. It is easily checked that fx, xg ∈ X_{0}, ⟨x, y⟩_{E_{0}} ∈ E_{0} and ⟨x, y⟩_{B_{0}} ∈ B_{0}. The map (f, x) → fx is a left action of E_{0} on X_{0} and is the integrated form of the covariant pair of left actions of Ind(A, α) and G on X_{0} given by

(ψx)(r) = ψ(r)x(r)
(sx)(r) = x(s^{−1}r)

(ψ ∈ Ind(A, α), s ∈ G, x ∈ X_{0}). The map (x, g) → xg is a right action of B_{0} on X_{0} and is the integrated form of the covariant pair of right actions of A and H on X_{0} given by

(xa)(r) = x(r)a
(xt)(r) = Δ_{G}(t)^{−1/2}Δ_{H}(t)^{−1/2}α_{t^{−1}}(x(rt^{−1}))

(a ∈ A, t ∈ H, x ∈ X_{0}).

The following theorem generalises [Gr, Proposition 3] and [Rie1, §7]. It is straightforward, if rather tedious, to write out a proof along the lines of those of [Gr] and [Rie1], though, as Siegfried Echterhoff has pointed out to us, the result is a corollary of Raeburn’s more general symmetric imprimitivity theorem [Rae, Theorem 1.1, special case 1.5]. We are grateful to Echterhoff for drawing our attention to the latter, and also for showing us how the result can, alternatively, be deduced directly from Green’s original imprimitivity theorem.

Theorem 2.1 With the structure defined above, X_{0} is an E_{0}–B_{0} equivalence (or imprimitivity) bimodule.

Remark If the action α is actually defined on the whole of G, so that the covariant systems {Ind(A, α), G, α̃} and {C_{0}(G/H, A), G, Δ^{α}} are equivariantly isomorphic, it is straightforward to verify that the E_{0}–B_{0} equivalence bimodule X_{0} is isomorphic to the C_{c}(G, C_{0}(G/H, A))–C_{c}(H, A) equivalence bimodule constructed by Green in [Gr], and that Theorem 2.1 reduces to [Gr,Proposition 3].

Let || ||_{u} be the universal C*-norm on C_{c}(H, A). If X_{A} is the completion of X_{0} with respect to the norm x → ||⟨x, x⟩_{B_{0}}||_{u}^{1/2}, the action of B_{0} extends canonically to a right action of its completion A ⋊_{α} H on X_{A}. Moreover the left action of E_{0} on X_{0} extends to a left action on X_{A} by bounded operators, the operator norm on E_{0} being a C*-norm, generally incomplete. If E is the completion of E_{0} with respect to this norm, there is a canonical left action of E on X_{A} extending that of E_{0}, and X_{A} is an E–(A ⋊_{α} H) equivalence bimodule [Rie2]. The C*-algebra E is a quotient of the full crossed product Ind(A, α) ⋊α̃ G. In Corollary 2.2 we shall show that the kernel of the quotient map is trivial, so that E ≅ Ind(A, α) ⋊α̃ G canonically. Since the proofs of the corollary and other later results use induced representations, we review the inducing process briefly.

Let A and B be strongly Morita equivalent pre-C*-algebras (i.e. normed *-algebras whose norms satisfy the C*-condition but are not necessarily complete), and let X be an A–B equivalence bimodule. If π is a contractive *- representation of B on a Hilbert space H, then the corresponding induced representation ^{X}π acts contractively and non-degenerately on the Hilbert space ^{X}H obtained by completing X ⊗_{B} H with respect to the semi-norm ∑ x_{i} ⊗ ξ_{i} → ||(π(⟨x_{i}, x_{i}⟩_{B})ξ_{i}|ξ_{i})||^{1/2} and for a ∈ A,

Xπ(a)(x ⊗ ξ) = ax ⊗ ξ (x ∈ X, ξ ∈ H).

Let X^{∗} be the B–A equivalence bimodule dual to X. Thus X^{∗} is the image of X by an antilinear bijection x → x^{∗} such that

λ̄x^{∗} = (λx)^{∗}, bx^{∗} = (xb^{∗})^{∗}, x^{∗}a = (a^{∗}x)^{∗}

for λ ∈ ℂ, a ∈ A, b ∈ B and x ∈ X, and the B- and A-valued inner products on X^{∗} are given by

⟨x^{∗}, y^{∗}⟩_{B} = ⟨y, x⟩_{B}, ⟨x^{∗}, y^{∗}⟩_{A} = ⟨y, x⟩_{A}

for x, y ∈ X. If σ is a contractive representation of A on a Hilbert space K, then ^{X^{∗}}σ is a contractive representation of B. The A–A equivalence bimodules X ⊗_{B} X^{∗} and A are isomorphic, and likewise there is an isomorphism between the B–B equivalence bimodules X^{∗} ⊗_{A} X and B. It follows that there are unitary equivalences ^{X^{∗}}(^{X}π) ≅ π and ^{X}(^{X^{∗}}σ) ≅ σ for any non-degenerate representations π and σ of B and A, respectively, so that there is a bijective correspondence between the equivalence classes of non-degenerate representations of A and B. In the rest of the paper all representations will be assumed non-degenerate.

If A and B are actually C*-algebras, by [Rie1] there are bijective correspondences between (i) ideals of A, (ii) closed A–B-invariant subspaces of X and (iii) ideals of B. If Y is a closed A–B-invariant subspace of X, the corresponding ideals of A and B are

A_{Y} = \overline{span}\{< y, x >_{A}: x ∈ X, y ∈ Y \}
B_{Y} = \overline{span}\{< x, y >_{B}: x ∈ X, y ∈ Y \},

respectively. In the opposite direction, if I and J are ideals in B and A, respectively, then the corresponding A–B-invariant subspaces of X are

Y_{I} = XI = \overline{span}\{xz : x ∈ X, z ∈ I\}

and

JY = JX = \overline{span}\{zx : z ∈ J, x ∈ X\}.

These correspondences clearly respect inclusion. When necessary we shall say that I and J correspond via X. It is straightforward to verify that if π is a representation of B, then the ideals ker^{X} π and ker π of A and B, respectively, correspond via X. In particular, ^{X}π is faithful if and only if π is faithful.

Corollary 2.2 The operator norm on E_{0} is the the universal C*-norm coming from the canonical embedding of E_{0} in Ind(A, α) ⋊α̃ G, and X_{A} is canonically an (Ind(A, α) ⋊α̃ G)–(A ⋊_{α} H) equivalence bimodule.

Proof: Let {π, U} be a covariant pair of representations of the system {A, H, α} on a Hilbert space H. Writing X for X_{A}, let^{X}π and^{X}U denote the restrictions of the induced representation ^{X}(π ⋊ U) to Ind(A, α) and G, respectively. We shall say that the covariant pair {^{X}π,^{X} U} is induced from the pair {π, U}. Similarly, given a covariant pair {σ, V } of representations of {Ind(A, α), G, α̃}, we obtain a covariant pair {^{X^{∗}}σ,^{X^{∗}} U} of representations of {A, H}. By the above discussion, the pairs {π, U} and {^{X^{∗}}(^{X}π),^{X^{∗}} (^{X}U)} are unitarily equivalent, as are the pairs {σ, V } and {^{X}(^{X^{∗}}σ),^{X} (^{X^{∗}}V)}.

Now let {π, U} and {σ, V } be universal covariant pairs for {A, H, α} and {Ind(A, α), G, α̃}, respectively. Replacing {π, U} and {σ, V } by the pairs {π ⊕^{X^{∗}} σ, U ⊕^{X^{∗}} V } and {σ ⊕^{X} π, V ⊕^{X} U}, respectively, we can assume, since the inducing process respects direct sums, that

σ =^{X} π, V =^{X} U, π =^{X^{∗}} σ, U =^{X^{∗}} V.

Now the representation σ ⋊ V of Ind(A, α) ⋊α̃ G is universal, hence faithful. Also, the representation ^{X}(π ⋊ U) of E has restrictions σ and V to Ind(A, α) and G, respectively. This implies that σ ⋊ V factorises via the quotient map Ind(A, α) ⋊α̃ G → E, which implies that the quotient map is injective, so that Ind(A, α) ⋊α̃ G ≅ E as required. □

Remark If {π, U} is any universal covariant pair of representations of {A, H, α}, then π ⋊ U is a faithful representation of A ⋊_{α} H, and so ^{X_{A}}(π ⋊ U) is a faithful representation of Ind(A, α) ⋊α̃ G. Thus {^{X_{A}}π,^{X_{A}} U} is universal for {Ind(A, α), G, α̃}.

Let G be a locally compact group with closed normal subgroup N. For (A, α) ∈ C_{G}^{∗} a twisting map for N is a strictly continuous homomorphism τ of N into the unitary group U(A) of M(A) such that for n ∈ N, s ∈ G,

τ(n)aτ(n)^{−1} = α_{n}(a)

and

τ(sns^{−1}) = α_{s}(τ(n)).

The pair (α, τ) is called a twisted action of of G on A relative to N, and, provided A is nonzero, {A, G, α, τ} is referred to as a twisted covariant system. A covariant pair of representations {π, V } of {A, G} on a Hilbert space H is τ-covariant or twist-preserving if

π̄(τ(n)) = V_{n}

for n ∈ N, where π̄ denotes the canonical extension of π to the multiplier algebra M(A).

Let I_{τ} be the closed, two-sided ideal ⋂_{{π,V }} ker(π ⋊ V) of the full crossed product A ⋊_{α} G, where the supremum is over all τ-covariant pairs of representations of {A, G, α}. The full twisted crossed product A⋊_{α,τ} G is the C*-algebra (A ⋊_{α} G)/I_{τ}. It has the universal property that if {π, V } is a τ-covariant pair of representations of {A, G, α}, then I_{τ} ⊆ ker(π ⋊ V), so that π ⋊ V is the composition of a representation π ⋊_{τ} V of A ⋊_{α,τ} G with the quotient map A ⋊_{α} G → A ⋊_{α,τ} G.

Although it is not made explicit in [Gr], for a given twisted covariant system {A, G, α, τ} it is always possible to find a twist-preserving covariant pair of representations {π, V } with π faithful. In §3 we shall construct for a given faithful representation π of A on a Hilbert space H a τ-covariant pair of representations {π_{α,τ} , λ_{τ} } of {A, G, α, τ} on a Hilbert space L_{τ}^{2}(G, H) canonically associated with π with π_{α,τ} faithful. In the case when N is trivial, this pair reduces to the usual regular pair {π_{α}, λ^{G}}. It follows that we can find a faithful representation π of A ⋊_{α,τ} G on a Hilbert space H, with restrictions {π_{A}, π_{G}} to {A, G} such that π_{A} is injective and for a ∈ A, g ∈ G, π_{A}(a) and π_{G}(g) are multipliers of π(A ⋊_{α,τ} G). If we identify A ⋊_{α,τ} G with its image under π and the multiplier algebra M(A ⋊_{α,τ} G) with a *-subalgebra of the weak closure of this image, π_{A} and π_{G} are respectively a *-monomorphism of A and a group homomorphism of G with kernel contained in N into M(A⋊_{α,τ} G). With these identifications, π_{A} and π_{G} are independent of π, and will be referred to as the canonical morphisms. It then follows that for any faithful representation of A ⋊_{α,τ} G, π_{A} is injective. A twisted covariant pair {π, V } will be called universal if the representation π ⋊_{τ} V of A ⋊_{α,τ} G is faithful.

Now let G be a locally compact group with a closed normal subgroup N and let H be a closed subgroup of G containing N. If (A, α) ∈ C_{H}^{∗}, let τ : N → M(A) be a twisting map for N. A homomorphism τ̃ : N → U(Ind(A, α)) is defined by

(τ̃(n)ψ)(s) = τ(s^{−1}ns)ψ(s) (ψ ∈ Ind(A, α)).

It is straightforward to verify that {Ind(A, α), G, α̃, τ̃} is a twisted covariant system relative to N.

Proposition 2.3 Let {π, V } be a covariant pair of representations of the covariant system {A, H, α, τ} on a Hilbert space H. The pair {^{X_{A}}π,^{X_{A}} V } is τ̃-covariant if and only if the pair {π, V } is τ-covariant.

Proof: Let π̃ =^{X_{A}} π and U =^{X_{A}} V. For n ∈ N, f, g ∈ C_{c}(G, A) ⊆ X_{A} and ξ, η ∈ H,

(U_{n}(f ⊗ ξ)|g ⊗ η)
= (π(⟨g, nf⟩_{B})ξ|η)
= ∫_{H} (π(⟨g, nf⟩_{B}(t))V_{t}ξ|η) dm_{H}(t)
= ∫_{H} ∫_{G} δ(t)(π(g(s)^{∗}α_{t}(f(n^{−1}st)))V_{t}ξ|η) dm_{G}(s) dm_{H}(t)
= ∫_{G} ∫_{H} δ(t)(π(g(s)^{∗}α_{t}(f(s(s^{−1}n^{−1}st))))V_{t}ξ|η) dm_{H}(t) dm_{G}(s)

=^{t→s^{−1}nst}∫_{G} ∫_{H} δ(t)(π(g(s)^{∗}α_{s^{−1}nst}(f(st)))V_{s^{−1}ns}V_{t}ξ|η) dm_{H}(t) dm_{G}(s)

(Since Δ_{G}|N = Δ_{H}|N = Δ_{N}, N being a normal subgroup)

= ∫_{G} ∫_{H} δ(t)(π(g(s)^{∗})V_{s^{−1}ns}π(α_{t}(f(st)))V_{t}ξ|η) dm_{H}(t) dm_{G}(s)

and (τ̃(n)f ⊗ ξ|g ⊗ η)

= ∫_{G} ∫_{H} δ(t)(π(g(s)^{∗}α_{t}(τ(t^{−1}s^{−1}nst)f(st)))V_{t}ξ|η) dm_{H}(t) dm_{G}(s)
= ∫_{G} ∫_{H} δ(t)(π(g(s)^{∗})π̄(τ(s^{−1}ns))π(α_{t}(f(st)))V_{t}ξ|η) dm_{H}(t) dm_{G}(s).

If the pair {π, V } is τ-covariant, it follows that

(U_{n}(f ⊗ ξ)|g ⊗ η) = (τ̃(n)f ⊗ ξ|g ⊗ η),

so that U_{n} = π̃(τ̃(n)), which means that the pair {π̃, U} is τ̃-covariant. If, conversely, {π̃, U} is τ̃-covariant, then

(U_{n}(f ⊗ ξ)|g ⊗ η) = (τ̃(n)f ⊗ ξ|g ⊗ η),

and, by the above calculations,

∫_{G}∫_{H}δ(t)(π(g(s)^{∗})(V_{s^{−1}ns} − π̄(τ(s^{−1}ns)))π(α_{t}(f(st)))V_{t}ξ|η) dm_{H}(t) dm_{G}(s)=0

(∗).

Let a, b ∈ A, let ε > 0 and let V be a symmetric compact neighbourhood of the identity in G such that for s, t ∈ V^{2}

|(π(b^{∗})(V_{s^{−1}ns}−π̄(τ(s^{−1}ns)))π(α_{t}(a))V_{t}ξ|η)−(π(b^{∗})(V_{n}−π̄(τ(n)))π(a)ξ|η)|≤ε.

Letting h be a continuous positive function with support in V such that

∫_{G} ∫_{H} δ(t)h(s)h(st) dm_{H}(t) dm_{G}(s) = 1

and taking f and g to be the functions s → h(s)a and s → h(s)b, respectively, a simple calculation shows that the difference between π(b^{∗})(V_{n} −π̄(τ(n)))π(a) and the integral on the left-hand side of (∗) has modulus less than or equal to ε. Since ε is arbitrary, this implies that

π(b^{∗})(V_{n} − π̄(τ(n)))π(a) = 0

for a, b ∈ A, so that π̄(τ(n)) = V_{n}, by the nondegeneracy assumption on π, which implies that the pair {π, V } is τ-preserving. □

Let I_{τ} be the kernel of the canonical quotient map A ⋊_{α} H → A ⋊_{α,τ} H. If Ĩ is the ideal of Ind(A, α) ⋊α̃ G corresponding to I_{τ} via X_{A}, let E_{τ} be the quotient (Ind(A, α) ⋊α̃ G)/Ĩ. Then X_{A,τ} = X_{A}/X_{A}I_{τ} is an E_{τ} –(A ⋊_{α,τ} H) equivalence bimodule. The following theorem generalises [Gr,Corollary 5].

Theorem 2.4 The C*-algebra E_{τ} is canonically isomorphic to Ind(A, α)⋊_{α̃,τ̃} G and X_{A,τ} is an (Ind(A, α) ⋊_{α̃,τ̃} G)–(A ⋊_{α,τ} H) equivalence bimodule.

Proof: The proof is very similar to that of Corollary 2.2. Let {π, V } and {σ, U} be universal twist-covariant pairs of representations of {A, H, α, τ} and {Ind(A, α), G, α̃, τ̃}, respectively. Then {σ, U} is unitarily equivalent to the pair {^{X_{A}}(^{X_{A}^{∗}}σ),^{X_{A}} (^{X_{A}^{∗}}U)} and, by Proposition 2.3, the pairs {^{X_{A}}π,^{X_{A}} V } and {^{X_{A}^{∗}}σ,^{X_{A}^{∗}} U} are τ̃- and τ-covariant, respectively. Replacing {π, V } and {σ, U} by {π ⊕^{X_{A}^{∗}} σ, V ⊕^{X_{A}^{∗}} U} and {σ ⊕^{X_{A}} π, U ⊕^{X_{A}} V }, respectively, we can assume that the pair of representations of {Ind(A, α), G, α̃, τ̃} induced from {π, V } is universal. By our earlier discussion the ideal Ĩis the kernel of the representation (^{X_{A}}π)⋊(^{X_{A}}V), which is the kernel of the canonical quotient map Ind(A, α)⋊α̃ G →Ind(A, α)⋊_{α̃,τ̃} G. It follows that X_{A,τ} is an (Ind(A, α)⋊_{α̃,τ̃} G)–(A⋊_{α,τ} H) equivalence bimodule. □

Remarks 2.5 1. It follows, by reasoning similar to that of the remark following the proof of Corollary 2.2, that if {π, V } is any universal τ-covariant pair of representations of {A, H, α, τ}, then the τ̃-covariant pair {^{X_{A}}π,^{X_{A}} V } is universal for {Ind(A, α), G, α̃, τ̃}.

2. Suppose that H = N, so that α is a continuous action of N on A, and that τ : N → U(A) is a twisting map. The pair of homomorphisms {id, τ} of {A, N}, where id : A → M(A) is the canonical embedding, is τ-covariant. If we represent M(A) faithfully on a Hilbert space in such a way that the restriction of the representation to A is non-degenerate, we can regard this pair of maps as a τ-covariant pair of representations, and the integrated form of {id, τ} is a *-homomorphism Φ of A ⋊_{α,τ} N into M(A). For f ∈ C_{c}(N, A),

Φ(f) = ∫_{N} f(n)τ(n) dm_{N}(n) ∈ A.

It follows, by taking limits, that Φ(A ⋊_{α,τ} N) ⊆ A, and, by considering f with suitably small support containing the identity of N, that the image of Φ is dense in A. Thus Φ(A ⋊_{α,τ} N) = A.

Let {π, V } be a τ-covariant pair of representations of the twisted system {A, N, α, τ} on a Hilbert space H. Then π̄(τ(n)) = V_{n} for n ∈ N and for f ∈ C_{c}(N, A),

(π ⋊ V)(f) = ∫_{N} π(f(n))π̄(τ(n)) dm_{N}(n)
= π (∫_{N} f(n)τ(n) dm_{N}(n))
= π(Φ(f)).

Thus π ⋊ V = π ∘ Φ. If {π, V } is a universal pair for {A, N, α}, then π ⋊_{τ} V and π are faithful. This implies that Φ is an isomorphism, i.e. A ⋊_{α,τ} N ≅ A. If π is a representation of A, then {π, π̄ ∘ τ} is a τ-covariant pair for {A, N} and π ⋊ (π̄ ∘ τ) = π ∘ Φ. If π is faithful, this implies that the pair {π, π̄ ∘ τ} is universal for {A, N, α}.

3. The reduced twisted crossed product.

Let G be a locally compact group and let (A, α) ∈ C_{G}^{∗}. Let N be a closed normal subgroup of G and let {π, V } be a covariant pair of representations of {A, N, α|N} on a Hilbert space H such that π ⋊ V is a faithful representation of the full crossed product A ⋊_{α|N} N. If we identify A ⋊_{α|N} N with its image under π ⋊ V, a twisted action (γ, τ) of G on A ×_{α} N relative to N is defined by

γ_{s} (∫_{N} π(f(t))V_{t} dm_{N}(t)) = V_{s} (∫_{N} π(f(t))V_{t} dm_{N}(t)) V_{s}^{−1}
= ∫_{N} \frac{Δ_{G}(s)}{Δ_{G/N}(sN)}π(α_{s}(f(s^{−1}ts))V_{t} dm_{N}(t)

for f ∈ C_{c}(N, A) and s ∈ G, and

τ(n) = V_{n} (n ∈ N)

(cf. [Ech, proof of Theorem 1, et seq.]). This twisted action has the fundamental property that there is a natural isomorphism

A ⋊_{α} G ≅ (A ⋊_{α} N) ⋊_{γ,τ} G

[Gr]. In §5 we shall need an analogous isomorphism with A ⋊_{α,r} G on the left and A ⋊_{α,r} N on the right. To formulate such a result we give a definition of reduced twisted crossed product appropriate to the present context. Although there are various definitions of reduced twisted crossed product in the literature for cocycle twistings, our definition seems to be new.

For the definition we require a twisted version of the left regular representation of a crossed product, which we construct as follows. Let π be a not necessarily faithful representation of A on a Hilbert space H and let C_{c}(G, H, τ) be the set of those continuous H-valued functions f on G whose supports have relatively compact image in G/N and which satisfy

π̄(τ(n))f(s) = f(sn^{−1})  (∗)

(or, equivalently, π̄(τ(s^{−1}ns))(f(ns)) = f(s)) for s ∈ G, n ∈ N. For f ∈ C_{c}(G, H, τ) the nonnegative-valued real function s → ||f(s)|| is constant on each coset of N, and if we denote the common value on the coset sN by ||f(sN)||, the function sN → ||f(sN)|| is in C_{c}(G/N, ℝ). Let

||f||_{2}^{τ} = (∫_{G/N} ||f(sN)||^{2}dm_{G/N}(sN))^{1/2} .

Then || ||_{2}^{τ} is a norm on C_{c}(G, H, τ) and the completion L_{τ}^{2}(G, H) is a Hilbert space. It is not difficult to see that L_{τ}^{2}(G, H) is precisely the family of equivalence classes modulo null sets of H-valued measurable functions f on G satisfying (∗) and such that

(∫_{G/N} ||f(sN)||^{2}dm_{G/N}(sN))^{1/2} < ∞.

For a ∈ A, ξ ∈ L_{τ}^{2}(G, H) and g, s ∈ G let

(π_{α,τ} (a)ξ)(s) = π(α_{s^{−1}}(a))ξ(s)

and

(λ_{τ,g}ξ)(s) = ξ(g^{−1}s).

It is readily checked that π_{α,τ} (a)ξ and λ_{τ,g}ξ are in L_{τ}^{2}(G, H), so that π_{α,τ} and λ_{τ} are representations of A and G, respectively. When necessary we shall write λ_{τ}^{G} to make it clear which group is involved.

Lemma 3.1 1. The pair {π_{α,τ} , λ_{τ} } is τ-covariant. If π is faithful, so is π_{α,τ}.

2. If λ̇ denotes the representation of G on L^{2}(G/N) obtained by composing the quotient map G → G/N with the left regular representation of G/N, then {π_{α,τ} ⊗1_{L^{2}(G/N)}, λ_{τ} ⊗λ̇} is a τ-covariant pair of representations of {A, G, α, τ} which is unitarily equivalent to the pair {π_{α,τ} ⊗ 1_{L^{2}(G/N)}, λ_{τ} ⊗ 1_{L^{2}(G/N)}}.

3. If {π, V } is a τ-covariant pair of representations of {A, G, α, τ} on a Hilbert space H, then {π ⊗ 1_{L^{2}(G/N)}, V ⊗ λ̇} is a τ-covariant pair which is unitarily equivalent to the pair {π_{α,τ} , λ_{τ} }.

Proof: 1. It follows readily from the definitions that, for a ∈ A and g ∈ G,

λ_{τ,g}π_{α,τ} (a)λ_{τ,g}^{−1} = π_{α,τ} (α_{g}(a)),

so that {π_{α,τ} , λ_{τ} } is a covariant pair for (A, α). Also, if n, s ∈ N, ξ ∈ C_{c}(G, H, τ),

(λ_{τ,n}ξ)(s) = ξ(n^{−1}s)
= π̄(τ(s^{−1}ns))ξ(s)
= π̄(α_{s^{−1}}(τ(n)))ξ(s)
= (π̄_{α,τ} (τ(n))ξ)(s),

i.e. {π_{α,τ} , λ_{τ} } is τ-preserving.

For f ∈ C_{c}(G, H) let f̄ be given by

f̄(s) = ∫_{N} π̄(τ(m))f(sm)dm_{N}(m).

Then

f̄(sn^{−1}) = ∫_{N} π̄(τ(m))f(sn^{−1}m)dm_{N}(m)
= ∫_{N} π̄(τ(n))π̄(τ(n^{−1}m))f(sn^{−1}m)dm_{N}(m)
= π̄(τ(n))f̄(s).

Also, if supp f ⊆ C for some compact subset C of G, then supp f̄ ⊆ CN, so that f̄ is in L_{τ}^{2}(G, H).

Suppose that π is faithful. Let a be a nonzero element of A and let ξ ∈ H such that π(a)ξ ≠ 0. For ε > 0 let C be a compact neighbourhood of the identity e in G such that ||τ(m^{−1})ξ − ξ|| ≤ ε for m ∈ C ∩ N. Taking f a continuous nonnegative-valued real function on G with support in C^{−1} such that ∫_{N} f(n)dm_{N}(n) = 1 and defining F ∈ C_{c}(G, H) by F (s) = f(s)ξ,

||(π_{α,τ} (a)F̄)(e) − π(a)ξ|| = || ∫_{N} f(m)π(a)(π(τ(m))ξ − ξ)dm_{N}(m)||
≤ sup_{n∈C∩N} ||π(a)(π(τ(n))ξ − ξ)|| ∫_{N} f(m^{−1})dm_{N}(m)
≤ ε||a||

which implies, since ε is arbitrary and π is faithful, that π_{α,τ} (a) ≠ 0, i.e. π_{α,τ} is faithful.

2. Regarding elements of L_{τ}^{2}(G, H) ⊗ L^{2}(G/N) as equivalence classes of H-valued functions on G × (G/N), it is straightforward to show that a unitary operator U on L_{τ}^{2}(G, H) ⊗ L^{2}(G/N) is defined by

(Uξ)(r, sN) = ξ(r, r^{−1}sN).

Then

U(π_{α,τ} (a) ⊗ 1)U^{∗} = π_{α,τ} (a) ⊗ 1

and

U(λ_{τ,g} ⊗ 1)U^{∗} = λ_{τ,g} ⊗ λ̇_{gN},

for a ∈ A and g ∈ G, i.e. U implements the stated equivalence.

3. For ξ ∈ C_{c}(G/N, H) let W ξ be the H-valued function on G given by

(W ξ)(s) = V_{s^{−1}}ξ(sN).

Then for n ∈ N

(W ξ)(sn^{−1}) = V_{ns^{−1}}ξ(sN)
= (π̄(τ(n))W ξ)(s),

so that W ξ ∈ L_{τ}^{2}(G, H). It is also immediate that ||W ξ||_{2}^{τ} = ||ξ||_{2}, the latter norm being that on L^{2}(G/N, H). Thus W extends to an isometry of L^{2}(G/N, H) into L_{τ}^{2}(G, H). For ξ ∈ L_{τ}^{2}(G, H) the H-valued function s → V_{s}ξ(s) on G is constant on each coset of N. Letting W_{1}ξ be the H-valued function on G/N given by

(W_{1}ξ)(sN) = V_{s}ξ(s),

W_{1} is an isometry from L_{τ}^{2}(G, H) to L^{2}(G/N, H), and W_{1} = W^{−1}. Hence W is bijective. Moreover

(λ_{τ,g}W ξ)(s) = (W ξ)(g^{−1}s)
= V_{s^{−1}}V_{g}ξ(g^{−1}sN)
= V_{s^{−1}}((V_{g} ⊗ λ̇_{g})ξ)(sN)
= (W (V_{g} ⊗ λ̇_{g})ξ)(s)

and

(π_{α,τ} (a)W ξ)(s) = π(α_{s^{−1}}(a))V_{s^{−1}}ξ(sN)
= V_{s^{−1}}π(a)ξ(sN)
= (W (π(a) ⊗ 1)ξ)(s).

This shows that the pairs {π ⊗ 1_{L^{2}(G/N)}, V ⊗ λ̇} and {π_{α,τ} , λ_{τ} } are unitarily equivalent.

□

By analogy with the untwisted case, we would like to define the reduced twisted crossed product A ⋊_{α,τ,r} G to be the image of A ⋊_{α} G under the representation π_{α,τ} ⋊ λ_{τ}, where π is some faithful representation of A. First, however, it is necessary to show that the resulting quotient of A ⋊_{α} G does not depend on the choice of π. This is achieved in what follows by showing that, for a given π, the τ-covariant pair {π_{α,τ} , λ_{τ}^{G}} is obtained by inducing the representation π of A ≅ A ⋊_{α,τ} N to M(Ind(A, α|N) ⋊_{α̃,τ̃} G) via the equivalence bimodule X_{A} of §2 and composing the induced representation with canonical morphisms from A and G into this multiplier algebra.

Let H be a closed subgroup of G containing N, let {A, α} ∈ C_{H}^{∗} and let Ind(A, α) be the associated C*-algebra defined in §2. Let π be a faithful representation of A on H and for ψ ∈ Ind(A, α) and ξ ∈ L_{τ}^{2}(G, H) let

(π̃_{α,τ} (ψ)ξ)(s) = π(ψ(s))ξ(s).

Then

((π̃_{α,τ} (ψ)ξ)(sn^{−1}) = π(ψ(sn^{−1}))π̄(τ(n))ξ(s)
= π̄(τ(n))π(α_{n^{−1}}(ψ(sn^{−1})))ξ(s)
= π̄(τ(n))π(ψ(s))ξ(s)
= π̄(τ(n))((π̃_{α,τ} (ψ)ξ)(s),

so that π̃_{α,τ} (ψ)ξ ∈ L_{τ}^{2}(G, H), and π̃_{α,τ} is a representation of Ind(A, α) on L_{τ}^{2}(G, H). For g ∈ G,

(λ_{τ,g}^{G}π̃_{α,τ} (ψ)λ_{τ,g^{−1}}^{G}ξ)(s) = π(ψ(g^{−1}s))ξ(s)
= (π̃_{α,τ} (α̃_{g}(ψ))ξ)(s),

so that {π̃_{α,τ} , λ_{τ}^{G}} is a covariant pair of representations of the covariant system {Ind(A, α), G, α̃}. The proof of Lemma 3.1 (1) shows that

λ_{τ,n}^{G} = ¯ π̃_{α,τ} (τ̃(n))

for n ∈ N, which implies that the pair {π̃_{α,τ} , λ_{τ}^{G}} is τ̃-covariant.

Proposition 3.2 The τ̃-covariant pair of representations of {Ind(A, α), G, α̃, τ̃} induced from the τ-covariant pair of representations {π_{α,τ} , λ_{τ}^{H}} of {A, H, α, τ} via the equivalence bimodule X_{A} of §2 is unitarily equivalent to the pair {π̃_{α,τ} , λ_{τ}^{G}}.

Proof: We shall also assume, as we may, that the left Haar measures m_{G}, m_{H}, m_{N} and m_{G/N} have been chosen so that

∫_{G} f(s) dm_{G}(s) = ∫_{G/N} ∫_{N} f(sn) dm_{N}(n)dm_{G/N}(sN)  (1)

and

∫_{H} g(t) dm_{H}(t) = ∫_{H/N} ∫_{N} g(tn) dm_{N}(n)dm_{H/N}(sN)  (2)

for f ∈ C_{c}(G), g ∈ C_{c}(H).

For f, g ∈ X_{0} = C_{c}(G, A) and ξ, η ∈ L_{τ}^{2}(H, H), we calculate the inner product (f ⊗ ξ|g ⊗ η) in ^{X_{A}}L_{τ}^{2}(H, H). To prevent the notation becoming too cumbersome, we regard H as an M(A)-module via π̄, so that, for a ∈ M(A) and ζ ∈ H, aζ will denote (π̄(a))ζ, and similarly regard L_{τ}^{2}(H, H) as an M(A⋊_{α}H)- module via \overline{π_{α,τ} ⋊ λ_{τ}^{H}}. Then

(f ⊗ ξ|g ⊗ η)
=  (⟨g, f⟩_{B_{0}}ξ|η)

=  ∫_{H/N} ∫_{H} ∫_{G} δ(t)Δ_{G}(s)^{−1}(α_{r^{−1}}(g(s^{−1})^{∗})α_{r^{−1}t}(f(s^{−1}t))ξ(t^{−1}r)|η(r))
×dm_{G}(s)dm_{H}(t)dm_{H/N}(rN)

=  ∫_{H/N} ∫_{H} ∫_{G} δ(t)Δ_{G}(s)^{−1}(α_{r^{−1}t}(f(s^{−1}t))ξ(t^{−1}r)|α_{r^{−1}}(g(s^{−1}))η(r))

×dm_{G}(s)dm_{H}(t)dm_{H/N}(rN)

\begin{matrix}t→rt \\ s→rs \\ = \end{matrix} ∫_{H/N} ∫_{H} ∫_{G} δ(t)δ(r)Δ_{G}(r)^{−1}Δ_{G}(s)^{−1}

×(α_{t}(f(s^{−1}t))ξ(t^{−1})|α_{r^{−1}}(g(s^{−1}r^{−1}))η(r)) ×dm_{G}(s)dm_{H}(t)dm_{H/N}(rN)

=^{s→s^{−1}}  ∫_{H/N} ∫_{H} ∫_{G} δ(t)δ(r)Δ_{G}(r)^{−1}(α_{t}(f(st))ξ(t^{−1})|α_{r^{−1}}(g(sr^{−1}))η(r))

×dm_{G}(s)dm_{H}(t)dm_{H/N}(rN)

=  ∫_{H/N} ∫_{H} ∫_{G/N} ∫_{N} δ(t)δ(r)Δ_{G}(r)^{−1}

×(α_{t}(f(smt))ξ(t^{−1})|α_{r^{−1}}(g(smr^{−1}))η(r)) ×dm_{N}(m)dm_{G/N}(sN)dm_{H}(t)dm_{H/N}(rN)

(by (1))

=^{t→m^{−1}t} ∫_{H/N} ∫_{H} ∫_{G/N} ∫_{N} δ(m)^{−1}δ(t)δ(r)Δ_{G}(r)^{−1}

×(α_{m^{−1}t}(f(st))ξ(t^{−1}m)|α_{r^{−1}}(g(smr^{−1}))η(r)) ×dm_{N}(m)dm_{G/N}(sN)dm_{H}(t)dm_{H/N}(rN)

=^{m→m^{−1}}  ∫_{H/N} ∫_{H} ∫_{G/N} ∫_{N} δ(t)δ(r)Δ_{G}(r)^{−1}Δ_{N}(m)^{−1}

×(α_{mt}(f(st))ξ(t^{−1}m^{−1})|α_{r^{−1}}(g(sm^{−1}r^{−1}))η(r)) ×dm_{N}(m)dm_{G/N}(sN)dm_{H}(t)dm_{H/N}(rN)

(since δ(m) = 1 for m ∈ N)

=  ∫_{H/N} ∫_{H} ∫_{G/N} ∫_{N} δ(t)δ(r)Δ_{G}(r)^{−1}Δ_{N}(m)^{−1}

×(τ(m)α_{t}(f(st))τ(m^{−1})ξ(t^{−1}m^{−1})|α_{r^{−1}}(g(sm^{−1}r^{−1}))τ(m)η(rm))

×dm_{N}(m)dm_{G/N}(sN)dm_{H}(t)dm_{H/N}(rN)

=  ∫_{H/N} ∫_{H} ∫_{G/N} ∫_{N} δ(t)δ(r)Δ_{G}(rm)^{−1}

×(α_{t}(f(st))ξ(t^{−1})|α_{m^{−1}r^{−1}}(g(sm^{−1}r^{−1}))η(rm))
×dm_{N}(m)dm_{G/N}(sN)dm_{H}(t)dm_{H/N}(rN)

=  ∫_{G/N} ∫_{H} ∫_{H} δ(t)δ(r)Δ_{G}(r)^{−1}(α_{t}(f(st))ξ(t^{−1})|α_{r^{−1}}(g(sr^{−1}))η(r))
×dm_{H}(t)dm_{H}(r)dm_{G/N}(sN)

(by (2))

=^{r→r^{−1}}  ∫_{G/N} ∫_{H} ∫_{H} δ(t)δ(r)(α_{t}(f(st))ξ(t^{−1})|α_{r}(g(sr))η(r^{−1}))
×dm_{H}(t)dm_{H}(r)dm_{G/N}(sN).

Let T (f ⊗ ξ) be the A-valued function on G given by

(T (f ⊗ ξ))(s) = ∫_{H} δ(t)α_{t}(f(st))ξ(t^{−1}) dm_{H}(t).

Then

T (f ⊗ ξ)(sn^{−1}) = ∫_{H} δ(t)α_{t}(f(sn^{−1}t))ξ(t^{−1}) dm_{H}(t)
= ∫_{H} δ(t)α_{nt}(f(st))ξ(t^{−1}n^{−1}) dm_{H}(t)
= τ(n)((T (f ⊗ ξ))(s)),

since δ(n) = 1 for n ∈ N, and if K_{1} is the support of f in G and K_{2} is the support of ξ in H, then the support of T (f ⊗ ξ) is contained in the set K_{1}K_{2}. The latter set has relatively compact image in G/N since the same is true of K_{2} and K_{1} is compact. It follows that T (f ⊗ ξ) ∈ C_{c}(G, H, τ). By the above calculation, T is an isometric linear map from a dense subspace of ^{X_{A}}L_{τ}^{2}(H, H) into L_{τ}^{2}(G, H). Standard arguments involving partitions of unity show that the image of T is dense in L_{τ}^{2}(G, H). Thus T has an extension to an isometry U from ^{X_{A}}L^{2}(H, H) onto L_{τ}^{2}(G, H).

For ψ ∈ Ind(A, α), f ∈ X_{0}, ξ ∈ H and g, s ∈ G,
[T (ψf ⊗ ξ)](s) = ∫_{H} γ(t)α_{t}(ψ(st)f(st))ξ(t^{−1}) dm_{H}(t)
= ψ(s)(T (f ⊗ ξ))(s)
= [π̃_{α,τ} (ψ)(T (f ⊗ ξ))](s)

and

[T (gf ⊗ ξ)](s) = ∫_{H} γ(tα_{t}(f(g^{−1}st))ξ(t^{−1}) dm_{H}(t)
= [λ_{τ,g}(T (f ⊗ ξ))](s),

from which it follows that U implements the desired unitary equivalence. □

In the following corollary we assume that H = N, so that α is an action of N on A by inner automorphisms.

Corollary 3.3 If the representation π is faithful, then the integrated form representation π̃_{α,τ} ⋊τ̃ λ_{τ} of Ind(A, α) ⋊_{α̃,τ̃} G is faithful.

Proof: By Proposition 3.2, π̃_{α,τ} ⋊τ̃ λ_{τ} is the representation of Ind(A, α) ⋊_{α̃,τ̃} G induced from the representation π ⋊_{τ} λ_{τ}^{N} of A ⋊_{α,τ} N via X_{A}. The Hilbert space L_{τ}^{2}(N, H) is just the space of continuous H-valued functions f such that

f(n) = π̄(τ(n^{−1}))f(e)

for n ∈ N, with norm ||f(e)||, and the map f → f(e) is an isometry of L_{τ}^{2}(N, H) onto H. This map implements a unitary equivalence between the τ-covariant pairs {π_{α,τ} , λ_{τ}^{N}} and {π, π̄ ∘ τ}. By Remark 2.5 (2), the latter pair is universal for {A, N, α, τ}, since π is faithful. Hence {π_{α,τ} , λ_{τ}^{N}} is universal for {A, N, α, τ}, so that π_{α,τ} ⋊_{τ} λ_{τ}^{N} is a faithful representation of A ⋊_{τ} N (≅ A, by Remark 2.5 (2)). Since faithful representations induce faithful representations, the result follows. □

We are now ready to define the reduced twisted crossed product. Let G be a locally compact group with a closed normal subgroup N. Let (A, α) ∈ C_{G}^{∗} and let τ : N → U(A) be a twisting map relative to α. Let π be a faithful representation of A on a Hilbert space H. Letting E = Ind(A, α|N), we note in passing that by the discussion of §2 the pair (E, α̃) is G-equivariantly isomorphic to C_{0}(G/N, A) with G acting by left translation. By Corollary 3.3, the representation π̃_{α,τ} ⋊τ̃ λ_{τ} of E ⋊_{α̃,τ̃} G on L_{τ}^{2}(G, H) is faithful, and hence so is its canonical extension \overline{π̃_{α,τ} ⋊τ̃ λ_{τ}} to M(E ⋊_{α̃,τ̃} G). Identifying M(E ⋊_{α̃,τ̃} G) with its image under \overline{π̃_{α,τ} ⋊τ̃ λ_{τ}}, for a ∈ A, ψ ∈ E and ξ ∈ L_{τ}^{2}(G, H),

(π_{α,τ} (a)π̃_{α,τ} (ψ)ξ)(s) = π(α_{s^{−1}}(a))π(ψ(s))ξ(s)
= (π̃_{α,τ} (aψ)ξ)(s),

where aψ is the element of E given by

(aψ)(s) = α_{s^{−1}}(a)ψ(s).

For f ∈ C_{c}(G,E) and a ∈ A, let af be the element of C_{c}(G,E) given by (af)(s) = af(s). Then

π_{α,τ} (a)(π̃_{α,τ} ⋊τ̃ λ_{τ})(f) = π_{α,τ} (a) ∫_{G} π̃_{α,τ} (f(s))λ_{τ,s} dm_{G}(s)
= ∫_{G} π̃_{α,τ} (af(s))λ_{τ,s} dm_{G}(s)
= (π̃_{α,τ} ⋊τ̃ λ_{τ})(af)
∈ Ind(A, α) ⋊_{α̃,τ̃} G,

and, by taking limits of sequences of such f, it follows that π_{α,τ} (a)(π̃_{α,τ} ⋊τ̃ λ_{τ})(x) ∈ E ⋊_{α̃,τ̃} G for all x ∈ E ⋊_{α̃,τ̃} G. Similarly, if g ∈ G and ψ ∈ E, let gψ be the element of E given by

(gψ)(s) = α_{g}(ψ(g^{−1}s)),

and for f ∈ C_{c}(G, E) let gf be the element of C_{c}(G, E) such that (gf)(s) = gf(s). A similar calculation shows that λ_{τ,g} multiplies E ⋊_{α̃,τ̃} G. There are thus canonical homomorphisms π_{0} and λ_{0} from A and G into M(E ⋊_{α̃,τ̃} G) given by

π_{0}(a)f = af, λ_{0}(g)g = gf

for f ∈ C_{c}(G, E). Moreover π_{α,τ} = (\overline{π̃_{α,τ} ⋊τ̃ λ_{τ}})∘π_{0} and λ_{τ} = (\overline{π̃_{α,τ} ⋊τ̃ λ_{τ}})∘λ_{0}, from which it follows that π_{0} is an isomorphism, and {π_{0}, λ_{0}} is a τ-covariant pair.

Definition 3.4 The reduced twisted crossed product A ⋊_{α,τ,r} G is the image of A ⋊_{α} G in M(Ind(A, α|N) ⋊_{α̃,τ̃} G) under the *-homomorphism π_{0} ⋊ λ_{0}.

In the next proposition we consider the natural class of mappings between twisted covariant systems with respect to given G and N. Let {A, G, α, τ} and {B, G, β, τ′} be two such systems and let θ : A → B be a G-equivariant *-homomorphism. We shall say that θ is twist-equivariant (with respect to τ and τ′) if θ(τ(n)a) = τ′(n)θ(a) for n ∈ N and a ∈ A.

Proposition 3.5 1. Let π be a representation of A on a Hilbert space H. Then the representation π_{α,τ} ⋊_{τ} λ_{τ} of A ⋊_{α,τ} G is the composition of a representation π_{α,τ} ⋊_{τ,r} λ_{τ} of A ⋊_{α,τ,r} G with the canonical quotient map A ⋊_{α,τ} G → A ⋊_{α,τ,r} G. If π is faithful, then so is π_{α,τ} ⋊_{τ,r} λ_{τ}.

2. Let {A, G, α, τ} and {B, G, β, τ′} be twisted covariant systems with respect to the closed normal subgroup N of G, and let θ : A → B be a *- homomorphism which is twist-equivariant with respect to the given actions and twisting maps. Then there is a unique *-homomorphism θ_{N,r} : A ⋊_{α,τ,r} G → B ⋊_{β,τ′,r} G such that the diagram

\xymatrix{A ⋊_{α} G \ar[r]^{θ_{u}} \ar[d] & B ⋊_{β} G \ar[d] \\ A ⋊_{α,τ,r} G \ar[r]^{θ_{N,r}} & B ⋊_{α,τ′,r} G}

commutes, the vertical arrows denoting the canonical *-homomorphisms. The morphism θ_{N,r} is injective (resp. surjective) if and only if θ is injective (resp. surjective). If Im θ is an ideal of B, then Im θ_{N,r} is an ideal of B ⋊_{β,τ′,r} G.

Proof: 1. This follows immediately from the factorisations π_{α,τ} = (\overline{π̃_{α,τ} ⋊τ̃ λ_{τ}})∘ π_{0} and λ_{τ} = (\overline{π̃_{α,τ} ⋊τ̃ λ_{τ}})∘λ_{0}, and the fact that, if π is faithful, then \overline{π̃_{α,τ} ⋊τ̃ λ_{τ}} is a faithful representation of M(Ind(A, α) ⋊_{α̃,τ̃} G).

2. If {π, V } is a τ′-covariant pair of representations of {B, G, β}, then {π ∘ θ, V } is a covariant pair of representations of {A, G, α} on a Hilbert space H, and for n ∈ N and a ∈ A,

\overline{(π ∘ θ)}(τ(n))(π ∘ θ)(a) = π(θ(τ(n)a))
= \overline{π}(τ′(n))(π ∘ θ)(a).

If θ is surjective, this shows that \overline{(π ∘ θ)}(τ(n)) = V_{n}, so that the pair \{π∘θ, V \} is τ-covariant. By part 1, there is a canonical *-epimorphism θ_{N,r} : A ⋊_{α,τ,r} G → B ⋊_{β,τ′,r} G such that

(π ⋊_{τ′,r} V) ∘ θ_{N,r} = (π ∘ θ) ⋊_{τ,r} V,  (∗)

where (π ∘ θ) ⋊_{τ,r} V and π ⋊_{τ′,r} V are the representations of A ⋊_{α,τ,r} G and B ⋊_{β,τ′,r} G associated with π ∘ θ and π, respectively.

If, on the other hand, θ is injective, let H_{1} be the closure in H of π(θ(A))H and let E be the projection onto H_{1}. Then by the covariance condition, EV_{g} = V_{g}E for g ∈ G. Letting W_{g} = EV_{g}|H_{1}, the above identity implies that

\overline{(π ∘ θ)}(τ(n)) = W_{n}E

for n ∈ N. Defining σ by σ(a) = π(θ(a))|H_{1}, it follows that {σ, W } is a τcovariant pair for {A, G, α, τ}. It is easily seen that the images of A⋊_{α} G under the representations (π ∘ θ) ⋊ V and σ ⋊ W are isomorphic. If π is faithful, so is σ, and the latter image is canonically isomorphic to A ⋊_{α,τ,r} G by part 1. This implies that there is a canonical *-monomorphism θ_{N,r} : A ⋊_{α,τ,r} G → B ⋊_{β,τ′,r} G for which (∗) holds.

Combining these two cases, the existence of θ_{N,r} satisfying (∗) for arbitrary θ follows. The commutativity of the given diagram is a simple consequence of (∗). □

We return now to the general situation where N is a closed normal subgroup of G, H is a closed subgroup of G containing N, (A, α) ∈ C_{H}^{∗} and τ : N → U(A) is a twisting map relative to α. Let X_{A} be the (Ind(A, α)⋊α̃ G)– (A ⋊_{α} H) equivalence bimodule constructed in §2, and let π be a faithful representation of A on a Hilbert space H. The kernel J_{τ,r} of the canonical quotient map A ⋊_{α} H → A ⋊_{α,τ,r} H is the kernel of the representation π_{α,τ} ⋊ λ_{τ}^{H}. By Proposition 3.2 the representation of Ind(A, α) ⋊α̃ G induced from this representation via X_{A} is the integrated form of the pair {π̃_{α,τ} , λ_{τ}^{G}}. Let I be the kernel of this representation. By Lemma 3.1, the τ̃-covariant pair {π̃_{α,τ} ⊗ 1_{L^{2}(G/N)}, λ_{τ}^{G} ⊗ 1_{L^{2}(G/N)}} is unitarily equivalent to the pair {π̃_{α,τ} ⊗ 1_{L^{2}(G/N)}, λ_{τ}^{G} ⊗ λ̇}, which is in turn unitarily equivalent to the pair {(π̃_{α,τ})_{α̃,τ̃}, λ_{τ} }. Since π, and hence π̃_{α,τ}, are faithful, the kernel of the integrated form of the pair {(π̃_{α,τ})_{α̃,τ̃}, λ_{τ} } is the ideal I_{τ̃,r}, the kernel of the canonical *-homomorphism from Ind(A, α) ⋊α̃ G to Ind(A, α) ⋊_{α̃,τ̃,r} G, by Proposition 3.5 (1). It follows that that I coincides with I_{τ̃,r} and corresponds to J_{τ,r} via X_{A}. By [Rie2] X_{A,τ,r} = X_{A}/X_{A}J_{τ,r} is an (Ind(A, α) ⋊_{α̃,τ̃,r} G)– (A ⋊_{α,τ,r} H) equivalence bimodule. In fact, X_{A,τ,r} is obtained from the E_{0}–B_{0} equivalence bimodule X_{0} = C_{c}(G, A) of §2 by completing with respect to the semi-norm f → ||(π_{α,τ} ⋊ λ_{τ}^{H})(< f, f >_{B})||^{1/2}. This proves

Theorem 3.6 The C*-algebras Ind(A, α)⋊_{α̃,τ̃,r} G and A⋊_{α,τ,r} H are strongly Morita equivalent via the equivalence bimodule X_{A,τ,r}.

Remark 3.7 When H = N, it follows by Remark 2.5 (2) that if π is a faithful representation of A, the pair {π_{α,τ} , λ_{τ}^{N}} is universal for {A, N}, and the kernels of the canonical quotient maps of A⋊_{α} N onto A⋊_{α,τ} N and A⋊_{α,τ,r} N coincide. Since these kernels correspond via X_{A} to the kernels of the canonical quotient maps of Ind(A, α) ⋊α̃ G onto Ind(A, α) ⋊_{α̃,τ̃} G and Ind(A, α) ⋊_{α̃,τ̃,r} G, respectively, the latter kernels coincide, and there is a canonical isomorphism

Ind(A, α) ⋊_{α̃,τ̃} G ≅ Ind(A, α) ⋊_{α̃,τ̃,r} G.

Specialising to the case where A = ℂ, N = {1}, we recover the well known isomorphism

C_{0}(G) ⋊_{λ} G ≅ C_{0}(G) ⋊_{λ,r} G,

where λ is the action of G on C_{0}(G) by left translation. As is well-known, the reduced crossed product on the right-hand side is isomorphic to the space of compact operators on L^{2}(G).

If (A, α), (B, β) ∈ C_{H}^{∗}, and θ : A → B is an H-equivariant *-homomorphism, an associated *-homomorphism θ̃:Ind(A, α) →Ind(B, β) is defined by

(θ̃(f))(h) = θ(f(h)).

Let (A, α) ∈ C_{H}^{∗} and let I be an α_{H}-invariant ideal of A. If ι : I → A and q : A → A/I denote the inclusion and quotient maps, respectively, then ι and q are equivariant when I has the restriction action α|I and A/I the quotient action α̇.

Lemma 3.8 The sequence

\xymatrix{0 \ar[r] & Ind (I, α|) \ar[r]^{ι̃} & Ind (A, α) \ar[r]^{q̃} & Ind (A/I, α̇) \ar[r] & 0}

is exact.

Proof: If f ∈ ker q̃, then

0 = (q̃(f))(h) = q(f(h)) ⇒ f(h) ∈ I

for h ∈ H, i.e. f ∈Ind(I, α|). □

Let (A, α) ∈ C_{H}^{∗} and let τ : N → U(A) be a twisting map relative to α. If I is an α_{H}-invariant ideal of A, there are unital *-homomorphisms ῑ: M(I) → M(A) and q̄ : M(A) → M(A/I) given by ῑ(u)x = ux and q̄(u)q(a) = q(ua) which extend ι and q, respecively. Twisting maps τ_{I} and τ_{A/I} relative to the restriction action α|I and the quotient action α̇ are given by τ_{I} = ῑ ∘ τ and τ_{A/I} = q̄ ∘ τ, respectively, relative to which ι and q are twist-equivariant. To simplify the notation, we write τ for both τ_{I} and τ_{A/I}. By the same token, there are twisting maps relative to N, which will be denoted by τ̃, on Ind(I, α|) and Ind(A/I, α̇), relative to which the induced morphisms ι̃ and q̃ are twistequivariant.

The following theorem is the main technical result for §§4 and 5.

Theorem 3.9 The sequence

\xymatrix{0 \ar[r] & I ⋊_{α|,τ,r} H \ar[r]^{ι_{N,r}} & A ⋊_{α,τ,r} H \ar[r]^{q_{N,r}} & (A/I) ⋊_{α̇,τ,r} H \ar[r] & 0} (∗)

is exact if and only if the sequence

\xymatrix{0 \ar[r] & Ind(I, α|) ⋊_{α̃|,τ̃,r} G \ar[r]^{ι̃_{N,r}} & Ind(A, α) ⋊_{α̃,τ̃,r} G}

[figure: q̃_{N,r}; Ind(A/I, α̇) ⋊_{˜α̇,τ̃,r} G; 0]  (∗∗)

is exact.

Proof: Let B_{I}, B_{A} and B_{A/I} denote the three C*-algebras in (∗) and E_{I}, E_{A} and E_{A/I} the three C*-algebras in (∗∗). Let J and J̃ be the kernels of the homomorphisms q_{N,τ} : B_{A} → B_{A/I} and q̃_{N,τ̃} : E_{A} → E_{A/I}. Let J = ker q_{N,τ} and J̃ = ker q̃_{N,τ̃}. Identifying B_{I} and E_{I} with their images in B_{A} and E_{A} under the embeddings ι_{N,τ} and ι̃_{N,τ̃}, respectively, it follows by Proposition 3.5 that B_{I} ⊆ J and E_{I} ⊆ J̃. To prove the proposition it suffices to show that the ideals B_{I} and J of B_{A} correspond to the ideals E_{I} and J̃, respectively, of E_{A} via the equivalence bimodule X_{A,τ,r}.

The natural embedding of C_{c}(G, I) in C_{c}(G, A) extends to an embedding of the E_{I}–B_{I} equivalence bimodule X_{I,τ,r} as an E_{I}–B_{I} submodule X of X_{A,τ,r}. It follows readily from the definitions that X is the norm closed linear span of both X_{A,τ,r}B_{I} and E_{I}X_{A,τ,r}, from which it follows that E_{I} and B_{I} correspond via X_{A,τ,r}.

Let σ be a faithful representation of A/I on a Hilbert space H, and let π = σ ∘ q. By Propositions 3.2 and 3.5, the representation of E_{A} induced from the representation π ⋊_{τ,r} λ_{τ}^{H} of B_{A} via X_{A,τ,r} is unitarily equivalent to

π̃_{α,τ} ⋊_{τ̃,r} λG ; τ̃. Since

π ⋊_{τ,r} λ_{τ}^{H} = (σ ⋊_{τ,r} λ_{τ}^{H}) ∘ q_{N,r}

and

π̃_{α,τ} ⋊_{τ̃,r} λG ; τ̃ = (σ̃_{α̇,τ} ⋊_{τ̃,r} λG ; τ̃) ∘ q̃_{N,r}

the images of π ⋊_{τ,r} λ_{τ}^{H} and π̃_{α,τ} ⋊_{τ̃,r} λG ; τ̃ are canonically isomorphic to B_{A/I} and E_{A/I} and their kernels are J and J̃, respectively. By the discussion in §2 it follows that J and J̃ correspond via X_{Aτ,r}. Since correspondence of ideals respects inclusion, this implies that J = B_{I} if and only if J̃ = E_{I}, i.e. (∗) is exact if and only if (∗∗) is exact. □

In §5 we shall consider a continuous action α of H on A which extends to a continuous action, also denoted by α, of G on A, and such that the ideal I is α_{G}-invariant, so that α| and α̇ also extend to continuous actions α| and α̇ of G on I and A/I, respectively. As noted in §2, there are then natural isomorphism

θ_{I} : Ind(I, α|) → C_{0}(G/H, I),
θ_{A} : Ind(A, α) → C_{0}(G/H, A),

and

θ_{A/I} : Ind(A/I, α̇) → C_{0}(G/H, A/I).

A twisting map τ̃ : N → M(C_{0}(G/H, A)) is defined by

(τ̃(n)f)(sH) = τ(n)f(gH) (f ∈ C_{0}(G/H, A)).

Relative to this twisting map and that on Ind(A, α), the map θ_{A} is twist equivariant. Defining twisting maps τ̃ : N → M(C_{0}(G/H, I)) and τ̃ : N → M(C_{0}(G/H, A/I)) similarly, the maps θ_{I} and θ_{A/I} are twist-equivariant, and it follows straightforwardly that the diagram

\xymatrix{0 \ar[r] & Ind(I, α|) \ar[d]^{θ_{I}} \ar[r]^{ι̃} & Ind(A, α) \ar[d]^{θ_{A}} \ar[r]^{q̃} & Ind(A/I, α̇) \ar[d]^{θ_{A/I}} \ar[r] & 0 \\ 0 \ar[r] & C_{0}(G/H, I) \ar[r]^{id⊗ι} & C_{0}(G/H, A) \ar[r]^{id⊗q} & C_{0}(G/H, A/I) \ar[r] & 0}

commutes (we have identified C_{0}(G/H, A) with C_{0}(G/H) ⊗ A, etc., in the bottom row to define the horizontal maps). Taking reduced twisted crossed products by G, we obtain a commutative diagram

\xymatrix{0 \ar[d] & 0 \ar[d] \\ Ind(I, α|) ⋊_{α̃,τ̃,r} G \ar[r] \ar[d] & C_{0}(G/H, I) ⋊_{Δ^{α},τ̃,r} G \ar[d] \\ Ind(A, α) ⋊_{α̃,τ̃,r} G \ar[r] \ar[d] & C_{0}(G/H, A) ⋊_{Δ^{α},τ̃,r} G \ar[d] \\ Ind(A/I, α̇) ⋊_{˜α̇,τ̃,r} G \ar[r] \ar[d] & C_{0}(G/H, A/I) ⋊_{Δα̇,τ̃,r} G \ar[d] \\ 0 & 0}

Since the maps denoted by the horizontal arrows are bijections, by Proposition 3.5 (2), the left-hand column is exact if and only if the same is true of the right-hand column. The following corollary is now an immediate consequence of Theorem 3.9.

Corollary 3.10 The sequence

\xymatrix{0 \ar[r] & I ⋊_{α|,τ,r} H \ar[r]^{ι_{N,r}} & A ⋊_{α,τ,r} H \ar[r]^{q_{N,r}} & (A/I) ⋊_{α̇,τ,r} H \ar[r] & 0}

is exact if and only if the sequence

\xymatrix{0 \ar[r] & (C_{0}(G/H, I) ⋊_{Δ^{α|},τ̃,r} G \ar[r]^{(id⊗ι)_{N,r}} & (C_{0}(G/H, A) ⋊_{Δ^{α},τ̃,r} G}

[figure: (id⊗q)_{N,r}; (C_{0}(G/H, A/I)) ⋊_{Δα̇,τ̃,r} G; 0]

is exact.

4. Closed subgroups.

Throughout this section H will denote a closed subgroup of the locally compact group G. The following theorem is the first of the two main results of this section.

Theorem 4.1 If G is exact, then so is H.

Proof: Let (A, α) ∈ C_{H}^{∗} and let I be an α_{H}-invariant ideal of A. By Lemma 3.8, the associated sequence

0 → Ind(I, α|) → Ind(A, α) → Ind(A/I, α̇) → 0

is exact. Since G is exact, the corresponding sequence

0 → Ind(I, α|) ⋊_{α̃,r} G → Ind(A, α) ⋊_{α̃,r} G → Ind(A/I, α̇) ⋊_{˜α̇,r} G → 0

is exact. By Theorem 3.9, the sequence

\xymatrix{0 \ar[r] & I ⋊_{α|,r} H \ar[r]^{ι_{r}} & A ⋊_{α,r} H \ar[r]^{q_{r}} & (A/I) ⋊_{α̇,r} H \ar[r] & 0}

is exact. Since (A, α) and I are arbitrary, this implies the exactness of H. □

For the rest of the section we assume that G is σ-compact and H is exact. By [Bo, Chap. VII, §2, Th. 2] there is a Borel measure on G/H which is quasiinvariant for the action of G on H given by left translation, though in general it is not possible to find a measure which is actually invariant for this action. If an invariant Borel probability measure on G/H exists, H is said to have finite covolume in G. We shall assume that this is the case for the rest of this section.

Let (A, α) ∈ G_{G}^{∗} and let ᾱ denote the canonical extension of α to a continuous action of G on the multiplier algebra M(A). The natural embedding of A in M(A) is G-equivariant relative to this action and the corresponding crossed product map is an embedding of A ⋊_{α,r} G as an ideal of M(A) ⋊_{ᾱ,r} G. It follows that there is a canonical *-homomorphism

M(A) ⋊_{ᾱ,r} G → M(A ⋊_{α,r} G)

extending the natural embedding of A ⋊_{α,r} G in M(A ⋊_{α,r} G). This *- homomorphism is, in fact, an embedding. To see this, let π be a faithful representation of A on a Hilbert space H. Then π_{α} ⋊_{r} λ is a faithful representation of A ⋊_{α,r} G on H ⊗ L^{2}(G) which extends to a faithful representation of M(A ⋊_{α,r} G) on H ⊗ L^{2}(G). Let π̄ be the canonical extension of π to M(A). Then π̄ is a faithful representation of M(A) on H and π̄ᾱ ⋊_{r} λ is a faithful representation of M(A) ⋊_{ᾱ,r} G on H ⊗ L^{2}(G). If we identify M(A ⋊_{α,r} G) with its image on H⊗L^{2}(G), the above *-homomorphism M(A)⋊_{ᾱ,r} G → M(A⋊_{α,r} G) is just the *-monomorphism π̄ᾱ ⋊_{r} λ.

Let (A, α) ∈ C_{G}^{∗} and let E_{A} = (C_{0}(G/H) ⊗ A) ⋊_{Δ^{α},r} G. Replacing (A, α) by the pair (C_{0}(G/H) ⊗ A, Δ^{α}) in the argument of the previous paragraph, we get a canonical embedding

κ : M(C_{0}(G/H) ⊗ A) ⋊_{Δ̄^{α},r} G → M(E_{A}),

where Δ̄^{α} is the canonical extension of the diagonal action Δ^{α} to M(C_{0}(G/H)⊗ A). If π is the embedding a → 1⊗a of A in M(C_{0}(G/H)⊗A), π is G-equivariant and the corresponding crossed product map

π_{r} : A ⋊_{α,r} G → M(C_{0}(G/H) ⊗ A) ⋊_{Δ̄^{α},r} G is an embedding. Let Φ_{A} denote the embedding

κπ_{r} : A ⋊_{α,r} G → M(E_{A}).

If I is an α_{G}-invariant ideal of A, and q : A → A/I is the quotient map, associated with the *-homomorphism q̃ = id ⊗ q : C_{0}(G/H) ⊗ A → C_{0}(G/H) ⊗ (A/I) is the surjective crossed product *-homomorphism

q̃_{r} : E_{A} → E_{A/I}.

This map extends to a *-homomorphism M(E_{A}) → M(E_{A/I}), also denoted by q̃_{r}, which need not be surjective.

Lemma 4.2 q̃_{r}Φ_{A} = Φ_{A/I}q_{r}.

Proof: Let π and λ be the canonical embeddings of C_{0}(G/H) ⊗ A and G in M(E_{A}), respectively, and let π̄ be the embedding of M(C_{0}(G/H) ⊗ A) in M(E_{A}) obtained by extending π as above. Then for a ∈ C_{c}(G, A),

Φ_{A}(a) = ∫_{G} π̄(1 ⊗ a(s))λ_{s} ds.

The linear span of the subset

{∫_{G} π(f ⊗ b(t))λ_{s} ds : f ∈ C_{0}(G/H), b ∈ C_{c}(G, A)}

of E_{A} is dense in E_{A}, and for a, b ∈ C_{c}(G, A), f ∈ C_{0}(G/H),

q̃_{r}(Φ_{A}(a))q̄_{r}(∫_{G} π(f ⊗ b(t)λ_{t} dt) = q̃_{r}(Φ_{A}(a) ∫_{G} π(f ⊗ b(t)λ_{t} dt)
= q̄_{r}(∫_{G} ∫_{G} π(f ⊗ a(s)α_{s}(b(t)))λ_{st} ds dt)
= ∫_{G} ∫_{G} π(f ⊗ q(a(s))q(α_{s}(b(t))))λ_{s}t ds dt
= Φ_{A/I}(q(a))q̄_{r}(∫_{G} π(f ⊗ b(t)λ_{t} dt).

Thus q̃_{r}Φ_{A} = Φ_{A/I}q. □

Lemma 4.3 For f ∈ C_{0}(G/H) and a ∈ A ⋊_{α,r} G, π̄(f ⊗ 1)Φ_{A}(a) ∈ E_{A}.

Proof: By the continuity of Φ_{A}, it is enough to check this for a in the dense subset C_{c}(G, A) of A ⋊_{α,r} G. Then

π̄(f ⊗ 1)Φ_{A}(a) = ∫_{G} π(f ⊗ a(s))λ_{s} ds ∈ E_{A}.

□

Fix an invariant probability measure μ on G/H. The map P : C_{0}(G/H)⊗ A → A given by

P (f ⊗ a) = (∫_{G/H} f dμ) a

is completely positive, contractive and G-equivariant. The corresponding complete contraction P_{r} : E_{A} → A ⋊_{α,r} G is given by

P_{r}(∫_{G} π(f ⊗ a(s))λ_{s} ds) = (∫_{G} f dμ) a

for f ∈ C_{0}(G/H), a ∈ C_{c}(G, A).

Lemma 4.4 (i) For f ∈ C_{0}(G/H) and a ∈ A ⋊_{α,r} G,

P_{r}(π̄(f ⊗ 1)Φ_{A}(a)) = (∫_{G} f dμ) a.

(ii) P_{r}(E_{I}) ⊆ I ⋊_{α|,r} G.

Proof: (i) If a ∈ C_{c}(G, A),

π̄((f ⊗ 1)Φ_{A}(a)) = ∫_{G} π(f ⊗ a(s))λ_{s} ds

and

P_{r}(π̄((f ⊗ 1)Φ_{A}(a)) = (∫_{G} f dμ) a,

by the definition of P_{r}. The identity for general a now follows by the continuity of Φ_{A} and P_{r}.

(ii) This is immediate from the definition of P_{r}.

Theorem 4.5 Let G be a σ-compact group. If G has a closed exact subgroup H which has finite covolume in G, then G is exact.

Proof: Let μ be an invariant probability measure on G/H. We must show that if A is a C*-algebra with an action α of G and I is an α_{G}-invariant ideal of A with quotient map q : A → A/I, then ker q_{r} ⊆ I ⋊_{α,r} G.

Let x ∈ ker q_{r}. By Lemma 4.2, if f ∈ C_{0}(G/H),

q̄_{r}(π̄(f ⊗ 1)Φ_{A}(x)) = π̄(f ⊗ 1)q̄_{r}(Φ_{A}(x)) = Φ̄_{A/I}(q_{r}(x)) = 0,

and so π̄(f ⊗ 1)Φ_{A}(x) ∈ ker q̄_{r}. Since H is exact, the sequence

0 → I ⋊_{α|,r} H → A ⋊_{α,r} H → (A/I) ⋊_{α̇,r} H → 0

is exact. By Corollary 3.10, the sequence

0 → E_{I} → E_{A} → E_{A/I} → 0

is exact, so that ker q̄_{r} = E_{I}. Thus π̄(f ⊗ 1)Φ_{A}(x) ∈ E_{I} and

(∫_{G} f dμ) x = P_{r}(π̄(f ⊗ 1)Φ_{A}(x)) ∈ I ⋊_{α|,r} G,

by Lemma 4.4 (ii). Since G is σ-compact, there is a compact subset K of G such that μ(K) ≠ 0. Choosing f ∈ C_{0}(G/H) such that f(g) = 1 for g ∈ K, it follows that ∫_{G} f dμ ≠ 0, which implies that x ∈ I ⋊_{α|,r} G, as required. □

5. Extension of an exact group by an exact group.

The main result of this section is

Theorem 5.1 Let G be a locally compact group and let N be a closed normal subgroup of G. If N and G/N are exact, then G is exact.

Let N be a closed normal subgroup of G and let (A, α) ∈ C_{G}^{∗}. As indicated at the beginning of §3, there are a twisted action (γ^{α}, τ) of G on A⋊_{α}N relative to N canonically associated with α and a natural isomorphism A ⋊_{α} G ≅ (A ⋊_{α} N) ⋊_{γ^{α},τ} G. If {π, U} is a universal covariant pair of representations of the system {A, G, α} on a Hilbert space H and A ⋊_{α} N is identified with its image under the representation π ⋊ U to B(H), γ^{α} and τ are given by

γ_{s}^{α}(x) = U_{s}xU_{s^{−1}}

and

τ_{n} = U_{n}

for x ∈ A ⋊_{α} N, s ∈ G and n ∈ N. That γ^{α} and τ have the stated properties follows from the proof of the reduced case, which is given in the following proposition. We define a twisted action of G on the reduced crossed product A ⋊_{α,r} N with analogous properties.

By Lemma 3.1 (3), the restriction of the representation (π ⊗1)⋊_{r} (U ⊗λ^{G}) on H⊗L^{2}(G) to A⋊_{α,r} N is faithful. Identifying A⋊_{α,r} N with its image under this representation, an action γ^{α,r} of G on A ⋊_{α,r} N is given by

γ_{s}^{α,r}(x) = (U_{s} ⊗ λ_{s})x(U_{s^{−1}} ⊗ λ_{s^{−1}})

for x in A ⋊_{α,r} N and s ∈ G, and a twisting map τ′ is given by

τ′(t) = U_{t} ⊗ λ_{t}

for t ∈ N. It is immediate that the canonical quotient map A⋊_{α} N → A⋊_{α,r} N is twist-equivariant relative to this action and twisting.

Proposition 5.2 There is an isomorphism Φ_{A} : (A ⋊_{α,r} N) ⋊_{γ^{α,r},τ′,r} G → A ⋊_{α,r} G which is natural in the sense that if {B, G, β} is another G-covariant system and θ : A → B is a G-equivariant *-homomorphism with associated homomorphisms

θ_{r} : A ⋊_{α,r} G → B ⋊_{β,r} G

and

θ̃: (A ⋊_{α,r} N) ⋊_{γ^{α,r},τ′,r} G → (B ⋊_{β,r} N) ⋊_{γ^{β,r},τ′,r} G,

then Φ_{B}θ̃ = θ_{r}Φ_{A}, i.e. the diagram

\xymatrix{(A ⋊_{α,r} N) ⋊_{γ^{α,r},τ′,r} G \ar[r]^{θ̃} \ar[d]_{Φ_{A}} & (B ⋊_{β,r} N) ⋊_{γ^{β,r},τ′,r} G \ar[d]_{Φ_{B}} \\ A ⋊_{α,r} G \ar[r]^{θ_{r}} & B ⋊_{β,r} G}

commutes.

Proof: Let{π, U} be a covariant pair of representations of {A, G, α} on the Hilbert space H with π faithful. The crossed product A⋊_{α,r} N can be identified with the C*-algebra generated by the operators on H ⊗ L^{2}(G) of form

∫_{G}(π(a(n)) ⊗ 1)(U_{n} ⊗ λ_{n})dm_{N}(n)

for a ∈ C_{c}(N, A). By Lemma 3.1 (3), (A ⋊_{α,r} N) ⋊_{α,τ′,r} G can be identified with the C*-algebra (in fact the closed linear span) generated by the set {T_{f} : f ∈ C_{c}(G × N, A)} of operators on H ⊗ L^{2}(G) ⊗ L^{2}(G/N), where

T_{f} = ∫_{G} ∫_{N}(π(f(s, n)) ⊗ 1 ⊗ 1)(U_{n} ⊗ λ_{n} ⊗ 1)(U_{s} ⊗ λ_{s} ⊗ λ̇_{sN})dm_{N}(n)dm_{G}(s)
= ∫_{G} ∫_{N}(π(f(n^{−1}s, n)) ⊗ 1 ⊗ 1)(U_{s} ⊗ λ_{s} ⊗ λ̇_{sN})dm_{N}(n)dm_{G}(s)
∈ A ⋊_{α,r} G

for f ∈ C_{c}(G × N, A), A ⋊_{α,r} G being identified here with its image under the integrated form representation of the pair {π ⊗ 1 ⊗ 1, U ⊗ λ ⊗ λ̇} (which is unitarily equivalent to the pair {(π ⊗ 1) ⊗ 1, (U ⊗ λ̇) ⊗ λ}).

Let g ∈ C_{c}(N) be a nonnegative real-valued function such that

∫_{N} g(n)dm_{N}(n) = 1,

let a ∈ C_{c}(G, A) and let f ∈ C_{c}(G × N, A) be given by

f(s, n) = a(s)g(n).

Choosing g with support in a suitable neighbourhood of the identity of N, the operator of the above form corresponding to this f can be made to approximate

∫_{G}(π(a(s)) ⊗ 1 ⊗ 1)(U_{s} ⊗ λ_{s} ⊗ λ̇_{sN})dm_{G}(s)

arbitrarily closely in norm. Thus if we define Φ_{A} on {T_{f} : f ∈ C_{c}(G × N, A)} by

Φ_{A}(T_{f}) = ∫_{G} ∫_{N}(π(f(n^{−1}s, n)) ⊗ 1)(U_{s} ⊗ λ_{s})dm_{N}(n)dm_{G}(s),

Φ_{A} extends uniquely to a *-isomorphism of (A⋊_{α,r}N)⋊_{γ^{α,r},τ′,r}G onto A⋊_{α,r}G. The naturalness of Φ_{A} is a straightforward consequence of this definition. □

Combes [Co] introduced a notion of Morita equivalence for actions of a locally compact group G. Let (A, α), (B, β) ∈ C_{G}^{∗} and let X be a B–A equivalence bimodule. A continuous action u of G on X is a set of bijective linear isometries {u_{s} : s ∈ G} of X such that the map s → u_{s} is strongly continuous and for each s

u_{s}(x < y, z >_{A}) = u_{s}(x) < u_{s}(y), u_{s}(z) >_{A}

for x, y, z ∈ X. The actions β and α, or more accurately the pairs (B, β) and (A, α), are Morita equivalent if there is a continuous action u of G on X such that for each s,

α_{s}(< x, y >_{A}) =< u_{s}(x), u_{s}(y) >_{A}

and

β_{s}(< x, y >_{B}) =< u_{s}(x), u_{s}(y) >_{B}

for x, y ∈ X. When (A, α) and (B, β) are Morita equivalent, there are Morita equivalences between A⋊_{α} G and B ⋊_{β} G, and between A⋊_{α,r} G and B ⋊_{β,r} G, by [Co, §3].

Echterhoff [Ech] has extended this idea to twisted actions as follows. Let N be a closed normal subgroup of G and let (α, τ) and (β, σ) be twisted actions of G on A and B, respectively, relative to N. Then (β, σ) and (α, τ) are Morita equivalent relative to the pair (X, u) if u satisfies the above identities and also

u_{n}x = σ_{n}xτ_{n}^{−1}

for n ∈ N and x ∈ X. Moreover, if {A, G, α, τ} and {B, G, β, σ} are Morita equivalent, then A⋊_{α,τ} G is Morita equivalent to B ⋊_{β,σ} G [Ech, p.174, Remark 2]. We shall show the corresponding result for the reduced crossed products.

Although the arguments that follow involving Morita equivalence could be expressed solely in terms of equivalence bimodules, we have found it easier to bring out some of the functorial aspects of the proof using the equivalent idea of a linking algebra, due to Brown, Green and Rieffel [BGR], which we now briefly recall. If C is a nonzero C*-algebra, a projection p ∈ M(C) is full if the linear span of the set CpC = {xpy : x, y ∈ C} is dense in C. Suppose that p, q ∈ M(C) are full projections such that p + q = 1, and let A = pCp, B = qCq and X = pCq. If we define A- and B- valued inner products on X by

< x, y >_{A}= xy^{∗}, < x, y >_{B}= x^{∗}y

and let A and B act on X by left and right multiplication, respectively, then X becomes an A–B equivalence bimodule. Conversely, if A and B are C*-algebras and X is an A–B equivalence bimodule, then we can find a C*-algebra C, known as a linking algebra for A and B, and full projections p, q ∈ M(C) such that p+q = 1, A ≅ pCp, B ≅ qCq and such that X and pCq are isomorphic as A–B equivalence modules. Passage from X to the corresponding C is functorial, in a sense to be made precise in what follows.

Let IMP be the category whose objects are triples {A, B, X} consisting of a Morita equivalent pair of C*-algebras A, B and an A–B equivalence bimodule X. Given two such triples {A, B, X} and {A_{1}, B_{1}, X_{1}}, a map between them is a triple {ϕ, ψ, ω}, consisting of *-homomorphisms ϕ : A → A_{1}, ψ : B → B_{1}, and a linear map ω : X → X_{1} satisfying

ω(ax) = ψ(a)ω(x), ω(xb) = ω(x)ϕ(b),
ω(x) < ω(y), ω(z) >_{B_{1}}=< ω(x), ω(y) >_{A_{1}} ω(z),
< ω(x), ω(y) >_{A_{1}}= ϕ(< x, y >_{A}), < ω(x), ω(y) >_{B_{1}}= ψ(< x, y >_{B})

for a ∈ A, b ∈ B and x, y, z ∈ X. Given two pairs A, A_{1} and B, B_{1} of C*- algebras, *-homomorphisms ϕ : A → A_{1} and ψ : B → B_{1} will be said to be Morita equivalent if there is an ω : X → X_{1} such that {ϕ, ψ, ω} is a map in IMP. This is consistent with the definition of Morita equivalence for actions when ϕ and ψ are automorphisms of A and B, respectively.

Let A and B be Morita equivalent C*-algebras and let X be an A–B equivalence bimodule. To see that the associated linking algebra is related to X functorially, we recall its construction (cf. [BGR, proof of Theorem 1]). Let X^{∗} be the B–A equivalence bimodule conjugate to X. There is a conjugate linear bijection x → x^{∗}; X → X^{∗} such that

bx^{∗} = (xb^{∗})^{∗}, x^{∗}a = (a^{∗}x)^{∗}, < x^{∗}, y^{∗} >_{A}=< y, x >_{A}, < x^{∗}, y^{∗} >_{B}=< y, x >_{B}

for a ∈ A, b ∈ B and x, y ∈ X. For x, y ∈ X let x^{∗}y and xy^{∗} be the elements < x, y >_{A} and < x, y >_{B} of A and B, respectively. The set of matrices

C = \{[ \begin{matrix}a & x \\ y^{∗} & b \end{matrix} ] : a ∈ A, b ∈ B, x, y ∈ X\},

with matrix addition and multiplication, is a *-algebra. Moreover X ⊕ B has left and right actions of C and B, respectively, and C- and B-valued inner products can be defined in such a way that X ⊕ B becomes a C–B equivalence bimodule. A norm on X ⊕ B is defined in terms of the B-valued inner product on X and the norm of B relative to which the associated C*-norm on C is complete. The projections

p = [ \begin{matrix}1 & 0 \\ 0 & 0 \end{matrix} ] , q = [ \begin{matrix}0 & 0 \\ 0 & 1 \end{matrix} ]

are in M(C), are both full, and A ≅ pCp and B ≅ qCq. Moreover pCq and X are isomorphic as A–B equivalence bimodules. Let {A, B, X}, {A_{1}, B_{1}, X_{1}} be objects of IMP and let {ϕ, ψ, ω} : {A, B, X} → {A_{1}, B_{1}, X_{1}} be a morphism. If C and C_{1} are the linking algebras constructed from {A, B, X} and {A_{1}, B_{1}X_{1}}, respectively, by this procedure, a *-homomorphsim ϕ = Φ_{{}ϕ, ψ, ω} : C → C_{1} is defined by

Φ_{\{}ϕ, ψ, ω\}([ \begin{matrix}a & x \\ y^{∗} & b \end{matrix} ]) = [ \begin{matrix}ϕ(a) & ω(x) \\ (ω(y))^{∗} & ψ(b) \end{matrix} ] .

Then Φ(p) = p, Φ(q) = q, Φ|pCp = ψ and Φ|qCq = ϕ. Let LINK be the category whose objects are triples {C, p, q} consisting of a C*-algebra C with full projections p, q ∈ M(C) such that p + q = 1. If {C, p, q} and {C_{1}, p_{1}, q_{1}} are objects of LINK, a morphism θ from {C, p, q} → {C_{1}, p_{1}, q_{1}} is a a *- homomorphism C → C_{1} such that θ(px) = p_{1}θ(x) for x ∈ C. Writing C_{{A,B,X}} for the linking algebra constructed from the triple {A, B, X}, the map {A, B, X} → C_{{A,B,X}} is a functor from IMP to LINK giving an equivalence of categories.

For strongly Morita equivalent C*-algebras A and B, the order-preserving correspondence between the ideals of A and B described in §2 can be expressed elegantly in terms of C. If I is an ideal of A or B, then I_{C}, the ideal of C generated by I, is just the closure of the linear span of the set CIC. If I is an ideal of C, let

I_{A} = pIp, I_{B} = qIq.

Then I_{A} = I ∩ pCp = I ∩ A and I_{B} = I ∩ qCq = I ∩ B, and it follows easily from the fullness of p and q that for any ideal I of C, I = (I_{A})_{C} = (I_{B})_{C}. For any ideal I of A, I = (I_{C})_{A}, and similarly for B. The map I → (I_{C})_{B} is thus an order-preserving bijection from the ideals of A to the ideals of B.

Lemma 5.3 Let {A, B, X}, {A_{1}, B_{1}, X_{1}} ∈ IMP, and let {ϕ, ψ, ω} : {A, B, X} → {A_{1}, B_{1}, X_{1}} be a morphism. Then ker ψ corresponds to ker ϕ under the above bijection.

Proof: If Φ : C_{{A,B,X}} → C_{{A_{1},B_{1},X_{1}}} is the *-homomorphism corresponding to {ϕ, ψ, ω}, then x ∈ ker Φ∩A ⇔ ϕ(x) = 0 ⇔ x ∈ ker ϕ. Hence ker ϕ = ker Φ∩A. Similarly ker ψ = ker Φ ∩ B. Thus ker ϕ, ker Φ and ker ψ are corresponding ideals. □

Let {A, G, α, σ} and {B, G, β, τ} be Morita equivalent twisted covariant systems relative to a pair (X, u) consisting of an A–B equivalence bimodule X with an action u of G, with twisting relative to the normal subgroup N of G. Then for each s ∈ G, {α_{s}, β_{s}, u_{s}} is a map in IMP, in fact an automorphism of {A, B, X}. If Γ_{s} denotes the corresponding automorphism of the linking algebra C = C_{{A,B,X}}, then

Γ_{s}([ \begin{matrix}a & x \\ y^{∗} & b \end{matrix} ]) = [ \begin{matrix}α_{s}(a) & u_{s}x \\ (u_{s}y)^{∗} & β_{s}(b) \end{matrix} ] .

It is immediate that s → Γ_{s} is a continuous action of G on C_{{A,B,X}}, Γ fixes p and q, Γ|A = α and Γ|B = β. For n ∈ N

κ_{n} = [ \begin{matrix}σ_{n} & 0 \\ 0 & τ_{n} \end{matrix} ]

is in M(C) and the map κ : n → κ_{n} is a twisting map for the action Γ.

The canonical embedding of C in M(C ⋊_{Γ} G) extends to an embedding of M(C) in M(C ⋊_{Γ} G), where x ∈ M(C) is identified with the element of M(C ⋊_{Γ} G) which sends f in C_{c}(G, C) to xf. With this identification, p and q are in M(C ⋊_{Γ} G), are full projections for C ⋊_{Γ} G, and there are canonical isomorphisms

p(C ⋊_{Γ} G)p ≅ A ⋊_{α} G, q(C ⋊_{Γ} G)q ≅ B ⋊_{β} G,

by [Co, §6]. In fact, if π_{0} : C → M(C ⋊_{Γ} G) and U_{0} : G → M(C ⋊_{Γ} G) are the canonical embeddings and A is identified with pCp, then {π_{0}|A, U_{0}} is a covariant pair of representations of {A, G, β}, and the integrated form of this pair is a *-homomorphism θ_{A} : A⋊_{α} G → M(C ⋊_{Γ} G). The image of A⋊_{α} G under θ_{α} is just p(C ⋊_{Γ} G)p. To see that θ_{A} is injective, let {π, U} be a universal covariant pair of representations of {A, G, α} on a Hilbert space H. By the equivariant form of Stinespring’s theorem, there are a Hilbert space K containing H and a covariant pair of representations {π_{1}, U_{1}} of {C, G, Γ} on K such that π̄_{1}(p) is the projection onto H and the pair {π̄_{1}(p)(π_{1}|B)π̄_{1}(p), π̄_{1}(p)U_{1}π̄_{1}(p)} is a covariant pair of representations of {A, G, α} unitarily equivalent to π ⋊ U. Then π̄_{1}(p) ((π_{1} ⋊ U_{1}) ∘ θ_{A}) π̄_{1}(p) is unitarily equivalent to π ⋊ U, which is faithful. Thus θ_{A} is faithful.

The canonical quotient map q_{C} : C ⋊_{Γ} G → C ⋊_{Γ,κ,r} G extends to a *- homomorphism q̄_{C} : M(C ⋊_{Γ} G) → M(C ⋊_{Γ,κ,r} G), and the covariant pair {q̄_{C} ∘ (π_{0}|B), q̄_{C} ∘ U} of homomorphisms of {A, G, α} is twist-preserving. It follows by Lemma 3.1 that q_{C} ∘ θ_{A} has a factorisation q_{C} ∘ θ_{A} = θ̄_{A} ∘ q_{A}, where q_{A} : A⋊_{α} G → A⋊_{α,σ,r} G is the quotient map and θ̄_{A} : A⋊_{α,σ,r} G → C ⋊_{Γ,κ,r} G is a *-isomorphism with image p̄(C ⋊_{Γ,κ,r} G)p̄, where p̄ is the projection q̄_{C}(p). Applying analogous considerations to the crossed products involving B, we obtain *-monomorphisms

θ_{B} : B ⋊_{β} G → q(C ⋊_{Γ} G)q

and

θ̄_{B} : B ⋊_{β,σ,r} G → q̄(C ⋊_{Γ,κ,r} G)q̄

such that q_{C} ∘ θ_{B} = θ̄_{B} ∘ q_{B}, where q̄ = q̄_{C}(q). Since the fullness of the projections p and q for C ⋊_{Γ} G implies the fullness of p̄ and q̄ for C ⋊_{Γ,κ,r} G, we obtain

Proposition 5.4 If {A, G, α, τ} and {B, G, β, σ} are Morita equivalent twisted covariant systems relative to N, then the C*-algebras A ⋊_{α,τ,r} G and B ⋊_{β,σ,r} G are strongly Morita equivalent. Moreover the diagram

\xymatrix{A ⋊_{α} G \ar[r]^{q_{A}} \ar[d]^{θ_{A}} & A ⋊_{α,τ,r} G \ar[d]^{θ̄_{A}} \\ C ⋊_{Γ} G \ar[r]^{q_{C}} & C ⋊_{Γ,κ,r} G \\ B ⋊_{β} G \ar[r]^{q_{B}} \ar[u]_{θ_{B}} & B ⋊_{β,σ,r} G \ar[u]_{θ̄_{B}}}

commutes, so that the *-homomorphisms q_{A} and q_{B} are Morita equivalent.

If {A, G, α, τ} is a twisted covariant system relative to N, then there is an action β of G on E_{A} = C_{0}(G/N, A) ⋊_{Δ^{α},τ̃} G such that the twisted covariant systems {A, G, α, τ} and {E_{A}, G, β, 1_{N}} are Morita equivalent, where 1_{N} is the trivial twisting map n → 1 ∈ M(B) [Ech, Theorem 1].

An equivalence bimodule Y_{A} giving this Morita equivalence is obtained by applying the mapping {ν_{A}, id, ᾱ} of IMP to the triple {Ind(A, α) ⋊α̃ G, A, X_{A}} of §2, with H = N, where ν_{A} : Ind(A, α) → C_{0}(G/N, A) is the map isomorphism of §2 and (ᾱ(x))(s) = α_{s}(x(s)) for x ∈ C_{c}(G, A). Letting X_{1} = C_{c}(G, A), E_{0} = C_{c}(G, C_{0}(G/N, A))) and B_{0} = C_{c}(N, A), with the convolution products relative to the actions Δ^{α} and α, respectively, the algebras E_{0} and B_{0} having the C*-norms and positive cones coming from their canonical embeddings in C_{0}(G/N, A) ⋊_{Δ^{α}} G and A ⋊_{α} N, respectively, the resulting E_{0}–B_{0} equivalence bimodule structure on X_{1} is given by

(fx)(r) = ∫_{G} f(s, rN)α_{s}(x(s^{−1}r))dm_{G}(s)
(xg)(r) = ∫_{H} x(rt)α_{rt}(g(t^{−1}))dm_{H}(t)
⟨x, y⟩_{E_{0}}(s, rN) = ∫_{H} Δ_{G}(s^{−1}rt)x(rt)α_{s}(y(s^{−1}rt)^{∗})dm_{H}(t)
⟨x, y⟩_{B_{0}}(t) = δ(t) ∫_{G} α_{s^{−1}}(x(s)^{∗}y(st))dm_{G}(s).

The action β of G on E_{A} is given on an element f of the dense *-subalgebra C_{c}(G, C_{0}(G/N, A)) of E_{A} by

(β_{s}(f))(r, tN) = f(r, tsN),

where f is considered as a function on G × (G/N), and an action u of G on X_{1} is given by

(u_{s}x)(t) = Δ_{G}(s)Δ_{G/N}(s)^{−1/2}x(ts).

Recalling that, by Remark 2.5 (2), there is a canonical isomorphism A ≅ A ⋊_{α,1_{n}} N, the equivalence bimodule Y_{A} is then obtained by completing X_{1} with respect to the norm x → || < x, x >_{A} ||^{1/2} and extending the left and right actions to E_{A} and A, respectively, by continuity.

Before proving Theorem 5.1 we require one further observation. Let β be a twisted action of G on a C*-algebra D relative to N with trivial twisting map 1_{N}, so that β_{n} = id_{D} for n ∈ N. Then an action β̄ of G/N on D is given by

β̄_{sN}(x) = β_{s}(x).

The analogue of the following result for full twisted crossed products is implicit in [Ech].

Lemma 5.5 There is an isomorphism Ψ_{D} : D ⋊_{β̄,r} (G/N) → D ⋊_{β,1_{N} ,r} G which is natural in the sense that, if {D, G, β, 1_{N}} and {D_{1}, G, β_{1}, 1_{N}} are twisted covariant systems and θ : D_{1} → D is a G-equivariant twist-preserving *-homomorphism, then the diagram

\xymatrix{D_{1} ⋊_{β̄_{1},r} (G/N) \ar[r]^{θ_{r}} & D ⋊_{β̄,r} (G/N) \\ D_{1} ⋊_{β_{1},1_{N} ,r} G \ar[u]_{Ψ_{D_{1}}} \ar[r]^{θ_{N,r}} & D ⋊_{β,1_{N} ,r} G \ar[u]_{Ψ_{D}}}  (∗)

commutes, where the horizontal maps are the *-homomorphisms corresponding canonically to θ.

Proof: Let π_{0} : D → M(D ⋊_{β̄,r} (G/N)) and λ_{0} : G/N → M(D ⋊_{β̄,r} (G/N)) be the canonical monomorphisms. If λ̇_{0} denotes the composition of λ_{0} with the quotient homomorphism G → G/N, and M(D ⋊_{β̄,r} (G/N)) is represented faithfully on some Hilbert space H, the pair {π_{o}, λ_{0}} is a covariant pair of representations of the system {D, G, β} which is twist-covariant for the twisting map 1_{N}. The representation π_{0} ⋊_{1_{n}} λ_{0} of D ⋊_{β,1_{N}} G in M(D ⋊_{β̄,r} (G/N)) has image D⋊_{β̄,r}(G/N), and, by Proposition 3.5 (1), has a factorisation π_{0}⋊_{1_{N}} λ_{0} = ψ_{D} ∘ q, where ψ_{D} is a *-homomorphism from D ⋊_{β,1_{N} ,r} G onto D ⋊_{β̄,r} (G/N) and q : D ⋊_{β,1_{N}} G → D ⋊_{β,1_{N} ,r} G is the canonical quotient map. Since π_{0} is faithful, ψ_{D} is injective.

If θ : D_{1} → D is a G-covariant twist-preserving *-homomorphism, let θ̄_{N,r} = ψ_{D}^{−1}θ_{r}ψ_{D_{1}}. By the definition of ψ_{D} and ψ_{D_{1}}, the diagram

\xymatrix{D_{1} ⋊_{β_{1}} G \ar[d] \ar[r]^{θ_{u}} & D ⋊_{β} G \ar[d] \\ D_{1} ×_{β_{1},1_{N} ,r} G \ar[r]^{θ̄_{N,r}} & D ⋊_{β,1_{N} ,r} G}

commutes, where the vertical arrows are the canonical quotient maps. By Proposition 3.5 (2), θ̄_{N,r} = θ_{N,r}, from which the commutativity of (∗) is immediate. □

Proof of Theorem 5.1. Assume that N and G/N are exact. Let (I, α|), (A, α), (B, α̇) ∈ C_{G}^{∗} and let ι : I → A and q : A → B be G-equivariant *-homomorphisms such that im ι = ker q, i.e. such that the sequence

\xymatrix{0 \ar[r] & I \ar[r]^{ι} & A \ar[r]^{q} & B \ar[r] & 0}  (1)

is exact. To prove the theorem we must show that the sequence

\xymatrix{0 \ar[r] & I ⋊_{α|,r} G \ar[r]^{ι_{r}} & A ⋊_{α,r} G \ar[r]^{q_{r}} & B ⋊_{α̇,r} G \ar[r] & 0}  (2)

is exact. In what follows we shall identify I with its image in A.

Since N is exact, the sequence

\xymatrix{0 \ar[r] & I ⋊_{α|,r} N \ar[r]^{ι_{r}} & A ⋊_{α,r} N \ar[r]^{q_{r}} & B ⋊_{α̇,r} N \ar[r] & 0}  (3)

is exact. Let I_{N} = I ⋊_{α|,r} N, A_{N} = A ⋊_{α,r} N and B_{N} = B ⋊_{α̇,r} N. By Proposition 5.2 and the discussion preceding it, there are N-twisted actions (γ_{I}, τ), (γ_{A}, τ) and (γ_{B}, τ) of G on I_{N}, A_{N} and B_{N}, respectively, and isomorphisms

Φ_{I} : I_{N} ⋊_{γ_{I},τ,r} G → I ⋊_{α|,r} G,
Φ_{A} : A_{N} ⋊_{γ_{A},τ,r} G → A ⋊_{α,r} G

and

Φ_{B} : B_{N} ⋊_{γ_{B},τ,r} G → B ⋊_{α̇,r} G

such that the diagram

\xymatrix{0 \ar[r] & I_{N} ⋊_{γ_{I},τ,r} G \ar[r]^{ι_{N,r}} \ar[d]_{Φ_{I}} & A_{N} ⋊_{γ_{A},τ,r} G \ar[r]^{q_{N,r}} \ar[d]_{Φ_{A}} & B_{N} ⋊_{γ_{B},τ,r} G \ar[r] \ar[d]_{Φ_{B}} & 0 \\ 0 \ar[r] & I ⋊_{α|,r} G \ar[r]^{ι_{r}} & A ⋊_{α,r} G \ar[r]^{q_{r}} & B ⋊_{α̇,r} G \ar[r] & 0}

commutes, where ι_{N,r} and q_{N,r} are the crossed product homomorphisms corresponding to the *-homomorphisms ι_{r}|I_{N} and q_{r}|A_{N}, then latter G-equivariant and twist preserving relative to the twisted actions on I_{N}, A_{N} and B_{N}, as follows readily from the definitions. The exactness of (2) is thus equivalent to that of the sequence

\xymatrix{0 \ar[r] & I_{N} ⋊_{γ_{I},τ,r} G \ar[r]^{ι_{τ,r}} & A_{N} ⋊_{γ_{A},τ,r} N \ar[r]^{q_{τ,r}} & B_{N} ⋊_{γ_{B},τ,r} N \ar[r] & 0} (4)

By Corollary 3.10, A_{N} and E_{A} = (C_{0}(G/N, A_{N}) ⋊_{Δ^{γ},τ̃} G are Morita equivalent via the E_{A}–A_{N} equivalence bimodule obtained by completing X_{0} = C_{c}(G, A_{n}) with respect to the seminorm || || given by

||x|| = || < x, x >_{A_{N}} ||^{1/2}.

Letting C_{A} = C_{{A,B,X}}, we identify E_{A} with pC_{A}p and A_{N} with qC_{A}q. If β_{A} is the action of G on E_{A} defined earlier such that (β_{A}, 1_{N}) is Morita equivalent to (γ_{A}, τ) via (X_{A}, u_{A}), where u_{A} is the canonically defined action of G on X_{A}, let (Γ_{A}, κ) be the corresponding twisted action of G on C_{A}. We define X_{I}, X_{B}, C_{I}, C_{B}, β_{I}, β_{B}, Γ_{I} and Γ_{B} similarly. If q_{X} : C_{c}(G, A_{N}) → C_{c}(G, B_{N}) is the natural map given by

(q_{X}(f))(s) = q_{r}(f(s)),

for s ∈ G, then

||q_{X}(f)|| = || < q_{X}(f), q_{X}(f) >_{B_{N}} ||^{1/2} = ||q_{r}(< f, f >)_{A_{N}} ||^{1/2} ≤ ||f||.

If ι_{X} : C_{c}(G, I_{N}) → C_{c}(G, A_{N}) is defined similarly starting from ι, since ι_{r} : I_{N} → A_{N} is an isometric embedding, the same is true of ι_{X}. It follows that ι_{X} and q_{X} extend to an isometric embedding of X_{I} in X_{A} and a contraction of X_{A} onto X_{B}, respectively, which we will still denote by ι_{X} and q_{X}. Identifying X_{I} with its image in X_{A} we get a corresponding embedding ι_{C} : C_{I} → C_{A}. It is straightforward to verify that the diagram

\xymatrix{E_{I} \ar[r]^{ι_{E}} \ar[d] & E_{A} \ar[d] \\ C_{I} \ar[r]^{ι_{C}} & C_{A} \\ I_{N} \ar[u] \ar[r]^{ι_{r}} & A_{N} \ar[u]}

commutes. Similarly the surjection q_{X} gives rise to a *-epimorphism q_{C} : C_{A} → C_{B} given by

q_{C}([ \begin{matrix}b & x \\ y^{∗} & a \end{matrix} ]) = [ \begin{matrix}q_{E}(b) & q_{X}(x) \\ q_{X}(y)^{∗} & q_{r}(a) \end{matrix} ] .

Again it is routine to verify that the diagram

\xymatrix{E_{A} \ar[r]^{q_{E}} \ar[d] & E_{B} \ar[d] \\ C_{A} \ar[r]^{q_{C}} & C_{B} \\ A_{N} \ar[u] \ar[r]^{q_{r}} & B_{N} \ar[u]}

commutes. All the maps in these two diagrams are twist-preserving relative to the respective twistings for each algebra. Hence if we take the twisted crossed products of all the algebras by G, we get commuting diagrams

\xymatrix{E_{I} ⋊_{β_{I},1_{N} ,r} G \ar[r]^{(ι_{E})_{r}} \ar[dd] & E_{A} ⋊_{β_{A},1_{N} ,r} G \ar[dd] &  & E_{A} ⋊_{β_{A},1_{N} ,r} G \ar[dd] \ar[r]^{(q_{E})_{r}} & E_{B} ⋊_{β_{B},1_{N} ,r} G \ar[dd] \\  &  & and \\ C_{I} ⋊_{Γ_{I},κ,r} G \ar[r]^{(ι_{C})_{r}} & C_{A} ⋊_{Γ_{A},κ,r} G &  & C_{A} ⋊_{Γ_{A},κ,r} G \ar[r]^{(q_{C})_{r}} & C_{B} ⋊_{Γ_{B},κ,r} G \\ I_{N} ⋊_{γ_{I},τ,r} G \ar[u] \ar[r]^{ι_{r}} & A_{N} ⋊_{γ_{A},τ,r} G \ar[u] &  & A_{N} ⋊_{γ_{A},τ,r} G \ar[u] \ar[r]^{q_{r}} & B_{N} ⋊_{γ_{B},τ,r} G \ar[u]}

In these diagrams the vertical arrows denote the embedding maps resulting from the identifications E_{I} ⋊_{β_{I},1_{N} ,r} G = p(C_{I} ⋊_{Γ_{I},κ,r} G)p, I_{N} ⋊_{γ_{I},τ,r} G = q(C_{I} ⋊_{Γ_{I},κ,r} G)q, etc. From the left-hand diagram it is apparent that the ideals E_{I} ⋊_{β_{I},1_{N} ,r} G of E_{A} ⋊_{β_{A},1_{N} ,r} G and I_{N} ⋊_{γ_{I},τ,r} G of A_{N} ⋊_{γ_{A},τ,r} G correspond. From the right-hand diagram the *-homomorphisms (q_{E})_{r} and q_{r} are seen to be Morita equivalent, so that, by Lemma 5.3, their kernels are corresponding ideals of E_{A}⋊_{β_{A},1_{N} ,r}G and A_{N} ⋊_{γ_{A},τ,r}G, respectively. Thus ker(q_{E})_{r} = E_{I} ⋊_{β_{I},1_{N} ,r}G if and only if ker q_{r} = I_{N} ⋊_{γ_{I},τ,r} G, that is, the sequence (4) is exact if and only if the sequence

\xymatrix{0 \ar[r] & E_{I} ⋊_{β_{I},1_{N} ,r} G \ar[r]^{(ι_{E})_{r}} & E_{A} ⋊_{β_{A},1_{N} ,r} G \ar[r]^{(q_{E})_{r}} & E_{B} ⋊_{β_{B},1_{N} ,r} G \ar[r] & 0}

(5)

is exact.

Lemma 5.5 implies that the sequence (5) is exact if and only if the sequence

\xymatrix{0 \ar[r] & E_{I}⋊_{β̄_{I},r}(G/N) \ar[r]^{(ι_{E})_{r}} & E_{A}⋊_{β̄_{A},r}(G/N) \ar[r]^{(q_{E})_{r}} & E_{B}⋊_{β̄_{B},r}(G/N) \ar[r] & 0}

(6)

is exact.

Since the sequence (3) is exact by assumption, the sequence

\xymatrix{0 \ar[r] & E_{I} \ar[r]^{ι_{E}} & E_{A} \ar[r]^{q_{E}} & E_{B} \ar[r] & 0}

is exact, by Corollary 3.10. Since G/N is exact by assumption, this implies that the sequence (6) is exact, from which the exactness of the sequences (5),

(4) and (2) follow successively. Hence G is exact. □

6. Examples of exact groups.

a. Amenable Groups. The following result is essentially well-known.

Proposition 6.1 Let G be an amenable locally compact group. Then G is exact.

Proof: Let (A, α) ∈ C_{G}^{∗}, and let I be an α_{G}-invariant ideal of A, so that the sequence

0 → I → A → (A/I) → 0

is exact. Then the diagram

\xymatrix{0 \ar[r] & I ⋊_{α|} G \ar[r] \ar[d] & A ⋊_{α} G \ar[r] \ar[d] & (A/I) ⋊α̇ G \ar[r] \ar[d] & 0 \\ 0 \ar[r] & I ⋊_{α|,r} G \ar[r] & A ⋊_{α,r} G \ar[r] & (A/I) ⋊_{α̇,r} G \ar[r] & 0}

is commutative, where the vertical arrows are the canonical *-homomorphisms, and the top row is exact. Since G is amenable, the canonical *-homomorphisms A ⋊_{α} G → A ⋊_{α,r} G, etc., are injective, from which it follows that the lower row of the diagram is exact. Hence G is exact. □

b. Discrete subgroups of semisimple Lie groups. Our goal here is to show that the discrete groups SL_{n}(ℤ) are exact for n = 1, 2, …, though we shall, in fact, prove a more general result. The following fact is undoubtedly known, although we lack a reference.

Proposition 6.2 Let A and B be Morita equivalent C*-algebras. Then A is nuclear if and only if B is nuclear.

Proof: Let (C, e, f) be a linking algebra for A and B, so that e, f are full projections in M(C) such that e + f = 1, A ≅ eCe and B ≅ fCf. Let C^{∗∗} be represented as a von Neumann algebra on a Hilbert space H. If we regard M(C) as canonically embedded in C^{∗∗}, the fullness conditions imply that e and f have central support 1 in C^{∗∗}, from which it follows that e(C^{∗∗})′e ≅ (C^{∗∗})′ ≅ f(C^{∗∗})′f. Now A^{∗∗} ≅ eC^{∗∗}e and A is nuclear if and only if A^{∗∗} is injective. Since a von Neumann algebra is injective if and only if its commutant is, it follows that A is nuclear if and only if (C^{∗∗})′, and hence C^{∗∗}, are injective. Since by the same reasoning B is nuclear if and only if C^{∗∗} is injective, the result follows. □

We believe that the Morita equivalence technique used in the proof of the next proposition is essentially due to Alain Connes (unpublished), though we lack a precise attribution.

Proposition 6.3 Let G be a locally compact group which has a closed amenable subgroup H such that G/H is compact. Then any closed discrete subgroup of G is exact.

Proof: Assume first that H and K are just closed subgroups of G, and let A = C_{0}(G/K). The continuous action of G on G/K by left multiplication gives rise to a continuous action α of G on A. Restricting α to H, the algebras C_{0}(G/H, A) ⋊_{Δ^{α},r} G ≅ C_{0}((G/H) × (G/K)) ⋊_{Δ,r} G and A⋊_{α,r} H ≅ C_{0}(G/K)⋊_{α,r} H are Morita equivalent, by Theorem 3.6, where Δ is the action of G on C_{0}((G/H) × (G/K)) arising from the diagonal left action of G on (G/H)×(G/K). Interchanging H and K, it follows that C_{0}(G/K)⋊_{α,r} H and C_{0}(G/H) ⋊_{β,r} K are Morita equivalent, where β is the action of K coming from left multiplication on G/H.

Now assume that H is amenable, G/H is compact and K is discrete. Then C_{0}(G/H) = C(G/H) and C_{r}^{∗}(K) ⊆ C(G/H) ⋊_{β,r} K canonically. Also C_{0}(G/K) ⋊_{α,r} H, being a crossed product of a nuclear C*-algebra by an amenable group, is nuclear. This implies, by Proposition 6.2, that C(G/H) ⋊_{β,r} K is nuclear, so that C_{r}^{∗}(K) is exact, so that, by the equivalence of exactness for K and C_{r}^{∗}(K) [KW, Theorem 5.2], K is exact. □

Corollary 6.4 Any closed discrete subgroup of a connected semisimple Lie group is exact.

Proof: Let G be a connected semisimple Lie group. The centre Z of G is discrete, and, applying the Iwasawa decomposition [Kn], G = KAN, where K, A and N are connected closed subgroups of G, Z ⊆ K, K/Z is compact, N is nilpotent, and A is abelian. Moreover A normalises N and AN is a connected solvable Lie group. Since AN has a composition series with abelian quotients, it follows that AN, and hence ZAN, are amenable, and G/ZAN is homeomorphic to K/Z. The result now follows from Theorem 6.3. □

c. Closed linear groups. Since SL_{n}(ℝ) is semisimple [Kn] and contains SL_{n}(ℤ) as a closed discrete subgroup, it is an immediate consequence of Corollary 6.4 that SL_{n}(ℤ) is exact for n = 1, 2, . . .. In fact the group SL_{n}(ℤ) is a lattice in SL_{n}(ℝ), i.e. a closed discrete subgroup of finite covolume [Rag,Theorem 10.5], so that SL_{n}(ℝ) is exact by Theorem 4.5.

Proposition 6.5 For n ∈ ℕ any closed subgroup of GL_{n}(ℝ) is exact.

Proof: The determinant gives a continuous homomorphisms of GL_{n}(ℝ) onto the multiplicative groups ℝ \ {0} with kernel SL_{n}(ℝ). Thus GL_{n}(ℝ) is an extension of an exact group by an abelian group, hence is exact by Proposition 6.1 and Theorem 5.1. The result now follows by Theorem 4.5. □

d. Connected locally compact groups.

Proposition 6.6 Any connected real semisimple Lie group is exact.

Proof: Let G be a connected semisimple Lie group with centre Z and Lie algebra 𝔤. If Ad is the adjoint representation of G on 𝔤, we have a Lie group isomorphism G/Z ≅ Ad(G), and Ad(G) coincides with Aut_{0}(𝔤), the connected component of the identity of the Lie group Aut(𝔤) [Kn]. Since the latter group is a closed subgroup of GL(𝔤), and 𝔤 is finite-dimensional, it follows by Proposition 6.5 that G/Z is exact. Since Z is abelian, hence exact, the result now follows by Theorem 5.1. □

An alternative proof of this proposition follows by the technique of §4, since, by a deep theorem of Borel [Rag, Theorem 14.1], any connected noncompact semisimple Lie group contains a lattice.

Proposition 6.7 Any connected real Lie group is exact.

Proof: Let G be a connected Lie group with Lie algebra 𝔤. If rad 𝔤 is the radical of 𝔤, rad 𝔤 is a solvable ideal of 𝔤 and 𝔤/rad 𝔤 is semisimple. If R is the closed normal subgroup of G with Lie algebra rad 𝔤, then R is solvable, hence exact, by earlier discussion, and G/R has Lie algebra 𝔤/rad 𝔤. Thus G/R is semisimple, hence exact by Proposition 6.6, and the result follows using Theorem 5.2. □

Theorem 6.8 Any connected locally compact group is exact.

Proof: Let G be a connected locally compact group. By [MZ, Theorem 4.6] G has a closed normal compact subgroup K such that G/K is a real Lie group. Since G/K is connected, it is exact by Proposition 6.7. The result now follows by Theorem 5.2, since K, being amenable, is exact. □

Recall that a locally compact group G is almost-connected if the quotient group G/G_{0} of G by the connected component G_{0} of the identity is compact. The following corollary is an immediate consequence of Proposition 6.1, Theorem 6.8 and Theorem 5.1.

Corollary 6.9 Any almost-connected group is exact.

e. Exactness of certain discrete groups. By [KW, Theorem 5.2], a discrete group G is exact if and only if C_{r}^{∗}(G) is exact. For certain groups G, C_{r}^{∗}(G) can be explicitly embedded as a C*-subalgebra of a nuclear C*-algebra. For these groups, C_{r}^{∗}(G), being subnuclear, is exact, so that G is exact. Two classes for which C_{r}^{∗}(G) is known to be subnuclear are (a) the free groups and (b) the hyperbolic groups. The case of free groups was treated in [KW, Corollary 5.3], where it was shown, using a celebrated construction of Choi, that if G is a free group on at most countably many generators, then C_{r}^{∗}(G) can be embedded as a C*-subalgebra of the Cuntz algebra O_{2}. Very recently Dykema [D] has shown that a reduced amalgamated free product of exact C*- algebras is exact. Since C_{r}^{∗}(ℤ) is abelian, hence nuclear, and C_{r}^{∗}(𝔽_{Λ}), where 𝔽_{Λ} is the free group on the set Λ, is the reduced free product of copies of C_{r}^{∗}(ℤ) indexed by Λ, Dykema’s result together with our result just cited gives a new proof that free groups are exact. If G is a hyperbolic group, Adams [Ad] has shown that the natural action α of G on its Gromov boundary ∂G is amenable, which implies that the crossed product C(∂G) ⋊_{α,r} G is nuclear. Since G is discrete, C_{r}^{∗}(G) is a closed subalgebra of C(∂G) ⋊_{α,r} G, hence exact. Germain [Ger] has recently given a concise and fairly simple proof of Adams’ amenability result.

7. Concluding Remarks.

1. After we had completed most of this paper, Georges Skandalis pointed out that Corollary 6.9 can be obtained more directly by a different route. The various structure results used above together imply that if G is an almost connected group, then G contains a closed amenable subgroup H such that G/H with the quotient topology is compact. Corollary 6.9 is then an immediate consequence of Proposition 6.1 and the following theorem, which is closely related to Theorem 4.1, and has a similar, though simpler, proof.

Theorem Let G be a locally compact group with a closed exact subgroup H. If G/H is compact, then G is exact.

Proof: Let (A, α) ∈ C_{G}^{∗} and let I be an α_{G}-invariant ideal of A. If θ : A → C_{0}(G/H) ⊗ A ≅ C_{0}(G/H, A) is the embedding a → 1 ⊗ a, let Φ_{A} denote the crossed product map

Φ_{A} : A ⋊_{α} G → C_{0}(G/H, A) ⋊_{Δ^{α},r} G.

Then Φ_{A} is an embedding, and if corresponding embeddings

Φ_{I} : I ⋊_{α|,r} G → C_{0}(G/H, I) ⋊_{Δ^{α},r} G

and

Φ_{A/I} : (A/I) ⋊_{α̇,r} G → C_{0}(G/H, A/I) ⋊_{Δα̇,r} G

are defined similarly, then the diagram

\xymatrix{0 \ar[d] & 0 \ar[d] \\ I ⋊_{α|,r} G \ar[r] \ar[d] \ar[r]^{ϕ_{I}} \ar[d] & C_{0}(G/H, I) ⋊_{Δ^{α},r} G \ar[d] \\ A ⋊_{α,r} G \ar[r] \ar[d] \ar[r]^{Φ_{A}} \ar[d] & C_{0}(G/H, A) ⋊_{Δ^{α},r} G \ar[d] \\ (A/I) ⋊_{α̇,r} G \ar[r] \ar[d] \ar[r]^{Φ_{A/I}} \ar[d] & C_{0}(G/H, A/I) ⋊_{Δα̇,r} G \ar[d] \\ 0 & 0}

commutes. Since H is exact, Corollary 3.10 implies, just as in the proof of Theorem 4.5, that the right-hand column is exact. If x is in the kernel of the quotient map A ⋊_{α,r} G → (A/I) ⋊_{α̇,r} G, it follows that Φ_{A}(x) is in the kernel of the quotient map

C_{0}(G/H, A) ⋊_{Δ^{α},r} G → C_{0}(G/H, A/I) ⋊_{Δα̇,r} G,

which is C_{0}(G/H, I) ⋊_{Δ^{α},r} G. Thus

Φ_{A}(x) ∈ (A ⋊_{α,r} G) ∩ (C_{0}(G/H, I) ⋊_{Δ^{α},r} G).

Let {e_{μ}} be a bounded approximate identity for I. If we identify 1 ⊗ e_{μ} with its image in M((C_{0}(G/H, I) ⋊_{Δ^{α},r} G) under the canonical embedding of M(C_{0}(G/H, A)) discussed in §4, it is readily checked that for each μ, (1 ⊗ e_{μ})y ∈ Φ_{A}(I ⋊_{α|,r} G) for y ∈ Φ_{A}(A ⋊_{α,r} G), and lim_{μ}(1 ⊗ e_{μ})z = z for z ∈ C_{0}(G/H, I) ⋊_{Δ^{α},r} G. Then

Φ_{A}(x) = lim_{μ}(1 ⊗ e_{μ})x ∈ Φ_{A}(I ⋊_{α|,r} G),

which shows that x ∈ I ⋊_{α|,r} G. Thus the left-hand column of the above diagram is exact, which implies that G is exact. □

2. If G is a locally compact group, the quotient G/G_{0} by the connected component G_{0} of the identity is a totally disconnected group. By Corollary 6.9, G_{0} is exact. If G/G_{0} is exact, it then follows, by Theorem 5.1, that G is exact. Thus to resolve the question of whether all locally compact groups are exact, it is enough to consider only totally disconnected groups. Our feeling is that if there if there are groups which are not exact, then there will probably be a discrete example.

3. In [KW, Lemma 2.5] we showed that a group which has an increasing family of exact open subgroups with union the whole group is itself exact. We have been unable to show that exactness is preserved under general inductive limits. Likewise, we do not know if exactness is preserved on passing to a quotient. If this were the case, then all discrete groups would be exact, since any discrete group is a quotient of a free group, which is exact, as noted in §5 (e).

Added note: The Referee has informed us that, in a recent preprint [Y], Guoliang Yu has studied a combinatorial property, property A, of discrete groups which is preserved under semi-direct products. It seems that property A is formally stronger than C*-exactness, but we only became aware of [Y] in September 1999, and have not yet studied all possible connections with our results.

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Eberhard Kirchberg Institut für Mathematik Humboldt-Universität Unter den Linden 6 D-10099 Berlin Germany Simon Wassermann Department of Mathematics University of Glasgow Glasgow G12 8QW United Kingdom