Dilation Theory, Commutant Lifting

and Semicrossed Products

Kenneth R. Davidson^{1} and Elias G. Katsoulis^{2}

Received: July 14, 2011 Revised: September 16, 2011

Communicated by Joachim Cuntz

Abstract. We take a new look at dilation theory for nonself-adjoint operator algebras. Among the extremal (co)extensions of a representation, there is a special property of being fully extremal. This allows a refinement of some of the classical notions which are important when one moves away from standard examples. We show that many algebras including graph algebras and tensor algebras of C*-correspondences have the semi-Dirichlet property which collapses these notions and explains why they have a better dilation theory. This leads to variations of the notions of commutant lifting and Ando’s theorem. This is applied to the study of semicrossed products by automorphisms, and endomorphisms which lift to the C*-envelope. In particular, we obtain several general theorems which allow one to conclude that semicrossed products of an operator algebra naturally imbed completely isometrically into the semicrossed product of its C*- envelope, and the C*-envelopes of these two algebras are the same.

2010 Mathematics Subject Classification: 47L55, 47L40, 46L05, 37B20

Keywords and Phrases: dilation, extremal coextension, semicrossed product, commutant lifting, Fuglede property, C*-envelopes, Ando’s theorem Contents 1 Introduction 782 2 A review of dilations 787 3 Fully Extremal Coextensions 790 4 Semi-Dirichlet algebras 800 5 Commutant Lifting 807 6 A 2 × 2 matrix example. 818 7 Isometric Commutant Lifting and Ando’s Theorem 824 8 Incidence Algebras 834 9 The Fuglede Property 843 10 Completely Isometric Endomorphisms 846 11 A Review of Semicrossed Products 851 12 Dilating Covariance Relations 854 13 Further Examples 860

1First author partially supported by an NSERC grant.

2Second author was partially supported by a grant from ECU.

Documenta Mathematica 16 (2011) 781–868

References 865

1 Introduction

This paper is a general study of dilation theory for arbitrary nonself-adjoint operator algebras. It began with an attempt to formalize those properties need to obtain dilation theorems for covariant representations of an operator algebra and an endomorphism, in order to understand the semicrossed product and its C*-envelope. In this paper, we discuss versions of commutant lifting and Ando’s theorem and consider when they allow us to determine the structure of a semicrossed product and its C*-envelope. This forced us to revisit basic notions in dilation theory, and to introduce a notion stronger than that of extremal (co)extensions. We feel that certain notions in dilation theory are too closely modelled on what happens for the disk algebra. This algebra has been shown to have many very strong properties, and they are often not perfectly reflected in the general case. Certain refinements should be considered to clarify the various dilation properties in a general context.

Dilation theory. Dilation theory for a single operator has its roots in the seminal work of Sz.Nagy [57] which is developed in the now classical book that he wrote with Foiaş [58]. Dilation theory for more general operators was initiated by the deep work of Arveson [5, 6]. The ideas have evolved over the past six decades. The basic ideas are nicely developed in Paulsen’s book [45].

In formulating general properties related to commutant lifting and Ando’s theorem, we were strongly motivated, in part, by the general module formulation expounded by Douglas and Paulsen [26] and the important study by Muhly and Solel [40]. The language used there is a module theoretic approach, while we will mostly talk about representations instead. But the general constructs can, of course, be formulated in either language. Douglas and Paulsen focus on Shilov modules as a primary building block. Muhly and Solel adopt this view, but focus more on a somewhat stronger property of orthoprojective modules. They may have gone further, as we do, had they known what we do today. We will argue that these are more central to the theory.

Another important influence is the Dritschel–McCullough [27] proof of the existence of Arveson’s C*-envelope [5, 6], first established using different methods by Hamana [31]. They provide a proof strongly influenced by ideas of Agler [1]. What they show is: given a completely contractive representation of a unital operator algebra A, that among all dilations of this representation, there are always certain representations which are maximal in the sense that any further dilation can only be obtained by appending a direct sum of another representation. These dilations always exist, as they show, and they are precisely those representations which extend to a ∗-representation of the C*-envelope. It is in this manner that they establish that the existence of the C*-envelope.

This fact was anticipated by Muhly and Solel in [41] where they show, assuming Hamana’s theorem, that every representation has a dilation which is both orthoprojective and orthoinjective. It is easy to see that this is a reformulation of the maximal dilation property. Indeed, one can see that a representation ρ is orthoprojective if and only if it is maximal as a coextension (called an extremal coextension)—meaning that any coextension can be obtained only by appending a direct sum of another representation. Dritschel and McCullough proved that these exist as well. The dual version shows that orthoinjective representations coincide with the extremal extensions.

An extremal (co)extension of a representation ρ on H is called minimal provided that the whole space is the smallest reducing subspace containing H. This is a weaker notion than saying that H is cyclic. However, there can be many extremal coextensions which are minimal but H is not cyclic. Among extremal (co)extensions, there are some preferred (co)extensions which we call fully extremal because they satisfy a stronger maximality property. While in many classical cases, this notion reduces to the usual extremal property, we argue that in general they are preferred. The existence of fully extremal (co)extensions is established by an argument similar to Arveson’s proof [8] of the existence of maximal dilations.

Commutant lifting. The classical commutant lifting theorem was established by Sz.Nagy and Foiaş [59]. Many variations on this theorem have been established in various contexts for a variety of operator algebras. Douglas and Paulsen [26] formulate a version for arbitrary operator algebras, and we propose a modification of their definition.

Shilov representations of an operator algebra A are those which are obtained by taking a ∗-representation of the C*-envelope and restricting it to an invariant subspace for the image of A. All extremal coextensions (orthoprojective representations) are Shilov. The converse holds in some of the classical situations, but is not valid in general. As we will argue, the notions of commutant lifting are better expressed in terms of fully extremal coextensions rather than Shilov coextensions. Limiting the family of coextensions for which lifting occurs increases the family of algebras with this property. Indeed the strong version of commutant lifting can only hold when there is a unique minimal fully extremal coextension (of ρ).

The Douglas-Paulsen formulation of commutant lifting starts with a (completely contractive) representation ρ of an operator algebra A, an operator X in the commutant of ρ(A), and a Shilov coextension σ of ρ; and they ask for a coextension of X to an operator Y of the same norm commuting with σ(A). As remarked in the previous paragraph, this only holds when the minimal Shilov extension is unique. We show that this holds when A is semi-Dirichlet, meaning that

A^{∗}A ⊂ \overline{A + A^{∗}},

such as the disk algebra, the non-commutative disk algebras, and all tensor algebras of graphs and C*-correspondences. The fact that this large class of popular algebras has this remarkable property has perhaps kept us from looking further for an appropriate definition of commutant lifting in other contexts.

We were also influenced by a different approach of Paulsen and Power [46, 47] and subsequent work of theirs with the first author [22, 15]. In this version, the coextension σ is not specified, and one looks for common coextensions σ and Y which commute. We will use extremal coextensions only, rather than arbitrary Shilov coextensions, with the obvious parallel definitions. The first version will be called strong commutant lifting, and the latter commutant lifting. A crucial point is that strong commutant lifting turns out to be equivalent commutant lifting plus uniqueness of the minimal fully extremal coextension.

The intertwining version of commutant lifting proved to be challenging in this context. The resolution of this problem was critical to obtaining good dilation theorems for semicrossed products.

Ando’s theorem. Ando’s Theorem [2] states that if A_{1} and A_{2} are commuting contractions, then they have coextensions V_{i} which are commuting isometries. For us, an Ando theorem for an operator algebra A will be formulated as follows: given a (completely contractive) representation ρ of an operator algebra A and a contraction X in the commutant of ρ(A) , there is a fully extremal coextension σ of ρ and an isometric coextension V of X which commutes with it. Even in the case of the disk algebra, our formulation is stronger than the original, as it asks that one of the isometries, say V_{1}, should have the form

V_{1} ≃ V_{A_{1}} ⊕ U

where V_{A_{1}} is the minimal isometric coextension of A_{1} and U is unitary (see Corollary 7.11).

In the classical case of the disk algebra, the universal algebra of a contraction, the generator of a representation, A = ρ(z), plays a role parallel to the operator X which commutes with it. For this reason, commutant lifting can be applied recursively to A and X, alternating between them, in order to obtain Ando’s Theorem. So in this context, the Sz.Nagy-Foiaş Commutant Lifting Theorem [59] is equivalent to Ando’s Theorem. But for other algebras, there are two distinct aspects, dilating ρ to an extremal coextension and simultaneously coextending X to a commuting contraction, and on the other hand coextending X to an isometry and simultaneously coextending ρ to a commuting representation.

Paulsen and Power [47] formulate Ando’s theorem as a dilation result for A⊗_{min} A(𝔻), or equivalently that

A ⊗_{min} A(𝔻) = A ⊗_{max} A(𝔻).

Such a result holds for a wide class of CSL algebras [46, 22, 15]. The stronger version of commutant lifting only holds in a restricted class [40]. See [40, chapter 5] for a discussion of the differences. In our language, they start with a representation ρ and a commuting contraction X, and seek a maximal dilation π and a simultaneous dilation of X to a unitary U commuting with π(A). We show that this is equivalent to the weaker property of obtaining some coextension σ of ρ and an isometric coextension V of X which commute. This is only ‘half’ of Ando’s theorem in our formulation.

Another property that we will consider is an analogue of the Fuglede theorem: that the commutant of a normal operator is self-adjoint. We formulate this for an operator algebra A with C*-envelope C_{e}^{∗}(A) as saying that for any ∗-representation π of C_{e}^{∗}(A), the commutant of π(A) coincides with the commutant of π(C_{e}^{∗}(A)). We show that a number of operator algebras have this property including all function algebras, the non-commutative disk algebras and more generally the tensor algebras of all finite directed graphs.

Semicrossed products. If A is a unital operator algebra and α is a completely isometric endomorphism, then the semicrossed product

A ×_{α} ℤ_{+}

is the operator algebra that encodes the system (A, α) in the sense that its (completely contractive) representations are in bijective correspondence with the covariant representations of the dynamical system. Concrete versions of these algebras occur in work or Arveson [4, 11]. When A is a C*-algebra, the abstract semicrossed product was defined by Peters [48]. The extension to arbitrary nonself-adjoint operator algebras is straightforward.

The structure of these semicrossed products can often be better understood by showing that the C*-envelope is a full C*-algebra crossed product. Peters [49] does this for the semicrossed product that encodes a discrete dynamical system. The operator algebras of multivariable dynamical systems is developed in [16]. The C*-envelope is further explained in [25], extending Peter’s analysis to this context. More recently, the second author and Kakariadis [32] develop an important generalization of these techniques to very general operator algebras. They show that for nonself-adjoint operator algebras, one first should try to imbed a general semicrossed product into a C*-semicrossed product. They show how to accomplish this, and demonstrate that often the two operator algebras have the same C*-envelope.

When α is the identity map, the semicrossed product is closely tied to commutant lifting. What we show here is that commutant lifting theorems can be sufficient to understand other semicrossed products provided the algebra has some other nice properties. We concern ourselves only with endomorphisms that extend to ∗-endomorphisms of the C*-envelope. When A satisfies the Ando property, every semicrossed product by a completely isometric automorphism is isometrically isomorphic to a subalgebra of the semicrossed product of C_{e}^{∗}(A). These general techniques recover various results in the literature about the structure of crossed products, especially of the non-commutative disk algebras [17] and tensor algebras of C*-correspondences [32]. To our knowledge, all of these results used the strong commutant lifting property (SCLT), which implies uniqueness of fully extremal extensions. Indeed, the theorems relate to algebras with a row contractive condition, the most general of which are tensor algebras of C*=correspondences. Our new result requires only commutant lifting, and applies much more widely.

With a stronger commutant lifting property and the Fuglede property, we can do the same for endomorphisms which lift to the C*-envelope. This applies, in particular, for the disk algebra (which has all of the good properties studied here). This recovers our results [18] for the semicrossed product of A(𝔻) by an endomorphism of the form α(f) = f ∘ b, in the case where b is a non-constant finite Blaschke product. These general results that imbed a semicrossed product into a C*-algebra crossed product are actually dilation theorems. Typically one proves a dilation theorem first, and then deduces the structure of the C*- envelope. However the papers [32, 18] actually compute the C*-envelope first and deduce the dilation theorem afterwards. One of the original motivations for this paper was an attempt to identify the C*-envelope of a semicrossed product using general dilation properties such as commutant lifting. Three such theorems are obtained in section 12.

2 A review of dilations

In this paper, an operator algebra will be a unital abstract operator algebra A in the sense of Blecher, Ruan and Sinclair [12]. A representation of A will mean a unital completely contractive representation ρ on some Hilbert space H. An extension of ρ is a representation σ on a Hilbert space K = H^{⊥} ⊕ H which leaves H invariant, and thus has the form

σ(a) = [\begin{matrix}σ_{11}(a) & 0 \\ σ_{12}(a) & ρ(a) \end{matrix}] .

Dually, a coextension of ρ is a representation σ on a Hilbert space K = H ⊕H^{⊥} which leaves H^{⊥} invariant, and thus has the form

σ(a) = [\begin{matrix}ρ(a) & 0 \\ σ_{12}(a) & σ_{22}(a) \end{matrix}] .

A dilation of ρ is a representation σ on a Hilbert space K containing H so that ρ(a) = P_{H}σ(a)|_{H}. A familiar result of Sarason [55] shows that K decomposes as K = H_{−} ⊕ H ⊕ H_{+} so that

σ(a) = [\begin{matrix}σ_{11}(a) & 0 & 0 \\ σ_{21}(a) & ρ(a) & 0 \\ σ_{31}(a) & σ_{32}(a) & σ_{33}(a) \end{matrix}] .

A representation ρ is an extremal coextension if whenever σ is a coextension of ρ, it necessarily has the form σ = ρ⊕σ′. That is, if H is a subspace of K and σ is a representation of A on K which leaves H^{⊥} invariant and P_{H}σ(a)|_{H} = ρ(a) for a ∈ A, then H reduces σ. Similarly, a representation ρ is an extremal extension if whenever σ is an extension of ρ, it necessarily has the form σ = ρ ⊕ σ′. That is, if H is a subspace of K and σ is a representation of A on K which leaves H invariant and P_{H}σ(a)|_{H} = ρ(a) for a ∈ A, then H reduces σ. Finally, a representation ρ is an extremal representation or a maximal representation if whenever σ is a dilation of ρ, it necessarily has the form σ = ρ⊕σ′. That is, if H is a subspace of K and σ is a representation of A on K so that P_{H}σ(a)|_{H} = ρ(a) for a ∈ A, then H reduces σ. A dilation σ of ρ is an extremal dilation or a maximal dilation of ρ if it is a maximal representation.

Hilbert modules. In the module language espoused by Douglas and Paulsen in [26], a representation ρ makes the Hilbert space H into a left A module H_{ρ} by a · h := ρ(a)h for a ∈ A and h ∈ H. If K = M ⊕ H and σ is a representation of A on K which leaves M invariant, so that with respect to the decomposition K = H ⊕ M of σ is

σ(a) = [\begin{matrix}σ_{11}(a) & 0 \\ σ_{12}(a) & σ_{22}(a) \end{matrix}] ,

then K_{σ} is an A-module with M_{σ_{22}} as a submodule and H_{σ_{11}} as a quotient module, and there is a short exact sequence

0 → M_{σ_{22}} → K_{σ} → H_{σ_{11}} → 0.

Here all module maps are completely contractive. So an extension σ of σ_{22} on M corresponds to larger Hilbert module K_{σ} containing M_{σ_{22}} as a submodule; and a coextension σ of σ_{11} corresponds to the Hilbert module K_{σ} having H_{σ_{11}} as a quotient module.

A module P_{ρ} is orthoprojective if whenever there is an isometric short exact sequence of module maps

\xymatrix{0 \ar[r] & M_{σ_{22}} \ar[r]^{ι} & K_{σ} \ar[r]^{q} & P_{ρ} \ar[r] & 0},

meaning that ι is isometric and q is coisometric, then there is an isometric module map φ : P → K so that K_{σ} = M ⊕ φ(P) as an A-module. It is not difficult to see that this is equivalent to saying that ρ is an extremal coextension. The term orthoprojective was coined by Muhly and Solel [40], and we think that it is superior to the Douglas-Paulsen terminology of hypo-projective because of its more positive aspect. Similarly, one can define orthoinjective modules, and observe that they are equivalent to extremal extensions. A maximal dilation corresponds to a module which is both orthoprojective and orthoinjective.

The C*-envelope. Every unital operator algebra A has a completely isometric representation ι on a Hilbert space H so that the C*-algebra C^{∗}(ι(A)) =: C_{e}^{∗}(A) is minimal in the sense that if σ is any other completely isometric representation on a Hilbert space H′, then there is a unique ∗-homomorphism π of C^{∗}(σ(A)) onto C_{e}^{∗}(A) so that the following diagram commutes:

\xymatrix{A \ar[r]^{ι} \ar[d]_{σ} & C_{e}^{∗}(A) \\ C^{∗}(σ(A)) \ar[ur]^{π}}

The C*-envelope was described by Arveson [5, 6] in his seminal work on noncommutative dilation theory. Its existence was established by Hamana [31].

Muhly and Solel [41] show that maximal dilations exist by invoking Hamana’s theorem. They accomplish this by showing:

Theorem 2.1 (Muhly-Solel). A representation is maximal if and only if it is both orthoprojective and orthoinjective. Equivalently, a representation is maximal if and only if it is both an extremal coextension and an extremal extension.

Dritschel and McCullough [27] establish the existence of maximal dilations directly based on ideas of Agler [1]. In this way, they provide a new and more revealing proof of the existence of the C*-envelope. In fact, they show that every representation has an extension which is extremal; and dually also has a coextension which is extremal. In particular, the maximal representations of A are precisely those representations which extend to ∗-representations of C_{e}^{∗}(A).

Arveson [10] provides a refinement of this result in the separable case by showing that there are sufficiently many irreducible maximal representations, which are the boundary representations that Arveson introduced in [5] as an analogue of the Choquet boundary of a function algebra. We will not require this strengthened version.

Extremal versus Shilov. Douglas-Paulsen [26] and Muhly-Solel [40] focus on Shilov modules. One starts with a ∗-representation π of C_{e}^{∗}(A) on a Hilbert space K. Consider K_{π} as an A-module. A submodule H of K_{π} is a Shilov module. It is easy to deduce from the discussion above that every orthoprojective module is Shilov. Unfortunately, the converse is false. We provide an example below. In the language of representations, a Shilov module corresponds to a representation which has an extension to a maximal representation. However it may still have proper coextensions.

Shilov modules are useful because every completely contractive A-module M has a finite resolution of the form

0 → S_{1} → S_{2} → M → 0,

where S_{1} and S_{2} are Shilov. Using orthoprojective modules, one can obtain

S_{2} → M → 0

with S_{2} orthoprojective. But since submodules do not inherit this extremal property, one does not obtain a short exact sequence. Indeed, while this procedure can be iterated, there need be no finite resolution. This occurs, for example, in the theory of commuting row contractions due to Arveson [9, §9]. However Arveson also argues that, in his context, these are the natural resolutions to seek.

Our view is that it is the extremal coextensions rather than Shilov coextensions which play the role in dilation theory that best models the classical example of the unilateral shift as an isometric model of the disc algebra.

Example 2.2. Consider the non-commutative disk algebra 𝔄_{n}. It is the unital subalgebra of the Cuntz algebra O_{n} generated as a unital nonself-adjoint subalgebra by the canonical isometric generators 𝔰_{1}, …, 𝔰_{n} of O_{n}. A representation ρ of 𝔄_{n} is determined by A_{i} = ρ(𝔰_{i}), and it is completely contractive if and only if

A = [A_{1} … A_{n}]

is a contraction as an operator from H^{(n)} to H [51]. The Frazho-Bunce-Popescu dilation theorem [30, 13, 50] states that A has a coextension to a row isometry.

Conversely, it is clear that any coextension of a row isometry must be obtained as a direct sum. Thus these row isometric representations are precisely the extremal coextensions and correspond to orthoprojective modules. The Wold decomposition [23] shows that this row isometry decomposes as a direct sum of a Cuntz row unitary and a multiple of the left regular representation of the free semigroup 𝔽_{n}^{+} on Fock space. This representation generates the Cuntz-Toeplitz C*-algebra, and thus is not a maximal representation. It can be extended to a maximal dilation in many explicit ways [23]. It is clear in this case that every ∗-representation of O_{n} sends

𝔰 = [𝔰_{1} … 𝔰_{n}]

to a row unitary, and the restriction to any invariant subspace is a row isometry. Thus every Shilov module is orthoprojective.

Example 2.3. Let A_{n} be the universal algebra of a row contraction with commuting coefficients. This algebra was studied extensively by Arveson beginning in [8]. The basic von Neumann inequality was proven much earlier by Drury [28], but the full version of the dilation theorem was due to Müller and Vascilescu [43] and later, Arveson. Arveson further showed that the multipliers S_{1}, …, S_{n} on symmetric Fock space H_{n}^{2} in n variables form a canonical model for A_{n}. Also H_{n}^{2} is a reproducing kernel Hilbert space, and A_{n} is the algebra of continuous multipliers. The C*-algebra generated by these multipliers is the C*-envelope of A_{n} [8].

The dilation theorem shows that every commuting row contraction has a coextension to a direct sum S_{i}^{(α)}⊕U_{i} where α is some cardinal and U_{i} are commuting normal operators satisfying

∑_{i=1}^{n} U_{i}U_{i}^{∗} = I.

These are precisely the extremal coextensions and determine the orthoprojective modules. Surprisingly they are also the maximal representations. So while one can dilate in both directions to obtain a maximal dilation of a representation ρ, only coextensions are required.

However, no submodule of the symmetric Fock space is orthoprojective. They are all Shilov, but none model the algebra in a useful way. Davidson and Le [21, Example 4.1] provide an explicit example of this phenomenon in their paper on the commutant lifting theorem for A_{n}.

3 Fully Extremal Coextensions

There is a natural partial order ≺ on dilations: say that ρ ≺ σ if σ acts on a Hilbert space K containing a subspace H so that P_{H}σ|_{H} is unitarily equivalent to ρ. There is also a partial order on extensions ≺_{e}: say that ρ ≺_{e} σ if σ acts on a Hilbert space K containing an invariant subspace H so that σ|_{H} is unitarily equivalent to ρ. Similarly, for coextensions, say that ρ ≺_{c} σ if σ acts on a Hilbert space K containing a co-invariant subspace H so that P_{H}σ|_{H} is unitarily equivalent to ρ.

Dritschel and McCullough [27] establish the existence of extremals dominating ρ in each of these classes. We want something a little bit stronger. It is possible for an extremal coextension σ of ρ to have a proper extension which is also a coextension of ρ, so that σ is not extremal in the partial order ≺. We provide an example shortly. We will require knowing that ρ has a coextension which is extremal with respect to ≺.

Definition 3.1. If ρ is a representation of A, say that a coextension σ of ρ is fully extremal with respect to ρ if whenever σ ≺ τ and ρ ≺_{c} τ, then τ = σ ⊕ τ′. Similarly we define an extension σ of ρ to be fully extremal with respect to ρ if whenever σ ≺ τ and ρ ≺_{e} τ, then τ = σ ⊕ τ′.

Example 3.2. Fix an orthonormal basis e_{1}, …, e_{n} for ℂ^{n}, and let E_{ij} be the canonical matrix units. Consider the subalgebra A of 𝔐_{n} spanned by the diagonal algebra

𝔇_{n} = span{E_{ii} : 1 ≤ i ≤ n}

and

span{E_{ij} : |i − j| = 1, j odd }.

This is a reflexive operator algebra with invariant subspaces

ℂe_{2i} for 1 < 2i ≤ n

and

L_{2i+1} = span{e_{2i}, e_{2i+1}, e_{2i+2}} for 1 ≤ 2i + 1 ≤ n,

where we ignore e_{0} and e_{n+1} if they occur. The elements of A have the form

A = [\begin{matrix}a_{11} & 0 & 0 & 0 & 0 & … \\ a_{21} & a_{22} & a_{32} & 0 & 0 & … \\ 0 & 0 & a_{33} & 0 & 0 & … \\ 0 & 0 & a_{43} & a_{44} & a_{45} & … \\ 0 & 0 & 0 & 0 & a_{55} & ... \\ ⋮ & ⋮ & ⋮ & ⋮ & ... & ... \end{matrix}]

Consider the representation ρ(A) = a_{11}, the 1,1 matrix entry of A. Since ℂe_{1} is coinvariant, this is a representation. The compression σ_{2} of A to M_{2} = span{e_{1}, e_{2}} is a coextension of ρ given by

σ_{2}(A) = P_{M_{2}}A|_{M_{2}} = L_{1} [\begin{matrix}a_{11} & 0 \\ a_{21} & a_{22} \end{matrix}]

This is readily seen to be an extremal coextension of ρ. It is minimal in the sense we use: it contains no proper reducing subspace containing H_{ρ} = ℂe_{1}, and is also minimal in the sense that H_{σ} = σ(A)H_{ρ}.

However σ_{2} is not fully extremal. Let

M_{k} = span{e_{i} : 1 ≤ i ≤ k}

and set

σ_{k}(A) = P_{M_{k}}A|_{M_{k}}.

Then σ_{2i+1} is an extension of σ_{2i} and σ_{2i+2} is a coextension of σ_{2i+1}. All are coextensions of ρ. Each σ_{2i} is an extremal coextension of ρ, as is σ_{n} = id even if n is even. Moreover all are minimal in that they have no proper reducing subspace containing ℂe_{1}. Only σ_{n} is fully extremal. Note that to get from ρ to σ_{n}, one must alternately coextend and extend n − 1 times if at each stage, you take a classical minimal extension or coextension.

One can also define an infinite version of this algebra where it takes a countable number of steps to attain the fully extremal coextension.

Example 3.3. Let A(𝔻) be the disk algebra. A representation of A(𝔻) is determined by T = ρ(z), and it is completely contractive if and only if ∥T ∥ ≤ 1. Every contraction coextends to a unique minimal isometry. So the extremal coextensions must be isometries. But conversely, it is easy to see that any contractive coextension of an isometry is obtained by adding a direct sum. So when T is an isometry, ρ is an extremal coextension. The minimal isometric dilation V of T yields a fully extremal coextension because the range of V together with H_{ρ} spans the whole space. Any (contractive) dilation of V must map the new subspace orthogonal to the range of V. So if it is not a summand, the range will not be orthogonal to H_{ρ}, so it won’t be a coextension of ρ.

The extremal coextensions of ρ correspond to all isometric coextensions of T, namely V ⊕W where V is the minimal isometric dilation and W is any isometry. But if W isn’t unitary, it can be extended to a unitary. This extension is still a coextension of T. So the fully extremal coextensions correspond to V ⊕ U where U is unitary.

Similarly, the maximal dilations of ρ correspond to unitary dilations of A. The restriction of a unitary to an invariant subspace is an isometry. So a Shilov representations are extremal coextensions. In particular, a minimal Shilov dilation of ρ is a fully extremal coextension.

Example 3.4. Let A = A(𝔻^{2}) with generators z_{1} and z_{2}. Then a completely contractive representation is determined by a pair of commuting contractions A_{i} = ρ(z_{i}). By Ando’s Theorem [2], every commuting pair coextends to a pair of commuting isometries. It is clear that any coextension of isometries to a larger space can only be obtained by adding a direct summand. So the extremal coextensions are the commuting isometries. It is also clear that any restriction to an invariant subspace is still isometric.

Moreover every pair of commuting isometries extends to a pair of commuting unitaries. These are the maximal dilations, and determine a ∗-representation of

C_{e}^{∗}(A) = C(𝕋^{2}).

The restriction of a unitary to an invariant subspace is an isometry. So every Shilov module is orthoprojective (an extremal coextension).

What we wish to point out is that extremal coextensions of a representation ρ of A(𝔻^{2}) need not be fully extremal. Let

A_{1} = A_{2} = 0 acting on H = ℂ.

Identify H with ℂe_{0,0} in

K = span{e_{m,n} : m, n ≥ 0},

where {e_{m,n} : m, n ≥ 0} is an orthonormal basis. Then it is clear that there is a coextension of A_{i} to the commuting isometries

S_{1} = S ⊗ I and S_{2} = I ⊗ S,

where S is the unilateral shift. Let σ be the corresponding coextension of ρ. This is an extremal coextension because the S_{i} are isometries.

Enlarge this orthonormal basis further to obtain a space

L = span{e_{k,l} : max{k, l} ≥ 1 or k = l = 0}

containing K. Let T_{i} be the commuting isometries given by

T_{1}e_{m,n} = e_{m+1,n} and T_{2}e_{m,n} = e_{m,n+1}.

Let τ be the induced representation of A(𝔻^{2}). It is clear by inspection that H = ℂe_{00} is coinvariant, and hence ρ ≺_{c} τ. Moreover, τ is extremal because T_{i} are isometries. The subspace K is invariant for T_{1} and T_{2}, and T_{i}|_{K} = S_{i}. Therefore σ ≺_{e} τ. So σ is not fully extremal with respect to ρ.

We claim that τ is fully extremal with respect to ρ. Since it is extremal, it can only fail to be fully extremal if there is a larger space M ⊃ L and commuting isometries V_{i} on M extending T_{i} so that L is not coinvariant, but ℂe_{00} is. Hence one of the isometries, say V_{1}, has P_{L^{⊥}}V_{1}P_{L} ≠ 0. Let

N = (Ran T_{1} ∨ ℂe_{00})^{⊥} = span{e_{1,l} : l < 0}.

There must be a vector x ∈ N so that V_{1}^{∗}x ≠ 0. Equivalently, there are vectors y, z ∈ L^{⊥} so that V_{1}y = z + x. Write x = ∑_{l<0} a_{l}e_{1,l}, and let l_{0} be the least integer so that a_{−l_{0}} ≠ 0. Let

x′ = T_{2}^{l_{0}−1}x = ∑_{l<0} a_{l+1−l_{0}}e_{1,l} =: ∑_{l<0} a′_{l}e_{1,l};

so that a′_{−1} ≠ 0. Also set

y′ = V_{2}^{l_{0}−1}y and z′ = V_{2}^{l_{0}−1}z.

Then

V_{1}y′ = V_{1}V_{2}^{l_{0}−1}y = V_{2}^{l_{0}−1}V_{1}y
= V_{2}^{l_{0}−1}(z + x) = z′ + x′.

Moreover, z′ is orthogonal to the range of T_{2}^{l_{0}−1}, which contains N. Hence

⟨V_{2}V_{1}y′, e_{1,0}⟩ = ⟨V_{2}(z′ + x′), e_{1,0}⟩
= ⟨z′ + x′, e_{1.−1}⟩ = a′_{−1} ≠ 0.

Therefore

0 ≠ ⟨V_{1}V_{2}y′, e_{1,0}⟩ = ⟨V_{2}y′, e_{0,0}⟩.

This contradicts the fact that τ is a coextension of ρ. Thus τ must be fully extremal relative to ρ.

Now we turn to the issue of establishing that fully extremal coextensions (and extensions) always exist.

Theorem 3.5. Let A be a unital operator algebra, and let ρ be a representation of A on H. Then ρ has a fully extremal coextension σ.

If A and H are separable, then one can take σ acting on a separable Hilbert space.

Proof. Our argument is based on Arveson’s proof [10, Theorem 2.5] that maximal dilations exist. He works with the operator system generated by A, which is self-adjoint. As we will work directly with A, we need to consider adjoints as well. The goal is to construct a coextension σ of ρ on a Hilbert space K so that for every a ∈ A and k ∈ K,

∥σ(a)k∥ = sup{∥τ(a)k∥ : τ ≻ σ, τ ≻_{c} ρ}  (1)

and

∥σ(a)^{∗}k∥ = sup{∥τ(a)^{∗}k∥ : τ ≻ σ, τ ≻_{c} ρ}.  (2)

Once this is accomplished, it is evident that any dilation τ of σ which is a coextension of ρ must have K as a reducing subspace, as claimed.

To this end, choose a dense subset of A × H, and enumerate it as

{(a_{α}, h_{α}) : α ∈ Λ}

where Λ is an ordinal. Suppose that we have found coextensions σ_{α} of ρ for all α < α_{0} < Λ acting on K_{α}, where K_{β} ⊂ K_{α} when β < α, so that

∥σ_{α}(a_{β})h_{β}∥ = sup{∥τ(a_{β})h_{β}∥ : τ ≻ σ_{α}, τ ≻_{c} ρ}  (1′)

and

∥σ_{α}(a_{β})^{∗}h_{β}∥ = sup{∥τ(a)^{∗}h_{β}∥ : τ ≻ σ_{α}, τ ≻_{c} ρ}.  (2′)

for all β < α. This latter condition is automatic because each τ(a)^{∗} leaves H invariant, and agrees with ρ(a)^{∗} there. But we carry this for future use.

If α_{0} is a limit ordinal, we just form the natural direct limit of the σ_{α} for α < α_{0}, and call it σ_{α_{0}}. Note that it will now satisfy (1′) and (2′) for β < α_{0}.

Otherwise α_{0} = β_{0} + 1 is a successor ordinal. Choose a dilation τ_{1} ≻ σ_{β_{0}} on M_{1} ⊃ K_{β_{0}} such that τ_{1} ≻_{c} ρ and satisfies

∥τ_{1}(a_{β_{0}})h_{β_{0}}∥ ≥ sup{∥τ(a_{β_{0}})h_{β_{0}}∥ : τ ≻ σ_{α}, τ ≻_{c} ρ} − 2^{−1}

and

∥τ_{1}(a_{β_{0}})^{∗}h_{β_{0}}∥ ≥ sup{∥τ(a)^{∗}h_{β_{0}}∥ : τ ≻ σ_{α}, τ ≻_{c} ρ} − 2^{−1}.

Then choose recusively dilations τ_{n+1} of τ_{n} on M_{n+1} ⊃ M_{n} which are all coextensions of ρ so that

∥τ_{n+1}(a_{β_{0}})h_{β_{0}}∥ ≥ sup{∥τ(a_{β_{0}})h_{β_{0}}∥ : τ ≻ σ_{α}, τ ≻_{c} ρ} − 2^{−n−1}

and

∥τ_{n+1}(a_{β_{0}})^{∗}h_{β_{0}}∥ ≥ sup{∥τ(a)^{∗}h_{β_{0}}∥ : τ ≻ σ_{α}, τ ≻_{c} ρ} − 2^{−n−1}.

The inductive limit is a representation σ_{α_{0}} with the desired properties.

Once we reach Λ, we have constructed a representation σ̃_{1} on K̃_{1} coextending ρ and satisfying (1) and (2) for vectors h ∈ H. Now repeat this starting with σ̃_{1} and a dense subset of A × K̃_{1}, but still considering dilations which are coextensions of ρ. This time, the equations involving the adjoint are important. The result is a representation σ̃_{2} on K̃_{2} dilating σ̃_{1} and coextending ρ satisfying (1) and (2) for all vectors k ∈ K̃_{1}. Repeat recursively for all n ≥ 1 and in the end, we obtain the desired coextension.

If A and K are separable, a countable sequence of points suffices, and at each stage of this countable process, one obtains separable spaces. So the result is a separable representation.

Remark 3.6. It easily follows from the proof of existence of fully extremal coextensions that if σ is a coextension of ρ, then there is a dilation τ of σ which is a fully extremal coextension of ρ.

Remark 3.7. A proof of existence of extremal coextensions can be made along the same lines. It is only necessary to achieve σ ≻_{c} ρ on K such that:

∥σ(a)k∥ = sup{∥τ(a)k∥ : τ ≻_{c} σ}.

One can always achieve this by repeated coextension, and in this way one obtains an extremal coextension σ of ρ with the additional property that H is cyclic, i.e. K = \overline{σ(A)H}. This is evidently not the case in general for extremal coextensions, and in particular, for fully extremal coextensions. See the preceding examples and Remark 3.13.

The same result for extensions follows by duality.

Corollary 3.8. Let A be a unital operator algebra, and let ρ be a representation of A on H. Then ρ has a fully extremal extension σ.

Corollary 3.9. If ρ_{1} and ρ_{2} are representations of A, then σ_{1} and σ_{2} are fully extremal coextensions of ρ_{1} and ρ_{2}, respectively, if and only if σ_{1} ⊕ σ_{2} is a fully extremal coextension of ρ_{1} ⊕ ρ_{2}.

In particular, σ_{1} and σ_{2} are extremal coextensions of A if and only if σ_{1} ⊕ σ_{2} is an extremal coextension.

Proof. First suppose that σ_{1} and σ_{2}, acting on K_{1} and K_{2}, are fully extremal coextensions of ρ_{1} and ρ_{2}, respectively. Suppose that τ is a representation on

P = K_{1} ⊕ K_{2} ⊕ P′

such that

τ ≻ σ_{1} ⊕ σ_{2} and τ ≻_{c} ρ_{1} ⊕ ρ_{2}.

Then as τ ≻ σ_{i} and τ ≻_{c} ρ_{i}, we deduce that τ reduces K_{i} and hence reduces K_{1} ⊕ K_{2}. So σ_{1} ⊕ σ_{2} is a fully extremal coextension of ρ_{1} ⊕ ρ_{2}.

Conversely, if σ_{1} ⊕ σ_{2} is a fully extremal coextension of ρ_{1} ⊕ ρ_{2}, suppose that τ is a representation on P = K_{1} ⊕ P′ satisfies τ ≻ σ_{1} and τ ≻_{c} ρ_{1}. Then

τ ⊕ σ_{2} ≻ σ_{1} ⊕ σ_{2} and τ ⊕ σ_{2} ≻_{c} ρ_{1} ⊕ ρ_{2}.

It follows that τ ⊕ σ_{2} reduces K_{1} ⊕ K_{2}. So τ reduces K_{1}. Whence σ_{1} is fully extremal.

Applying this to ρ_{i} = σ_{i} yields the last statement.

If one starts with a representation ρ and alternately forms extremal extensions and coextensions, it may require a countable sequence of alternating extensions and coextensions in order to obtain a maximal dilation as in Example 3.2. One advantage of fully extremal extensions and coextensions is that only one is required to obtain a maximal dilation.

Proposition 3.10. Let ρ be a representation of A. If σ is an extremal coextension of ρ, and τ is a fully extremal extension of σ, then τ is a maximal dilation.

Proof. Since τ is an extremal extension, it suffices to show that it is also an extremal coextension. For then the Muhly-Solel result, Theorem 2.1, will show that τ is a maximal dilation.

Say that ρ, σ and τ act on H, K and L respectively. Suppose that π is a coextension of τ acting on P. Decompose

P = (L ⊖ K) ⊕ H ⊕ (K ⊖ H) ⊕ (P ⊖ L).

Then we have

π = [\begin{matrix}τ_{11} & 0 & 0 & 0 \\ τ_{21} & ρ & 0 & 0 \\ τ_{31} & σ_{32} & σ_{33} & 0 \\ π_{41} & π_{42} & π_{43} & π_{44} \end{matrix}] ,

where σ is represented by the middle 2 × 2 square, and τ is represented by the upper left 3×3 corner. The lower right 3×3 corner is a coextension of σ. Since σ is an extremal coextension, we obtain

π_{42} = 0 = π_{43}.

Thus we can rearrange the decomposition moving P ⊖ L to the first coordinate to obtain

π ≃ [\begin{matrix}π_{44} & π_{41} & 0 & 0 \\ 0 & τ_{11} & 0 & 0 \\ 0 & τ_{21} & ρ & 0 \\ 0 & τ_{31} & σ_{32} & σ_{33} \end{matrix}] .

This is a coextension of τ which is an extension of σ. By the fact that τ is a fully extremal extension of σ, we deduce that π_{41} = 0 and so

π ≃ π_{44} ⊕ τ.

Therefore τ is also an extremal coextension.

The dual result is obtained the same way.

Corollary 3.11. Let ρ be a representation of A. If σ is an extremal extension of ρ, and τ is a fully extremal coextension of σ, then τ is a maximal dilation.

The classical notion of minimal coextension is that the space is cyclic for A. However, it seems more natural that the original space merely generate the whole space as a reducing subspace. This is because fully extremal coextensions do not generally live on the cyclic subspace generated by the original space.

Definition 3.12. An extremal coextension σ on K of a representation ρ of A on H is minimal if the only reducing subspace of K containing H is K itself. Likewise we define minimality for fully extremal coextensions, extremal extensions and fully extremal extensions. This minimal (fully) extremal (co)extension is unique if any two of these objects are unitarily equivalent via a unitary which is the identity on H.

Say that a coextension σ on K of a representation ρ of A on H is cyclic if K = \overline{σ(A)H}.

Remark 3.13. These notions of minimality are subtle. Look at Example 3.2. In general, to generate the space on which a coextension acts, one must alternately take the cyclic subspace generated by σ(A) and σ(A)^{∗}, perhaps infinitely often, in order to obtain the reducing subspace generated by H.

In Example 3.4, the 1-dimensional zero representation ρ has an extremal coextension σ. It is minimal because

K = \overline{σ(A(𝔻^{2}))H},

i.e. H is cyclic for σ(A(𝔻^{2})). However the extension τ of σ is also an extremal coextension of ρ. While it is no longer true that \overline{τ(A(𝔻^{2}))H} is the whole space, it is nevertheless the smallest reducing subspace containing H, and so it is also minimal. Thus it is a minimal fully extremal coextension.

It is important to note that there are minimal fully extremal coextensions obtained in the natural way.

Proposition 3.14. Let ρ be a representation of A on H. Let σ be a fully extremal (co)extension of ρ on K. Let σ_{0} be the restriction of σ to the smallest reducing subspace K_{0} for σ(A) containing H. Then σ_{0} is fully extremal. Moreover, σ = σ_{0} ⊕ σ_{1} where σ_{1} is a maximal representation. Conversely, every (co)extension of this form is fully extremal.

Proof. The proof is straightforward. Since K_{0} reduces σ, we can write

σ = σ_{0} ⊕ σ_{1}

acting on K = K_{0}⊕K_{0}^{⊥}. Suppose that τ is a dilation of σ_{0} which is a coextension of ρ. Then τ ⊕ σ_{1} is a dilation of σ which is a coextension of ρ. Since σ is fully extremal, we have a splitting

τ ⊕ σ_{1} ≃ σ ⊕ τ_{1} = σ_{0} ⊕ σ_{1} ⊕ τ_{1}.

Hence

τ = σ_{0} ⊕ τ_{1}.

It follows that σ_{0} is a fully extremal coextension of ρ.

Any dilation of σ_{1} yields a dilation of σ which is a coextension of ρ. As σ is fully extremal, this must be by the addition of a direct summand. Hence σ_{1} is a maximal representation. Conversely, if σ_{0} is a (minimal) fully extremal coextension of ρ and σ_{1} is a maximal representation, then σ = σ_{0} ⊕ σ_{1} is a fully extremal coextension because any dilation of σ is a dilation of σ_{0} direct summed with σ_{1}.

The same argument works for extensions.

We refine Proposition 3.10. In light of Remark 3.7, we know that the coextensions asked for in the following proposition always exist.

Proposition 3.15. Let ρ be a representation of A on H. Let σ be a cyclic extremal coextension of ρ on K. Let π be a minimal fully extremal extension of σ. Then π is a minimal maximal dilation of ρ.

Proof. It suffices to show that the whole space, L, is the smallest reducing subspace for π(A) containing H. In particular, it contains

\overline{π(A)H} = \overline{σ(A)H} = K.

But the minimality of π as a fully extremal extension of σ ensures that there is no proper reducing subspace containing K. So π is minimal as a maximal dilation.

We require a result which is more subtle than Proposition 3.15 but is valid for fully extremal coextensions.

Theorem 3.16. Let ρ be a representation of A on H. Let σ be a minimal fully extremal coextension of ρ on K. Let π be a minimal fully extremal extension of σ. Then π is a minimal maximal dilation of ρ.

Moreover, the representation π determines σ, and thus two inequivalent minimal fully extremal coextensions of ρ yield inequivalent minimal maximal dilations of ρ.

Proof. Let π act on the Hilbert space L. Note that π is a maximal dilation of ρ by Proposition 3.10. Let

M = \overline{π(A)^{∗}H} ⊖ H.

Then M^{⊥} is the largest invariant subspace for π(A) in which H is coinvariant. Let τ denote the restriction of π to M^{⊥}. Since M^{⊥} contains K, we have

τ ≻ σ and τ ≻_{c} ρ.

Hence by the fully extremal property of σ, we deduce that

τ = σ ⊕ τ′ on M^{⊥} = K ⊕ (M + K)^{⊥}.

Now the smallest reducing subspace for π(A) containing H clearly contains M. Thus it contains the smallest τ(A) reducing subspace of M^{⊥} contaning H. But since τ = σ ⊕ τ′ and σ is minimal as a fully extremal coextension, the smallest τ(A) reducing subspace containing H is K. Then since π is a minimal fully extremal extension of σ, we see that L is the smallest reducing subspace containing K. So π is minimal.

From the arguments above, we see that σ is recovered from π by forming

M = \overline{π(A)^{∗}H} ⊖ H.

restricting π to M^{⊥} to get τ, and taking the smallest τ reducing subspace of M^{⊥} containing H. The restriction to this subspace is σ. Hence π determines σ. Consequently, two inequivalent fully extremal coextensions of ρ yield inequivalent minimal maximal dilations of ρ.

The following is immediate by duality.

Corollary 3.17. Let ρ be a representation of A on H. Let σ be a minimal fully extremal extension of ρ on K. Let π be a minimal fully extremal coextension of σ. Then π is a minimal maximal dilation of ρ.

Moreover, the dilation π determines σ. Thus two inequivalent minimal fully extremal extensions of ρ yield inequivalent minimal maximal dilations of ρ.

4 Semi-Dirichlet algebras

In this section, we consider a class of algebras where the theory is more like the classical one. The semi-Dirichlet property is a powerful property that occurs often in practice. From the point of view of dilation theory, these algebras are very nice.

Definition 4.1. Say that an operator algebra A is semi-Dirichlet if

A^{∗}A ⊂ \overline{A + A^{∗}}

when A is considered as a subspace of its C*-envelope.

A unital operator algebra (not necessarily commutative) is called Dirichlet if A + A^{∗} is norm dense in C_{e}^{∗}(A).

Notice that since A is unital, we always have A + A^{∗} ⊂ span(A^{∗}A), so semi-Dirichlet means that

\overline{span}(A^{∗}A) = \overline{A + A^{∗}}.

The interested reader can note that in the case of w*-closed algebras, the proofs below can be modified to handle the natural w*-closed condition which we call semi-σ-Dirichlet if

A^{∗}A ⊂ \overline{A + A^{∗}}^{w∗}.

Free semigroup algebras and free semigroupoid algebras of graphs and nest algebras all are semi-σ-Dirichlet.

The following simple proposition establishes a few elementary observations.

Proposition 4.2.

(i) A is Dirichlet if and only if A and A^{∗} are semi-Dirichlet.

(ii) If σ is a completely isometric representation of A on H, and

σ(A)^{∗}σ(A) ⊂ \overline{σ(A) + σ(A)^{∗}},

then A is semi-Dirichlet.

(iii) If σ is a Shilov representation of a semi-Dirichlet algebra A, then σ(A) is semi-Dirichlet.

Proof. (i) It is obvious that if A is Dirichlet, then both A and A^{∗} are semi-Dirichlet. For the converse, notice that if A is semi-Dirichlet, then an easy calculation shows that \overline{span}(AA^{∗}) is a C*-algebra [8]. Since A generates C_{e}^{∗}(A), this is the C*-algebra \overline{span}(AA^{∗}). Thus the semi-Dirichlet property for A^{∗} now shows that A + A^{∗} is norm dense in C_{e}^{∗}(A).

(ii) If σ is completely isometric, then 𝔄 = C^{∗}(σ(A)) is a C*-cover of A. By the minimal property of the C*-envelope, there is a quotient map q : 𝔄 → C_{e}^{∗}(A) so that qσ|_{A} is the identity map. If σ(A)^{∗}σ(A) is contained in \overline{σ(A) + σ(A)^{∗}}, then passing to the quotient yields the semi-Dirichlet property.

(iii) If σ is Shilov, then there is a ∗-representation π of C_{e}^{∗}(A) on K and an invariant subspace H so that σ(a) = π(a)|_{H}. The map

σ̃(x) = P_{H}π(x)|_{H} for x ∈ C_{e}^{∗}(A)

is a completely positive map extending σ. In particular,

σ̃(a^{∗}) = σ(a)^{∗} for a ∈ A.

For a, b ∈ A, we calculate

π(a^{∗}b) = [\begin{matrix}∗ & ∗ \\ ∗ & σ̃(a^{∗}b) \end{matrix}] = π(a)^{∗}π(b)

= [\begin{matrix}∗ & ∗ \\ 0 & σ(a)^{∗} \end{matrix}] [\begin{matrix}∗ & 0 \\ ∗ & σ(b) \end{matrix}] = [\begin{matrix}∗ & ∗ \\ ∗ & σ(a)^{∗}σ(b) \end{matrix}]

Hence

σ̃(a^{∗}b) = σ(a)^{∗}σ(b) for all a, b ∈ A.

Since A is semi-Dirichlet, we can write

a^{∗}b = lim_{n} c_{n}^{∗} + d_{n}

where c_{n}, d_{n} ∈ A. Thus,

σ(a)^{∗}σ(b) = σ̃(a^{∗}b) = lim σ(c_{n})^{∗} + σ(d_{n}).

That is,

σ(A)^{∗}σ(A) ⊂ \overline{σ(A) + σ(A)^{∗}}.

It now follows from (ii) that σ(A) is semi-Dirichlet.

Example 4.3. Observe that if A is a function algebra with Shilov boundary X = ∂A, then span(A^{∗}A) is a norm closed self-adjoint algebra which separates points. So by the Stone-Weierstrass Theorem, it is all of C(X). So the semi-Dirichlet property is just the Dirichlet property for function algebras.

Example 4.4. The non-commutative disk algebras 𝔄_{n} are semi-Dirichlet. This is immediate from the relations 𝔰_{j}^{∗}𝔰_{i} = δ_{ij}I.

Indeed, it is easy to see that all tensor algebras of directed graphs and tensor algebras of C*-correspondences are semi-Dirichlet. For those familiar with the terminology for the tensor algebra of a C*-correspondence E over a C*-algebra 𝔄, the algebra T^{+}(E) is generated by

σ(𝔄) and {T (ξ) : ξ ∈ E},

where σ and T are the canonical representations of 𝔄 and E, respectively, on the Fock space of E. The relation

T (ξ)^{∗}T (η) = σ(⟨ξ, η⟩)

yields the same kind of cancellation as for the non-commutative disk algebra to show that

T^{+}(E)^{∗}T^{+}(E) ⊂ \overline{T^{+}(E) + T^{+}(E)^{∗}}.

Example 4.5. There is no converse to Proposition 4.2(ii). Consider the disk algebra A(𝔻). The Toeplitz representation on H^{2} given by σ(f) = T_{f}, the Toeplitz operator with symbol f, is completely isometric. This is Shilov, and so has the semi-Dirichlet property. This is also readily seen from the identity

T_{f}^{∗}T_{g} = T_{f̄g} for all f, g ∈ A(𝔻).

However the representation

ρ(f) = T_{f(z̄)}

generated by ρ(z) = T_{z}^{∗} is also completely isometric. However

ρ(z)^{∗}ρ(z) = T_{z}T_{z}^{∗} = I − e_{0}e_{0}^{∗}.

This is not a Toeplitz operator, and so is a positive distance from

\overline{ρ(A(𝔻)) + ρ(A(𝔻))^{∗}} = \overline{\{T_{f̄+g} : f, g ∈ A(𝔻)\}}
= {T_{f} : f ∈ C(𝕋)}.

We will establish the following theorem.

Theorem 4.6. Suppose that A is a semi-Dirichlet unital operator algebra. Let ρ be a representation of A. Then ρ has a unique minimal extremal coextension σ, it is fully extremal and cyclic (i.e. K = \overline{σ(A)H}). Moreover, every Shilov representation is an extremal coextension.

We begin with a couple of lemmas.

Lemma 4.7. Suppose that A is a semi-Dirichlet unital operator algebra. Let ρ be a representation of A, and let σ be a cyclic extremal coextension of ρ on K. Then σ is fully extremal.

Proof. Suppose that that τ is an extremal coextension of ρ which is a dilation of σ. Say τ acts on L ⊃ K. Let π be a fully extremal extension of τ. Then π is a maximal dilation of ρ by Proposition 3.10. Moreover L is invariant for π(A) because τ is extremal, as is K for the same reason. Therefore

τ(a) = π(a)|_{L} and σ(a) = π(a)|_{K} = τ(a)|_{K}.

Also H is semi-invariant for π(A) and coinvariant for τ(A) in L.

If σ is not a direct summand of τ, then K is not coinvariant for τ(A). Thus there is a vector x ∈ L ⊖ K and a ∈ A so that

P_{K}τ(a)x ≠ 0.

This vector in K can be approximated by a vector σ(b)h for some b ∈ A and h ∈ H sufficiently well so that

⟨τ(a)x, σ(b)h⟩ ≠ 0.

Now a^{∗}b ∈ A^{∗}A can be written as

a^{∗}b = lim_{n} c_{n} + d_{n}^{∗} where c_{n}, d_{n} ∈ A.

Therefore

0 ≠ ⟨τ(a)x, σ(b)h⟩ = ⟨π(a)x, π(b)h⟩

= ⟨x, π(a^{∗}b)h⟩ = lim_{n→∞}⟨x, π(c_{n})h + π(d_{n})^{∗}h⟩

= lim_{n→∞}⟨x, π(d_{n})^{∗}h⟩ = lim_{n→∞}⟨x, τ(d_{n})^{∗}h⟩

Here we used the fact that

π(c_{n})h = σ(c_{n})h ∈ K,

which is orthogonal to x, and then the fact that the compression of π(d_{n})^{∗} to L is τ(d_{n})^{∗}. This calculation shows that H is not coinvariant for τ, contrary to our hypothesis. This means that τ does indeed have σ as a direct summand. So σ is fully extremal.

Lemma 4.8. Suppose that A is a semi-Dirichlet unital operator algebra. Let ρ be a representation of A. Then any two cyclic Shilov coextensions σ_{i} of ρ, i = 1, 2, on K_{i} are equivalent. Hence a cyclic Shilov coextension of ρ is fully extremal.

Proof. Let σ_{i}, i = 1, 2, be two minimal cyclic Shilov coextensions of ρ on K_{i}; so that K_{i} = \overline{σ_{i}(A)H}. Let π_{i} be the maximal dilations of ρ on L_{i} ⊃ K_{i} such that K_{i} is invariant and π_{i}(a)|_{K_{i}} = σ_{i}(a) for a ∈ A. The idea is to follow the standard proof by showing that there is a map U ∈ B(K_{1}, K_{2}) given by

Uσ_{1}(a)h = σ_{2}(a)h

which is a well defined isometry of K_{1} onto K_{2}. To this end, it suffices to verify that

⟨σ_{1}(a_{1})h_{1}, σ_{1}(a_{2})h_{2}⟩ = ⟨σ_{2}(a_{1})h_{1}, σ_{2}(a_{2})h_{2}⟩

for all a_{1}, a_{2} ∈ A and h_{1}, h_{2} ∈ H.

By hypothesis, we can find b_{n}, c_{n} ∈ A so that

a_{2}^{∗}a_{1} = lim_{n} b_{n} + c_{n}^{∗}.

We calculate

⟨σ_{i}(a_{1})h_{1}, σ_{i}(a_{2})h_{2}⟩ = ⟨π_{i}(a_{1})h_{1}, π_{i}(a_{2})h_{2}⟩

= ⟨π_{i}(a_{2}^{∗}a_{1})h_{1}, h_{2}⟩

= lim_{n→∞}⟨(π_{i}(b_{n}) + π_{i}(c_{n})^{∗})h_{1}, h_{2}⟩

= lim_{n→∞}⟨(ρ(b_{n}) + ρ(c_{n})^{∗})h_{1}, h_{2}⟩.

This quantity is independent of the dilation, and thus U is a well-defined isometry.

Since K_{i} = \overline{σ_{i}(A)H}, it follows that U is unitary. It is also evident that U|_{H} is the identity map. So U is the desired unitary equivalence of σ_{1} and σ_{2}.

Since A always has a cyclic extremal coextension σ, it must be the unique cyclic Shilov coextension. By Lemma 4.7, σ is fully extremal.

Proof of Theorem 4.6. Let τ be any minimal extremal coextension of ρ on L ⊃ H. Set

K = \overline{τ(A)H} and σ = τ|_{K}.

Also let π be a fully extremal extension of τ. By Proposition 3.10, π is a maximal dilation of ρ. Since L is invariant for π(A) and K is invariant for τ(A), it follows that K is invariant for π(A). Hence σ is Shilov. By Lemma 4.8, σ is fully extremal. It follows that τ = σ ⊕ τ′. However τ is minimal. So

τ = σ and L = K = \overline{τ(A)H}.

Hence τ is cyclic. By Lemma 4.8, τ is unique.

Now let σ be a Shilov representation of A. Let τ be a cyclic extremal coextension of σ. By Lemma 4.8, σ and τ are equivalent coextensions of σ. Therefore τ = σ. Thus σ is extremal.

The consequences for Dirichlet algebras are apparent.

Corollary 4.9. If A is a Dirichlet operator algebra, then every Shilov extension and every Shilov coextension is fully extremal; and the minimal extremal (co-)extension of a representation is unique. Moreover the minimal maximal dilation of a representation is unique.

Proof. The first statement is immediate from Theorem 4.6 and its dual. Let π be a minimal maximal dilation of a representation ρ on a Hilbert space L. Let K = \overline{π(A)H}. This is the minimal Shilov subspace containing H. Thus by Theorem 4.6, it is the unique minimal fully extremal coextension of ρ. Let L_{0} = π(A)^{∗}K. This is the minimal Shilov extension of σ. Hence by the dual of Theorem 4.6, this coincides with the unique minimal fully extremal extension of σ. By Corollary 3.10, the restriction of π to L_{0} is a maximal dilation. Since π is minimal, L_{0} = L. So π is obtained by taking the unique minimal extremal coextension of ρ to get σ, followed by the unique minimal extremal extension of σ. So π is uniquely determined.

While semi-Dirichlet algebras behave exceptionally well for coextensions, they are not nearly so well behaved for extensions.

Example 4.10. We consider extensions for representations of the noncommutative disk algebra 𝔄_{n}. Denote the generators by 𝔰_{1}, …, 𝔰_{n}, and write 𝔰 = [𝔰_{1} … 𝔰_{n}]. A representation ρ on H is determined by a row contraction

A = ρ(𝔰) = [ρ(𝔰_{1}) … ρ(𝔰_{n})] =: [A_{1} … A_{n}],

where ∥A∥ = ∥ ∑_{i} A_{i}A_{i}^{∗}∥^{1/2} ≤ 1. We have seen that A has a unique minimal coextension to a row isometry, and this is the unique minimal fully extremal coextension.

Now consider an extension σ of ρ acting on K. This correspond to simultaneous extensions of A_{i} to

B_{i} = σ(𝔰_{i}) = [\begin{matrix}A_{i} & B_{i,12} \\ 0 & B_{i,22} \end{matrix}]

such that B = [B_{1} … B_{n}] is a row contraction. It is straightforward to verify that it is extremal if and only if B is a coisometry. We claim that: an extension σ of ρ is fully extremal if and only if B is a coisometry such that

Ran B^{∗} ∨ H^{(n)} = K^{(n)}.

Indeed, if this condition holds, then there is no proper extension of B; so consider any row contractive coextension C of B which is an extension of A_{i}. Then C_{i}^{∗} are extensions of B_{i}^{∗} which are coextensions of A_{i}^{∗}. So

C^{∗} = [\begin{matrix}B^{∗} & X \\ 0 & Y \end{matrix}] = [\begin{matrix}A^{∗} & 0 & 0 \\ B_{12}^{∗} & B_{22}^{∗} & X_{2} \\ 0 & 0 & Y \end{matrix}] .

Since B^{∗} is an isometry, we require that Ran X be orthogonal to Ran B^{∗}. And since C is an extension of A, we have Ran X is orthogonal to H^{(n)}. Therefore by hypothesis, Ran X is orthogonal to K^{(n)}, and thus X = 0. Therefore C = B⊕Y is a direct sum.

Conversely, suppose that there is a unit vector x = (x_{1}, …, x_{n})^{t} in K^{(n)} which is orthogonal to Ran B^{∗} ∨ H^{(n)}. Define an extension of B_{i}^{∗} to K ⊕ ℂ by

C_{i}^{∗} = [\begin{matrix}B_{i}^{∗} & x_{i} \\ 0 & 0 \end{matrix}] .

Since x_{i} ∈ H^{⊥}, this is a coextension of A_{i}^{∗}. So C determines an extension of A which is a coextension of B. Clearly it does not split as a direct sum. Finally, C is a coisometry because C^{∗} = [ \begin{matrix}B^{∗} & x \\ 0 & 0 \end{matrix} ] is an isometry. In particular, C is a row contraction.

Next we observe that the minimal fully extremal extensions are far from unique in general by showing how to construct a fully extremal coextension.

Start with A which is not coisometric. Then

D = (I − ∑ A_{i}A_{i}^{∗})^{1/2} ≠ 0.

Consider a fully extremal extension B as above. Then B is a coisometry on K = H ⊕ K_{0}; whence [A B_{12}] is a coisometry in B(K, H). Therefore

I_{H} = [A B_{12}] [A B_{12}]^{∗} = AA^{∗} + B_{12}B_{12}^{∗}.

Hence

B_{12}B_{12}^{∗} = D^{2},

and therefore B_{12} = DX where X = [X_{1} … X_{n}] is a coisometry in B(K_{0}^{(n)}, H). Let R = Ran X^{∗}. Then to be fully extremal, we have that B_{22}^{∗} is an isometry from K_{0} onto R^{⊥} ⊂ K_{0}^{(n)}. Now let V be any isometry in B(K_{0}^{(n)}) with Ran V = R^{⊥}. Then V^{∗}B_{22}^{∗} is a unitary in B(K_{0}, K_{0}^{(n)}). Decompose the unitary

S := B_{22}V = [S_{1} … S_{n}]

where S_{i} ∈ B(K_{0}). Observe that S_{i} are isometries such that

∑_{i} S_{i}S_{i}^{∗} = I;

in other words they are Cuntz isometries. Since

B_{22} = B_{22}V V^{∗} = SV^{∗}

in B(K_{0}^{(n)}, K_{0}), we decompose this as

B_{22} = SV^{∗} = [T_{1} … T_{n}].

We obtain

B_{i} = [\begin{matrix}A_{i} & B_{i,12} \\ 0 & B_{i,22} \end{matrix}] = [\begin{matrix}A_{i} & DX_{i} \\ 0 & T_{i} \end{matrix}] .

Conversely, if we choose any coisometry X in B(K_{0}^{(n)}, H), we may define R = Ran X^{∗}, choose an isometry V in B(K_{0}^{(n)}) with Ran V = R^{⊥}, and a set of Cuntz isometries S_{i} in B(K_{0}), then the formulae above yield a fully extremal extension. This may not be minimal in general, but the restriction to the smallest reducing subspace containing H is a minimal fully extremal extension. This restriction will not change X. So if two minimal fully extremal extensions are equivalent, then at the very least, there is a unitary U ∈ B(K_{0}) so that XU = X′. It is easy to see that there are many inequivalent choices for X even if D is rank one.

5 Commutant Lifting

Many variants of the commutant lifting theorem have been established for a wide range of operator algebras. They differ somewhat in the precise assumptions and conclusions. The general formulation in Douglas-Paulsen [26] and Muhly-Solel [40] uses Shilov modules. But we will formulate it using only fully extremal coextensions. The second definition is motivated by the lifting results of Paulsen-Power [47].

Definition 5.1. An operator algebra A has the strong commutant lifting property (SCLT) if whenever ρ is a completely contractive representation of A on H with a fully extremal coextension σ on K ⊃ H, and X commutes with ρ(A), then X has a coextension Y in B(K) with ∥Y ∥ = ∥X∥ which commutes with σ(A).

An operator algebra A has the commutant lifting property (CLT) if whenever ρ is a completely contractive representation of A on H and X commutes with ρ(A), then ρ has a fully extremal coextension σ on K ⊃ H and X has a coextension Y in B(K) with ∥Y ∥ = ∥X∥ which commutes with σ(A).

An operator algebra A has the weak commutant lifting property (WCLT) if whenever ρ is a completely contractive representation of A on H and X commutes with ρ(A), then ρ has an extremal coextension σ on K ⊃ H and X has a coextension Y in B(K) with ∥Y ∥ = ∥X∥ which commutes with σ(A).

The important distinction is that in SCLT, the coextension is prescribed first, while in CLT, it may depend on X.

It is clear that the more that we restrict the class of coextensions for which we have strong commutant lifting, the weaker the property. Thus SCLT using only fully extremal coextensions is asking for less than using all extremal coextensions, which in turn is weaker than using all Shilov extensions. As we will want a strong commutant lifting theorem, it behooves us to limit the class of extensions. On the other hand, as we limit the class of coextensions, the property CLT becomes stronger.

Observe that for the SCLT and CLT, it suffices to consider minimal fully extremal extensions. This is because any fully extremal extension decomposes as σ = σ_{0} ⊕ τ where σ_{0} is minimal. Any operator Y commuting with σ(A) will have a 1,1 entry commuting with σ_{0}(A).

Example 5.2. The disk algebra has SCLT by the Sz.Nagy–Foiaş Commutant Lifting Theorem [59]. In fact, as noted above, any isometric dilation is an extremal coextension, but not all are fully extremal. So A(𝔻) has the SCLT with respect to the larger class of all extremal coextensions, and these are all of the Shilov extensions. The reason it works is that every isometric coextension splits as σ_{0} ⊕ τ where σ_{0} is the unique minimal isometric coextension.

Example 5.3. The non-commutative disk algebra 𝔄_{n} also has SCLT by Popescu’s Commutant Lifting Theorem [50]. As noted in Example 2.2, the extremal coextensions are the row isometric ones, and these are Shilov. As in the case of the disk algebra, it is only fully extremal if it is the direct sum of the minimal isometric coextension with a row unitary.

More generally, the tensor algebra of any C*-correspondence has SCLT by the Muhly-Solel Commutant Lifting Theorem [42].

Example 5.4. The bidisk algebra A(𝔻^{2}) does not have WCLT, since commutant lifting implies the simultaneous unitary dilation of three commuting contractions [60, 44].

Example 5.5. The algebra A_{n} of continuous multipliers on symmetric Fock space (see Example 2.3) has SCLT [21]. The extremal extensions are in fact maximal dilations, and so in particular are fully extremal.

The relationship between SCLT and CLT is tied to uniqueness of minimal coextensions. We start with an easy lemma.

Lemma 5.6. Let A be a unital operator algebra. Suppose that σ is a minimal dilation on K of a representation ρ on H, in the sense that K is the smallest reducing subpace for σ(A) containing H. If X is a contraction commuting with σ(A) such that P_{H}X|_{H} = I, then X = I.

Proof. Since ∥X∥ = 1 and P_{H}X|_{H} = I, X reduces H. Note that for all h ∈ H and a ∈ A,

Xσ(a)h = σ(a)Xh = σ(a)h

and

X^{∗}σ(a)^{∗}h = σ(a)^{∗}X^{∗}h = σ(a)^{∗}h.

So the restriction of X to σ(A)H is the identity. As X is a contraction, it reduces this space. Similarly, the restriction of X^{∗} to σ(A)^{∗}H is the identity; and X reduces this space as well. Recursively we may deduce that X is the identity on the smallest reducing subspace containing H, which is K.

Theorem 5.7. Let A be a unital operator algebra. Then A has SCLT if and only if it has CLT and unique minimal fully extremal coextensions.

Proof. Assume first that A has CLT and unique minimal fully extremal coextensions. Let ρ be a representation of A on H with a fully extremal coextension σ on K ⊃ H, and suppose that X commutes with ρ(A). By CLT, there is a fully extremal coextension τ on L ⊃ H and X has a coextension Z in B(L) with ∥Z∥ = ∥X∥ which commutes with τ(A). By uniqueness of minimal fully extremal coextensions, there is a fully extremal coextension μ on K_{0} so that

σ ≃ μ ⊕ σ′ and τ ≃ μ ⊕ τ′.

With respect to the latter decomposition, Z can be written as a 2 × 2 matrix commuting with μ(a) ⊕ τ′(a) for all a ∈ A. Moreover, the corner entry Z_{11} is a coextension of X. A simple calculation of the commutator shows that μ(a) commutes with Z_{11}. Thus Y ≃ Z_{11} ⊕ 0 is the desired coextension of X commuting with σ(A).

Conversely, suppose that A has SCLT. A fortiori, it will have CLT. Suppose that a representation ρ on H has two minimal fully extremal coextensions σ_{1} and σ_{2} on K_{1} = H ⊕ K′_{1} and K_{2} = H ⊕ K′_{2}, respectively. Then ρ ⊕ ρ has σ_{1} ⊕ σ_{2} as a fully extremal coextension. This can be seen, for example, because of the identities (1) and (2) in the proof of Theorem 3.5. The operator X = [ \begin{matrix}0 & I \\ I & 0 \end{matrix} ] commutes with (ρ ⊕ ρ)(A). So by SCLT, X has a coextension Y on K_{1} ⊕ K_{2} of norm 1 which commutes with (σ_{1} ⊕σ_{2})(A). Since P_{H⊕H}Y |_{H⊕H} = X is unitary, Y reduces H ⊕ H.

Now Y^{2} commutes with σ_{1} ⊕ σ_{2}(A) and its restriction to H ⊕ H is X^{2} = I. Thus by Lemma 5.6, Y^{2} = I. In particular, Y is unitary. Let

Y_{12} = P_{K_{1}}Y |_{K_{2}} and Y_{21} = P_{K_{2}}Y |_{K_{1}}.

Observe that for a ∈ A,

Y_{21}σ_{1}(a) = σ_{2}(a)Y_{21} and Y_{12}σ_{2}(a) = σ_{1}(a)Y_{12}.

Moreover the restriction of Y_{21} to H is X restricted to H ⊕ {0}, which is the identity map if we identify H ⊕ {0} and {0} ⊕ H with H. We deduce that Y_{12}Y_{21} commutes with σ_{1}(A) and coincides with I on H. So by Lemma 5.6,

Y_{12}Y_{21} = I.

Similarly,

Y_{21}Y_{12} = I.

Since they are contractions, Y_{12} is unitary and Y_{21} = Y_{12}^{∗}. The identities above now show that Y_{21} implements a unitary equivalence between σ_{1} and σ_{2} fixing H. Hence the minimal fully extremal coextension of ρ is unique.

We can weaken CLT to WCLT if we strengthen the uniqueness hypothesis to minimal extremal coextensions. This seems a fair bit stronger in comparison however.

Corollary 5.8. If A has WCLT and unique minimal extremal coextensions, then A has SCLT.

Proof. If ρ is a representation of A, then the unique minimal extremal coextension σ of ρ must be fully extremal, since by Theorem 3.5, there are fully extremal coextensions and hence there are minimal ones. These are also minimal as extremal coextensions. Thus A has unique minimal fully extremal coextensions. Moreover, as in the proof above, if a contraction X commutes with ρ(A), then WCLT provides a coextension to a contraction Y commuting with an extremal coextension τ. But τ = σ ⊕ τ′. So arguing as before, the compression Z of Y to K_{σ} commutes with σ(A) and is a coextension of X. Now if φ is an arbitrary extremal coextension of ρ, again split φ = σ ⊕ φ′. One extends Z to Z ⊕ 0 to commute with φ(A).

It is common to look for a version of commutant lifting for intertwining maps between two representations. In the case of WCLT and SCLT, this is straightforward. Such a version for CLT is valid here too, but some care must be taken.

Proposition 5.9. Suppose that A has SCLT. Let ρ_{i} be representations of A on H_{i} for i = 1, 2 with fully extremal coextensions σ_{i} on K_{i}. Suppose that X is a contraction in B(H_{2}, H_{1}) such that

ρ_{1}(a)X = Xρ_{2}(a) for all a ∈ A.

Then there is a contraction Y in B(K_{2}, K_{1}) so that

P_{H_{1}}Y = XP_{H_{2}}

and

σ_{1}(a)Y = Y σ_{2}(a) for all a ∈ A.

Proof. Let ρ = ρ_{1} ⊕ ρ_{2}. By Corollary 3.9, σ = σ_{1} ⊕ σ_{2} is a fully extremal coextension of ρ. Observe that

X̃ = [\begin{matrix}0 & X \\ 0 & 0 \end{matrix}]

commutes with ρ(A). Hence by SCLT, there is a coextension Ỹ of X̃ which commutes with σ(A).

Write Y as a matrix with respect to

K = K_{1} ⊕ K_{2} = H_{1} ⊕ H_{2} ⊕ (K_{1} ⊖ H_{1}) ⊕ (K_{2} ⊖ H_{2})

and rearrange this to the decomposition

K = H_{1} ⊕ (K_{1} ⊖ H_{1}) ⊕ H_{2} ⊕ (K_{2} ⊖ H_{2}).

We obtain the unitary equivalence

Ỹ = [\begin{matrix}X̃ & 0 \\ ∗ & ∗ \end{matrix}] = [ \begin{matrix}0 & X & 0 & 0 \\ 0 & 0 & 0 & 0 \\ ∗ & ∗ & ∗ & ∗ \\ ∗ & ∗ & ∗ & ∗ \end{matrix} ] ≃ [ \begin{matrix}0 & 0 & X & 0 \\ ∗ & ∗ & ∗ & ∗ \\ 0 & 0 & 0 & 0 \\ ∗ & ∗ & ∗ & ∗ \end{matrix} ]

Restricting to the upper right 2 × 2 corner, we obtain

Y := P_{K_{1}} Ỹ |_{K_{2}} = [\begin{matrix}X & 0 \\ ∗ & ∗ \end{matrix}] .

Then Y is a contraction, and as an operator in B(K_{2}, H_{1}) we have

P_{H_{1}}Y = [X 0] = XP_{H_{2}}.

Finally the commutation relations show that

σ_{1}(a)Y = Y σ_{2}(a) for all a ∈ A.

A similar argument shows the following:

Proposition 5.10. Suppose that A has WCLT. Let ρ_{i} be representations of A on H_{i} for i = 1, 2. Suppose that X is a contraction in B(H_{2}, H_{1}) such that

ρ_{1}(a)X = Xρ_{2}(a) for all a ∈ A.

Then there are extremal coextensions σ_{i} of ρ_{i} acting on K_{i} ⊃ H_{i} for i = 1, 2 and a contraction Y in B(K_{2}, K_{1}) so that

P_{H_{1}}Y = XP_{H_{2}}

and

σ_{1}(a)Y = Y σ_{2}(a) for all a ∈ A.

Proof. Again form ρ = ρ_{1} ⊕ ρ_{2} and X̃ as above. Use WCLT to coextend ρ to an extremal coextension σ and X̃ to a contraction Ỹ commuting with σ(A) on K = H_{1} ⊕ H_{2} ⊕ K′. Now notice that σ is an extremal coextension of both ρ_{i}. Considering Ỹ as a map from

H_{1} ⊕ (H_{2} ⊕ K′) to H_{2} ⊕ (H_{1} ⊕ K′),

one finds that it has a matrix form

Ỹ = [ \begin{matrix}X & 0 & 0 \\ 0 & 0 & 0 \\ ∗ & ∗ & ∗ \end{matrix} ] .

This has the desired form.

Remark 5.11. The issue with CLT and fully extremal coextensions is that a fully extremal coextension of ρ = ρ_{1} ⊕ρ_{2} need not even contain a fully extremal coextension of ρ_{i} as a summand. Consider the subalgebra A ⊂ 𝔐_{5} given by

A = span{E_{21}, E_{32}, E_{31}, E_{34}, E_{45}, E_{35}, E_{ii} : 1 ≤ i ≤ 5}.

Let

ρ_{1}(a) = E_{22}a|_{ℂe_{2}} and ρ_{2}(a) = E_{44}a|_{ℂe_{4}} .

The minimal fully extremal coextensions of ρ_{i} are

σ_{1}(a) = E_{11}^{⊥}a|_{(ℂe_{1})^{⊥}} and σ_{2}(a) = E_{55}^{⊥}a|_{(ℂe_{5})^{⊥}} .

However the minimal fully extremal coextension of ρ_{1} ⊕ ρ_{2} is

σ(a) = (E_{22} + E_{33} + E_{44})a|_{span{e_{2},e_{3},e_{4}}}.

Thus a proof of the following result must follow different lines. This proof has its roots in the work of Sz.Nagy and Foiaş. Notice that it allows a specification of one of the coextensions. Normally we will use a fully extremal coextension σ_{2} of ρ_{2}.

Theorem 5.12. Suppose that A has CLT. Let ρ_{i} be representations of A on H_{i} for i = 1, 2. Suppose that X is a contraction in B(H_{2}, H_{1}) such that

ρ_{1}(a)X = Xρ_{2}(a) for all a ∈ A.

Let σ_{2} be an extremal coextension of ρ_{2} on K_{2}. Then there is a fully extremal coextension σ_{1} of ρ_{1} acting on K_{1} and a contraction Y in B(K_{2}, K_{1}) so that

P_{H_{1}}Y = XP_{H_{2}}

and

σ_{1}(a)Y = Y σ_{2}(a) for all a ∈ A.

Proof. Let K_{2} = H_{2} ⊕ K′_{2}, and decompose

σ_{2}(a) = [\begin{matrix}ρ_{2}(a) & 0 \\ σ_{21}(a) & σ_{22}(a) \end{matrix}]

Observe that [X 0] ∈ B(K_{2}, H_{1}) satisfies

ρ_{1}(a)[X 0] = [ρ_{1}(a)X 0] = [X 0] [\begin{matrix}ρ_{2}(a) & 0 \\ σ_{21}(a) & σ_{22}(a) \end{matrix}] .

Therefore

X̃ = [ \begin{matrix}0 & X & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix} ]

commutes with the range of ρ = ρ_{1} ⊕ σ_{2}.

Now we apply the CLT property to ρ and X̃ to obtain a fully extremal coextension τ of ρ and contraction Ỹ coextending X̃ on L = H_{1} ⊕ K_{2} ⊕ L′ which commute. We may write

τ(a) = [\begin{matrix}ρ_{1}(a) & 0 & 0 \\ 0 & σ_{2}(a) & 0 \\ τ_{31}(a) & τ_{32}(a) & τ_{33}(a) \end{matrix}] .

Observe that the lower right 2 × 2 corner is a coextension of σ_{2}. Since σ_{2} is extremal, we see that τ_{32} = 0. Define

σ_{1}(a) = [\begin{matrix}ρ_{1}(a) & 0 \\ τ_{31}(a) & τ_{33}(a) \end{matrix}] .

To complete the proof, we need to establish that σ_{1} is a fully extremal coextension of ρ_{1}. Suppose that γ is a representation of A which dilates σ_{1} and coextends ρ_{1}. Then γ ⊕ σ_{2} dilates σ_{1} ⊕ σ_{2} = τ and coextends ρ_{1} ⊕ σ_{2}. Since τ is fully extremal, γ = σ_{1} ⊕ γ′ as desired.

We make a few more definitions. (Apologies for all the acronyms.)

Definition 5.13. If A^{∗} has SCLT, CLT or WCLT, we say that A has SCLT*, CLT* or WCLT*.

Say that A has maximal commutant lifting (MCLT) if for every representation ρ on H and contraction X commuting with ρ(A), there is a maximal dilation π of ρ on a Hilbert space K ⊃ H and a contraction Y commuting with π(A) such that

P_{H}π(a)Y^{n}|_{H} = ρ(a)X^{n} for all a ∈ A and n ≥ 0.

If the maximal dilation π can be specified a priori, then say that A has strong maximal commutant lifting (SMCLT).

It is clear that the commutant lifting properties for A^{∗} can be interpreted as lifting commutants to (fully) extremal extensions instead of coextensions. On rare occasions, one gets both. For example, the disk algebra A(𝔻) is completely isometrically isomorphic to its adjoint algebra. Hence it has both SCLT and SCLT*.

The property MCLT for A is equivalent to MCLT for A^{∗}. So we will not have a property MCLT*.

The definition of MCLT contains the information that the compression of the algebra generated by π(A) and Y to H is an algebra homomorphisms which sends π to ρ and Y to X. It follows that H is semi-invariant for this algebra; i.e. H is the difference of two subspaces which are invariant for both π(A) and Y.

Theorem 5.14. If A has WCLT and WCLT*, then A has MCLT.

Proof. One uses WCLT to coextend ρ on H to an extremal coextension σ_{1} on K_{1} and coextend X to a contraction Y_{1} ∈ B(K_{1}) commuting with σ_{1}(A). Then use WCLT* extend σ_{1} to an extremal extension τ_{1} on L_{1}, and lift Y_{1} to a contraction Z_{1} ∈ B(L_{1}) commuting with τ_{1}(A). Alternate these procedures, obtaining an extremal coextension σ_{n+1} of τ_{n} on K_{n+1} and a contractive coextension Y_{n+1} ∈ B(K_{n+1}) in the commutant of σ_{n+1}(A); and then extending σ_{n+1} to an extremal τ_{n+1} on L_{n+1} and extending Y_{n+1} to a contraction Z_{n+1} ∈ B(L_{n+1}) in the commutant of τ_{n+1}(A). It is easy to see that at every stage, the original space H is semi-invariant for both the representation and the contraction—so that these are always simultaneous dilations of the representation and the contraction.

Moreover, we can write σ_{n+1} as a dilation of σ_{n} in the matrix form relative to

K_{n+1} = (L_{n} ⊖ K_{n}) ⊕ K_{n} ⊕ (K_{n+1} ⊖ L_{n})

as

σ_{n+1} = [\begin{matrix}∗ & 0 & 0 \\ ∗ & σ_{n} & 0 \\ ∗ & ∗ & ∗ \end{matrix}]

where the upper left 2 × 2 corner represents τ_{n}. The lower right 2 × 2 corner is a coextension of σ_{n}. Since σ_{n} is an extremal coextension, the 3, 2 entry is 0. Rearranging this as

K_{n+1} = K_{n} ⊕ (L_{n} ⊖ K_{n}) ⊕ (K_{n+1} ⊖ L_{n}),

we have

σ_{n+1} = [\begin{matrix}σ_{n} & ∗ & 0 \\ 0 & ∗ & 0 \\ 0 & ∗ & ∗ \end{matrix}] .

A similar analysis holds for the τ_{n}. Therefore, with respect to

K_{1} ⊕ (L_{1} ⊖ K_{1}) ⊕ (K_{2} ⊖ L_{1}) ⊕ (L_{2} ⊖ K_{2}) ⊕ …,

these representations have a tridiagonal form

[\begin{matrix}σ_{1} & ∗ & 0 & 0 & 0 & 0 & 0 & … \\ 0 & ∗ & 0 & 0 & 0 & 0 & 0 & … \\ 0 & ∗ & ∗ & ∗ & 0 & 0 & 0 & … \\ 0 & 0 & 0 & ∗ & 0 & 0 & 0 & … \\ 0 & 0 & 0 & ∗ & ∗ & ∗ & 0 & … \\ 0 & 0 & 0 & 0 & 0 & ∗ & 0 & … \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ... & ... & ... \end{matrix}] .

It follows that the direct limit π exists as a sot-∗ limit. Moreover, as π is a limit of extremal coextensions, it is an extremal coextension; and similarly it is an extremal extension. Thus it is a maximal dilation.

The operators Y_{n} and Z_{n} each leave the subspaces H semi-invariant, and the restriction of Z_{n} to K_{n} is Y_{n}, and the compression (actually a co-restriction) of Y_{n+1} to L_{n} is Z_{n}. Therefore the direct limit Y exists as a wot limit. It follows that Y is a contraction that commutes with π(A). To see this, let x_{n} ∈ K_{n} and y_{n} ∈ L_{n}. Then for a ∈ A,

π(a)x_{n} = σ_{k}(a)x_{n} for all k ≥ n.

Similarly,

π(a)^{∗}y_{n} = σ_{k}(a)^{∗}y_{n} for all k ≥ n + 1.

So we can compute:

⟨(π(a)Y − Y π(a))x_{n}, y_{n}⟩ = ⟨x_{n}, Y^{∗}π(a)^{∗}y_{n}⟩ − ⟨Y π(a)x_{n}, y_{n}⟩

= lim_{k→∞} ⟨x_{n}, Y_{k}^{∗}π(a)^{∗}y_{n}⟩ − ⟨Y_{k}π(a)x_{n}, y_{n}⟩

= lim_{k→∞} ⟨x_{n}, Y_{k}^{∗}σ_{k}(a)^{∗}y_{n}⟩ − ⟨Y_{k}σ_{k}(a)x_{n}, y_{n}⟩

= lim_{k→∞} ⟨(σ_{k}(a)Y_{k} − Y_{k}σ_{k}(a))x_{n}, y_{n}⟩ = 0.

So we have obtained the desired commutant lifting.

We can modify the proof of Theorem 5.7 characterizing SCLT to characterize SMCLT.

Theorem 5.15. The following are equivalent for A:

(i) A has SMCLT. (ii) A has MCLT and unique minimal maximal dilations.

Proof. Suppose that A has SMCLT. Then evidently it has MCLT. Moreover, suppose that a representation ρ has two minimal maximal dilations π_{1} and π_{2}. Then π_{1} ⊕ π_{2} is a maximal dilation of ρ ⊕ ρ. Now (ρ ⊕ ρ)(A) commutes with X = [ \begin{matrix}0 & I \\ I & 0 \end{matrix} ]. By SMCLT, this dilates to a contraction Y which commutes with (π_{1} ⊕ π_{2})(A) and H ⊕ H is jointly semi-invariant for Y and (π_{1} ⊕ π_{2})(A). Arguing exactly as in the proof of Theorem 5.7, we deduce that π_{1} and π_{2} are unitarily equivalent via a unitary which fixes H. So A has unique maximal dilations.

Conversely, it is routine to see that unique maximal dilations and MCLT yields SMCLT. So (i) and (ii) are equivalent.

Remark 5.16. One might suspect, as we did, that SMCLT is also equivalent to the following:

(iii) A has SCLT and SCLT*. (iv) A has MCLT and unique minimal fully extremal extensions and coexten-

sions.

But this is not the case.

If (ii) holds, then by Theorem 3.16 there is uniqueness of minimal fully extremal coextensions; and Corollary 3.17 yields uniqueness of minimal fully extremal extensions. So (iv) holds.

Also if (iii) holds, then there are unique minimal fully extremal extensions and coextensions by Theorem 5.7 and its dual result for SCLT*. Also MCLT holds by Theorem 5.14. So (iv) holds.

However, it is possible that SCLT and SCLT* hold, yet A does not have unique minimal maximal dilations. See the example of 2 × 2 matrices developed in the next section. Thus SMCLT fails to hold.

We do not know if SMCLT implies (iii).

In the case of semi-Dirichlet algebras, we have something extra. We do not know if all semi-Dirichlet algebras have SCLT. However, Muhly and Solel [42] show that the tensor algebra over any C*-correspondence has SCLT.

Proposition 5.17. Suppose that A is semi-Dirichlet and has MCLT. Then A has SCLT.

Proof. Let ρ be a representation of A on H commuting with a contraction X. Use MCLT to obtain a simultaneous dilation of ρ to a maximal dilation π and X to a contraction Y commuting with π(A). Let K be the common invariant subspace for π(A) and Y containing H. Since H is semi-invariant, the restriction of π to K is a coextension σ of ρ. The compression Z of Y to K is a contraction commuting with σ(A).

By Theorem 4.6, there is a unique minimal extremal coextension σ_{0} of ρ, and it must coincide with σ|_{K_{0}} where K_{0} = σ(A)H. Thus σ = σ_{0} ⊕ σ′. It follows that the compression Z_{0} of Z to K_{0} commutes with σ_{0}(A). Moreover, since H is coinvariant for Z, it is also coinvariant for Z_{0}. By Theorem 5.7, A has SCLT.

Corollary 5.18. If A is a Dirichlet algebra, the following are equivalent:

(i) A has MCLT (ii) A has SMCLT (iii) A has SCLT and SCLT* (iv) A has WCLT and WCLT*.

Finally we point out that there is also an intertwining version for MCLT.

Proposition 5.19. Suppose that A has MCLT. Let ρ_{i} be representations of A on H_{i} for i = 1, 2. Suppose that X is a contraction in B(H_{2}, H_{1}) such that

ρ_{1}(a)X = Xρ_{2}(a) for all a ∈ A.

Then there are maximal dilations π_{i} of ρ_{i} acting on K_{i} ⊃ H_{i} for i = 1, 2 and a contraction Y simultaneously dilating X in B(K_{2}, K_{1}) so that

π_{1}(a)Y = Y π_{2}(a) for all a ∈ A.

Proof. Again form ρ = ρ_{1} ⊕ ρ_{2} and X̃ as before. Use MCLT to dilate ρ to a maximal dilation π and X̃ to a contraction Ỹ commuting with π(A) on

K = K_{−} ⊕ H_{1} ⊕ H_{2} ⊕ K_{+}.

Write

Ỹ = [\begin{matrix}∗ & 0 & 0 & 0 \\ ∗ & 0 & X & 0 \\ ∗ & 0 & 0 & 0 \\ ∗ & ∗ & ∗ & ∗ \end{matrix}] .

Now notice that π is an maximal dilation of both ρ_{i}. Considering Ỹ as a map from

K_{−} ⊕ H_{1} ⊕ (H_{2} ⊕ K_{+}) to (K_{−} ⊕ H_{1}) ⊕ H_{2} ⊕ K_{+}

one finds that it has a matrix form

Ỹ = [ \begin{matrix}∗ & 0 & 0 & 0 \\ ∗ & 0 & X & 0 \\ ∗ & 0 & 0 & 0 \\ ∗ & ∗ & ∗ & ∗ \end{matrix} ] .

This is a dilation of X which commutes with π(A), where π is considered as a dilation of both ρ_{1} and ρ_{2} using the two decompositions of K.

6 A 2 × 2 matrix example.

Consider the algebra A = span{I_{2}, n} ⊂ 𝔐_{2} where

n = [\begin{matrix}0 & 0 \\ 1 & 0 \end{matrix}] .

Observe that A is unitarily equivalent to A^{∗}. It is not semi-Dirichlet. We will show that it has unique minimal extremal (co)extensions, which are always maximal dilations. But it does not have unique maximal dilations. It also has SCLT, SCLT* and MCLT, but does not have SMCLT.

Observe that a representation ρ of A is determined by N := ρ(n), which satisfies N^{2} = 0 and ∥N∥ ≤ 1. Conversely, any such N yields a completely contractive representation. It is easy to check that N is unitarily equivalent to an operator of the form

[\begin{matrix}0 & 0 \\ B & 0 \end{matrix}]

where B has dense range by setting H_{2} = \overline{Ran N} and decomposing H = H_{1} ⊕ H_{2}. This can be refined by using the polar decomposition of B to the form

[\begin{matrix}0 & 0 \\ A & 0 \end{matrix}] ⊕ 0 on H = Ran N^{∗} ⊕ Ran N ⊕ (ker N ∩ ker N^{∗}),

where A is a positive injective operator.

Clearly C_{e}^{∗}(A) = 𝔐_{2}. Thus a maximal representation π extends to a ∗- representation of 𝔐_{2}. Hence π(n) = N is a partial isometry such that

NN^{∗} + N^{∗}N = I and N^{2} = 0;

or equivalently,

(N + N^{∗})^{2} = I and N^{2} = 0.

In other words, there is a unitary U so that

N ≃ [\begin{matrix}0 & 0 \\ U & 0 \end{matrix}] ≃ [\begin{matrix}0 & 0 \\ I & 0 \end{matrix}] .

Geometrically, this says that N is a partial isometry such that

Ran N + Ran N^{∗} = H.

Proposition 6.1. The algebra A of 2 × 2 matrices of the form

[\begin{matrix}a & 0 \\ b & a \end{matrix}]

has unique minimal extremal coextensions (extensions). Moreover they are fully extremal coextensions (extensions), and in fact, are maximal dilations; and the original space is cyclic (cocyclic).

Proof. We first show that a representation ρ which is not maximal has a proper coextension. Use the matrix form

ρ(n) = N = [\begin{matrix}0 & 0 \\ B & 0 \end{matrix}]

on H = H_{1} ⊕ H_{2} where B has dense range. Suppose that B is not an isometry, and let

D_{B} = (I − B^{∗}B)^{1/2} and D_{B} = \overline{Ran D_{B}}.

Consider P_{D_{B}}D_{B} as an operator from H_{1} to D_{B}. Then B may be coextended to an isometry

V = [\begin{matrix}B \\ P_{D_{B}}D_{B} \end{matrix}]

mapping H_{1} to K_{2} = H_{2} ⊕ D_{B}. So ρ coextends to σ where

σ(n) = S = [\begin{matrix}0 & 0 \\ V & 0 \end{matrix}] = [\begin{matrix}0 & 0 & 0 \\ B & 0 & 0 \\ P_{D_{B}}D_{B} & 0 & 0 \end{matrix}]

on

K = H_{1} ⊕ K_{2} = H_{1} ⊕ H_{2} ⊕ D_{B}.

Hence when ρ is extremal, B is an isometry with dense range, so it is unitary. Thus ρ is a maximal representation.

The coextension constructed above is generally not extremal because V is not unitary. So one can coextend S again using the same procedure. K_{2} splits as R_{V} ⊕ D, where

R_{V} = Ran V and D = Ran(I_{K_{2}} − V V^{∗}).

We now decompose

K = H_{1} ⊕ D ⊕ R_{V} ,

and write

S = [ \begin{matrix}0 & 0 & 0 \\ 0 & 0 & 0 \\ P_{R_{V}} V & 0 & 0 \end{matrix} ]

Using the same dilation as the previous paragraph, but noting that

D_{[P_{R_{V}} V 0]} = (I_{K_{2}} − V V^{∗}),

we obtain the coextension τ of σ on a Hilbert space

L = H_{1} ⊕ D ⊕ R_{V} ⊕ D

by

τ(n) = W = [ \begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ P_{R_{V}} V & 0 & 0 & 0 \\ 0 & P_{D}(I_{K_{2}} − V V^{∗}) & 0 & 0 \end{matrix} ]

Documenta Mathematica 16 (2011) 781–868

The bottom left 2 × 2 corner is now a surjective isometry. So this is a maximal dilation.

To see that τ is minimal as a coextension, we need to verify that

L = AH = H ∨ W H.

To see this, we rewrite W with respect to the decomposition

L = H_{1} ⊕ H_{2} ⊕ D_{B} ⊕ D

to get

W = [ \begin{matrix}0 & 0 & 0 & 0 \\ B & 0 & 0 & 0 \\ P_{D_{B}}D_{B} & 0 & 0 & 0 \\ 0 & P_{D}(I_{K_{2}} − V V^{∗}) & 0 \end{matrix} ]

Since the range of D_{B} is dense in D_{B}, this is contained in H ∨ W H. For the space D, we expand the expression for I_{K_{2}} − V V^{∗} on K_{2} = H_{2} ⊕ D_{B}:

I_{K_{2}} − V V^{∗} = [\begin{matrix}I & 0 \\ 0 & I \end{matrix}] − [\begin{matrix}BB^{∗} & BD_{B} \\ D_{B}B^{∗} & I − B^{∗}B \end{matrix}]

= [\begin{matrix}I − BB^{∗} & −BD_{B} \\ −D_{B}B^{∗} & B^{∗}B \end{matrix}]

= [\begin{matrix}D_{B^{∗}}^{2} & −BD_{B} \\ −BD_{B^{∗}} & B^{∗}B \end{matrix}]

Thus one sees that

(I_{K_{2}} − V V^{∗})P_{H_{2}} = [ \begin{matrix}D_{B^{∗}} \\ −B \end{matrix}] D_{B^{∗}}.

Observe that D_{B^{∗}} maps H_{2} onto a dense subspace of ker(I − BB^{∗})^{⊥}, and

ker(I − BB^{∗}) = Ran V ∩ H_{2} ⊂ ker(I − V V^{∗}) ∩ H_{2}.

The range of

[ \begin{matrix}D_{B^{∗}} \\ −B \end{matrix}]

is easily seen to be the orthogonal complement of Ran V, so this is D. Restricting this map to the range of D_{B^{∗}} does not affect the closed range, since we only miss some of the kernel. Thus W H_{2} is dense in D. Therefore this is a minimal extremal coextension.

Now we consider uniqueness. To this end, suppose that τ′ is a minimal extremal coextension of ρ on

L′ = H_{1} ⊕ H_{2} ⊕ L_{3}.

Then we can write

τ′(n) = W′ = [\begin{matrix}0 & 0 & 0 \\ B & 0 & 0 \\ X & Y & Z \end{matrix}] .

This is a partial isometry satisfying

W^{′2} = 0 and W′W^{′∗} + W^{′∗}W′ = I.

In particular, H_{1} is orthogonal to the range of W′, so the restriction of W′ to H_{1} is an isometry. Therefore

X = UP_{D_{B}}D_{B}

where U is an isometry of D_{B} into L_{3}. Split

L_{3} = UD_{B} ⊕ L_{4} ≃ D_{B} ⊕ L_{4}.

By identifying the range of U with D_{B}, we have the refined form

W′ = [\begin{matrix}0 & 0 & 0 & 0 \\ B & 0 & 0 & 0 \\ P_{D_{B}}D_{B} & X_{1} & Y_{1} & Z_{1} \\ 0 & X_{2} & Y_{2} & Z_{2} \end{matrix}] .

The range of W′H_{1}^{⊥} is orthogonal to

W′H_{1} ∨ H_{2} = H_{2} ⊕ D_{B}.

So

X_{1} = Y_{1} = Z_{1} = 0.

Next note that minimality ensures that X_{2} has dense range in L_{4}. So L_{4} is in

Ran W′ = ker W^{′∗}.

Hence Z_{2} = 0. Observe that

Ran W′ = Ran V ⊕ L_{4},

and hence

Ran W^{′∗} = (Ran W′)^{⊥} = H_{1} ⊕ D.

Moreover, the operator

[X_{2} Y_{2}]

is an isometry of D onto L_{4}. Hence we may identify L_{4} with D in such a way that we obtain

[X_{2} Y_{2}] ≃ P_{D}(I_{K_{2}} − V V^{∗}).

This shows that W′ is unitarily equivalent to W via a unitary which fixes H. Therefore the minimal extremal coextension is unique.

The proof for extensions follows immediately since A^{∗} is unitarily equivalent to A.

Corollary 6.2. Every representation of the algebra A is Shilov.

Proof. By the previous theorem, one can extend ρ to a maximal dilation π. Thus ρ is obtained as the restriction of a maximal representation to an invariant subspace; i.e. it is Shilov.

Example 6.3. Take ρ to be the character representation on H = ℂ given by

ρ(aI_{2} + bn) = a.

This coextends to a maximal representation on K = H ⊕ ℂ as

σ(aI_{2} + bn) = [\begin{matrix}a & 0 \\ b & a \end{matrix}] .

It also extends to a maximal representation τ on K = ℂ ⊕ H where

τ(aI_{2} + bn) = [ \begin{matrix}a & 0 \\ b & a \end{matrix}] .

Note that σ and τ are not unitarily equivalent by a unitary which fixes H. So these are inequivalent minimal maximal dilations.

Corollary 6.4. The minimal maximal dilation of a representation ρ of A is not unique except when ρ is already maximal.

Proof. Let τ be the minimal extremal coextension of ρ, and let σ be the minimal extremal extension of ρ. Then in the first case, H is identified with a coinvariant subspace and in the latter with an invariant subspace. If these two dilations are unitarily equivalent via a unitary which fixes H, then H is reducing, and therefore by minimality, ρ = τ is maximal.

Next we show that A has commutant lifting.

Theorem 6.5. The algebra A of 2 × 2 matrices of the form

[\begin{matrix}a & 0 \\ b & a \end{matrix}]

has SCLT, SCLT* and MCLT, but not SMCLT.

Proof. It is enough to verify CLT. Since the minimal fully extremal coextensions are unique, it then has SCLT by Theorem 5.7. Since A^{∗} ≃ A, it has SCLT* as well. Thus by Theorem 5.14, it has MCLT. But by Theorem 5.15, it does not have SMCLT.

We make use of the construction of the minimal extremal extension in the proof of Theorem 6.1. Write

ρ(n) = N = [\begin{matrix}0 & 0 \\ B & 0 \end{matrix}]

as before on

H = H_{1} ⊕ H_{2} where H_{2} = \overline{Ran N}.

Suppose that it commutes with a contraction T. Then it is routine to check that

T = [\begin{matrix}X & 0 \\ Y & Z \end{matrix}]

such that

BX = ZB.

Coextend ρ to the coextension σ on

K = H_{1} ⊕ H_{2} ⊕ D_{B}

where

σ(n) = S = [\begin{matrix}0 & 0 & 0 \\ B & 0 & 0 \\ D_{B} & 0 & 0 \end{matrix}] .

We first find a coextension of T to T̃ which commutes with S and has norm one.

Consider the isometric dilation of N. Observe that

D_{N} = (I − N^{∗}N)^{1/2} = [\begin{matrix}D_{B} & 0 \\ 0 & I \end{matrix}] .

So the minimal isometric dilation acts on

(H_{1} ⊕ H_{2}) ⊕ (D_{B} ⊕ H_{2})^{(∞)},

and has the form

U = [ \begin{matrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & … \\ B & 0 & 0 & 0 & 0 & 0 & 0 & 0 & … \\ D_{B} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & … \\ 0 & I & 0 & 0 & 0 & 0 & 0 & 0 & … \\ 0 & 0 & I & 0 & 0 & 0 & 0 & 0 & … \\ 0 & 0 & 0 & I & 0 & 0 & 0 & 0 & … \\ ⋮ & ⋮ & ⋮ & ... & ... & ... & ... & ... & ... \end{matrix} ]

Notice that S is the upper left 3 × 3 corner. By the Sz.Nagy-Foiaş Commutant Lifting Theorem, we can coextend T to a contraction R commuting with U. It has the form

R = [\begin{matrix}X & 0 & 0 & … \\ Y & Z & 0 & … \\ C_{1} & C_{2} & C_{3} & … \\ ⋮ & ⋮ & ⋮ & ... \end{matrix}]

It is routine to verify that

T̃ = [\begin{matrix}X & 0 & 0 \\ Y & Z & 0 \\ C_{1} & C_{2} & C_{3} \end{matrix}]

commutes with S.

Now we repeat the argument with S and T̃. The same procedure was shown in Theorem 6.1 to yield the minimal (fully) extremal coextension τ of ρ. The operator T̃ is coextended once again to obtain a contraction commuting with τ(n) = W. This establishes SCLT.

Remark 6.6. We will show that in the commutant lifting theorem for A, it is not possible to coextend so that T̃ is an isometry. In the language of the next section, this will show that A does not have ICLT (isometric commutant lifting) nor the Ando property.

To see this, consider the identity representation

id(n) = N = [\begin{matrix}0 & 0 \\ 1 & 0 \end{matrix}]

on H = ℂ^{2}. Then id(A) commutes with T = N. Suppose that there were a coextension of id and T to σ and an isometry V on K so that σ(A) commutes with V. Since id is maximal, σ = id ⊕ τ. So

σ(n) = M = N ⊕ M_{0}

where M_{0}^{2} = 0. Let the canonical basis for H be e_{1}, e_{2}. Since T e_{2} = 0, we have V e_{2} = v is a unit vector in H^{⊥}; while

V e_{1} = T e_{1} = e_{2}.

Therefore

v = V e_{2} = V Me_{1} = MV e_{1} = Me_{2} = 0.

This contradiction shows that no such coextension is possible.

7 Isometric Commutant Lifting and Ando’s Theorem

Paulsen and Power [47] formulate commutant lifting and Ando’s theorem in terms of tensor products. In doing so, they are also able to discuss lifting commuting relations between two arbitrary operator algebras. They are interested in dilations which extend to the enveloping C*-algebra, which are the maximal dilations when this C*-algebra is the C*-envelope. The Paulsen-Power version of Ando’s theorem involves maximal dilations and a commuting unitary. The classical Ando Theorem, from our viewpoint, states that two commuting contractions have coextensions to commuting isometries. We will give a similar definition using extremal co-extensions instead which is actually stronger than the Paulsen-Power version.

Definition 7.1. We say that A has isometric commutant lifting (ICLT) if whenever ρ is a representation of A on H commuting with a contraction X, then there is a coextension σ of ρ and an isometric coextension V of X on a common Hilbert space K so that σ(A) and V commute.

Remark 7.2. We can see ICLT as a commutant lifting result for A(𝔻) relative to A. More generally, we can formulate a relative commutant lifting property for two operator algebras. We define this here, but will not pursue it except in the case above. Let A and B be unital operator algebras. Say that A has B-CLT (or commutant lifting with respect to B) if whenever α and β are (completely contractive) representations of A and B on a common Hilbert space H which commute:

α(a)β(b) = β(b)α(a) for all a ∈ A and b ∈ B,

then there exists an extremal coextension σ of α on a Hilbert space K and a coextension τ of β on K which commute.

Because there is no uniqueness for the classical Ando’s theorem, we do not seek a strong form. So we do not use the adjective weak either. Note that ICLT is not stronger than WCLT because the extremal condition is on the isometry, not on the coextension of A. Nevertheless it does imply a much stronger conclusion, as we show in Theorem 7.3 below.

It is often observed that Ando’s Theorem is equivalent to commutant lifting. However neither direction is completely trivial. From Ando’s theorem, one easily gets WCLT. So the uniqueness of the minimal isometric coextension, and the fact that this is fully extremal, then yields SCLT. Conversely, in deducing Ando’s theorem from WCLT, one is really using WCLT for both contractions. One iteratively dilates one contraction to an isometry and lifts the other to commute. The inductive limit is a pair of commuting isometries. We will see this more clearly for operator algebras other than the disk algebra.

The next result shows that the Paulsen-Power version of Ando’s theorem is equivalent to ICLT. It also shows why we consider A-CLT for A(𝔻) as a strong property for A, and makes it worthy of the term ICLT.

Theorem 7.3. For a unital operator algebra A, the following are equivalent:

(i) A had ICLT; i.e. if ρ is a representation of A on H commuting with a contraction X, then there is a coextension σ of ρ and an isometric coextension V of X on a common Hilbert space K which commute.

(ii) If ρ is a representation of A on H commuting with a contraction X, then there is a Shilov coextension σ of ρ and an isometric coextension V of X on a common Hilbert space K which commute.

(iii) If ρ is a representation of A on H commuting with a contraction X, then there is a simultaneous dilation of ρ to a maximal dilation π on K and of X to a unitary U commuting with π(A); i.e. there is a Hilbert space K ⊃ H, a ∗-representation π of C_{e}^{∗}(A) on K and a unitary operator U on K commuting with π(C_{e}^{∗}(A)) so that

P_{H}π(a)U^{n}|_{H} = ρ(a)X^{n}

for all a ∈ A and n ≥ 0.

Proof. It is evident that (iii) implies (ii) by restriction to the smallest invariant subspace containing H. And (ii) clearly implies (i). So assume that (i) holds. We will establish (iii).

First we dilate σ and V to τ and W so that W is unitary and commutes with τ(A). To accomplish this, consider the system with K_{n} = K and V considered as a map from K_{n} into K_{n+1}:

\xymatrix{K_{1} \ar[r]^{V} \ar[d]_{V} & K_{2} \ar[r]^{V} \ar[d]_{V} & K_{3} \ar[r]^{V} \ar[d]_{V} & … \ar[r] & P \ar[d]^{W} \\ K_{1} \ar[r]^{V} & K_{2} \ar[r]^{V} & K_{3} \ar[r]^{V} & … \ar[r] & P}

Then P is the Hilbert space direct limit of copies of K under V. Let J_{n} denote the canonical injection of K_{n} into P. Thus

J_{n} = J_{n+1}V for n ≥ 1.

The map V also determines isometries acting on each K_{n}, which we also denote by V. The direct limit of this system of maps is a unitary W on P such that its restriction to each K_{n} coincides with V. Hence

J_{n}V = W J_{n} for n ≥ 1.

In particular, W is an extension of V acting on K_{1}, which we identify with K.

We define a representation τ of A on P by

τ(a)J_{n}k = J_{n}σ(a)k for a ∈ A, k ∈ K_{n} and n ≥ 1.

Clearly each subspace J_{n}K_{n} is invariant for τ and the restriction of τ to K_{n} is equivalent to σ. In particular, τ is an extension of σ, where we identify K with K_{1}. Additionally, since τ is completely contractive when restricted to each J_{n}K_{n}, we see that τ is completely contractive. Finally, for a ∈ A and k ∈ K_{n} for n ≥ 2,

τ(a)W J_{n}(a)k = τ(a)J_{n}V k = τ(a)J_{n−1}k
= J_{n−1}σ(a)k = J_{n}V σ(a)k
= W J_{n}σ(a)k = W τ(a)J_{n}k.

Therefore, W commutes with τ(A).

The completely contractive map τ extends to a unique completely positive unital map on the operator system

\overline{A + A^{∗}} ⊂ C_{e}^{∗}(A).

By Fuglede’s Theorem, W commutes with \overline{τ(A) + τ(A)^{∗}}. The commutant 𝔑 of W is a type I von Neumann algebra, and therefore it is injective. Therefore by Arveson’s Extension Theorem, there is a completely positive extension of τ to C_{e}^{∗}(A) with range in 𝔑. By Stinespring’s Dilation Theorem, there is a minimal dilation to a ∗-representation π of C_{e}^{∗}(A) on a larger Hilbert space. Now a commutant lifting result of Arveson [5, Theorem 1.3.1] shows that there is a unique extension of W to an operator U commuting with π(C_{e}^{∗}(A)). This extension map is a ∗-homomorphism, so U is unitary. Moreover the fact that the restriction of π|_{A} to the space P is a homomorphism means that π is a maximal dilation of τ, and hence of ρ.

The following corollary is a consequence of (i) implies (iii) above.

Corollary 7.4. Property ICLT implies MCLT for A.

Another easy corollary is a consequence of the fact that (iii) is symmetric.

Corollary 7.5. Property ICLT is equivalent to ICLT*. So if A has ICLT, so does A^{∗}.

Example 7.6. Finite dimensional nest algebras have ICLT by Paulsen and Power [46, 47]. They actually prove variant (iii). They also claim that the minimal ∗-dilation is unique. This follows because finite dimensional nest algebras are Dirichlet. So by Theorem 5.15, they have SMCLT.

Dirichlet implies semi-Dirichlet for the algebra and its adjoint. So there are unique minimal fully extremal (co)extensions. The proof of ICLT in fact first coextends to an isometry in the commutant. Hence finite dimensional nest algebras have SCLT and SCLT*.

Nest algebras have the SCLT, SCLT* and MCLT for weak-∗ continuous completely contractive representations.

The first part of the following proposition uses exactly the same proof as Proposition 5.19 with the exception that the dilation Ỹ obtained can be taken to be a unitary operator when ICLT is invoked.

Proposition 7.7. Suppose that A has ICLT. Let ρ_{i} be representations of A on H_{i} for i = 1, 2. Suppose that X is a contraction in B(H_{2}, H_{1}) such that

ρ_{1}(a)X = Xρ_{2}(a) for all a ∈ A.

Then there are maximal dilations π_{i} of ρ_{i} acting on K_{i} ⊃ H_{i} for i = 1, 2 and a unitary operator U simultaneously dilating X in B(K_{2}, K_{1}) so that

π_{1}(a)U = Uπ_{2}(a) for all a ∈ A.

Consequently, there exist Shilov coextensions σ_{i} of ρ_{i} on L_{i} and a coextension of X to an isometry V ∈ B(L_{2}, L_{1}) so that

σ_{1}(a)V = V σ_{2}(a) for all a ∈ A.

Proof. We only discuss the second statement. Let π_{i} act on K_{i} = K_{i}^{−}⊕H_{i}⊕K_{i}^{+}, where L_{i} = H_{i} ⊕ K_{i}^{+} and K_{i}^{+} are invariant subspaces for π_{i}(A). With respect to these decompositions, we have the matrix forms

π_{i}(a) = [\begin{matrix}∗ & 0 & 0 \\ ∗ & ρ_{i}(a) & 0 \\ ∗ & π_{32}(a) & π_{33}(a) \end{matrix}] and U = [\begin{matrix}∗ & 0 & 0 \\ ∗ & X & 0 \\ ∗ & U_{32} & U_{33} \end{matrix}] .

Set

σ_{i}(a) = P_{L_{i}}π_{i}(a)|_{L_{i}} = [\begin{matrix}ρ_{i}(a) & 0 \\ π_{32}(a) & π_{33}(a) \end{matrix}] for i = 1, 2.

and

V = P_{L_{1}}U|_{L_{2}} = [\begin{matrix}X & 0 \\ U_{32} & U_{33} \end{matrix}]

Then σ_{i} are Shilov coextensions of ρ_{i}, V is an isometry, and

σ_{1}(a)V = P_{L_{1}}π_{1}(a)|_{L_{1}}P_{L_{1}}U|_{L_{2}}
= P_{L_{1}}π_{1}(a)UP_{L_{2}} = P_{L_{1}}Uπ_{2}(a)P_{L_{2}}
= P_{L_{1}}UP_{L_{2}}π_{2}(a)P_{L_{2}} = V P_{L_{2}}π_{2}(a)|_{L_{2}}
= V σ_{2}(a).

We ask a bit more for what we will call the Ando property. This is stronger than the classical Ando Theorem for A(𝔻). The weak version for A(𝔻) is just Ando’s Theorem.

Definition 7.8. A unital operator A satisfies the Ando property if whenever ρ is a representation of A on H and X ∈ B(H) is a contraction commuting with ρ(A), there is a fully extremal coextension σ of ρ on a Hilbert space K and a coextension of X to an isometry on K which commute.

Likewise say that A satisfies the weak Ando property if whenever ρ is a representation of A on H and X ∈ B(H) is a contraction commuting with ρ(A), there is an extremal coextension σ of ρ on a Hilbert space K and a coextension of X to an isometry on K which commute.

If A^{∗} has the (weak) Ando property, say that A has the (weak) Ando* property.

It is apparent that the Ando property implies CLT and ICLT for A; and the weak Ando property implies WCLT and ICLT. The converse of the latter fact follows the same lines as the classical deduction of Ando’s theorem from CLT. But the converse for the full Ando property is more difficult. The difference is that an extremal coextension of a coextension is extremal, but a fully extremal coextension of a coextension is generally not fully extremal. So more care has to be taken, and a Schaeffer type construction makes it work.

Proposition 7.9. A unital operator algebra A has the weak Ando property if and only if A has WCLT and ICLT.

Proof. Let ρ be a representation of A which commutes with a contraction X. Assume WCLT and ICLT. Coextend ρ and X to an extremal coextension σ_{1} and a commuting contraction Y_{1} using WCLT. Then coextend σ_{1} to ρ_{1} and Y_{1} to dilate to a commuting isometry by ICLT. Iterate these procedures recursively. The inductive limit has the desired properties.

Theorem 7.10. A unital operator algebra A has the Ando property if and only if A has CLT and ICLT.

Proof. Let ρ be a representation of A which commutes with a contraction X. Assume CLT and ICLT. Use CLT to coextend ρ and X to a fully extremal coextension σ and a commuting contraction Y on K ⊃ H. Then use ICLT to coextend this to a Shilov dilation τ and commuting isometry V on L = K ⊕ L′. Since σ is extremal, τ = σ ⊕ τ′. Write

V = [\begin{matrix}Y & 0 \\ Z & V′ \end{matrix}]

with respect to this decomposition of L. Then

τ′(a)Z = Zσ(a) for a ∈ A.

Since τ is Shilov, there is a maximal dilation π which is an extension of τ on a Hilbert space M containing L as an invariant subspace. So L′ is also invariant, as it reduces π(A)|_{L} = τ(A). Let P be the projection of M onto L′. Then

π(a)P = τ′(a)

and thus

π(a)(P Z) = P τ′(a)Z = (P Z)σ(a) for all a ∈ A.

The representation σ ⊕ π^{(∞)} is a fully extremal coextension of ρ. Moreover, X coextends to Y which coextends to

W = [\begin{matrix}Y & 0 & 0 & 0 & … \\ P Z & 0 & 0 & 0 & … \\ 0 & I & 0 & 0 & … \\ 0 & 0 & I & 0 & … \\ 0 & 0 & 0 & ... & ... \\ ⋮ & ⋮ & ⋮ & ... & ... \end{matrix}] .

It is easy to see that W is an isometry. The relations established in the previous paragraph ensure that W commutes with (σ ⊕π^{(∞)})(A). Thus we have verified that A has the Ando property.

This yields a strengthening of the classical Ando Theorem. The usual Ando Theorem verifies the weak Ando property, and hence ICLT. But the disk algebra has SCLT. So by Theorem 7.10, it has the Ando property. We provide a direct proof that is of independent interest.

Corollary 7.11. The disk algebra has the Ando property; i.e. if A_{1} and A_{2} are commuting contractions on H, then they have commuting isometric coextensions V_{i} on a common Hilbert space K. Moreover, we can arrange for V_{2} to be a fully extremal coextension (i.e. V_{A_{2}} ⊕ U, where V_{A_{2}} is the minimal isometric coextension and U is unitary).

Proof. If A is a contraction, let V_{A} denote the unique minimal isometric coextension of A. Let V_{i} be isometric dilations of A_{i}, i = 1, 2, acting on a common Hilbert space K = H ⊕ K′. (The minimal dilations may not have additional subspaces of the same dimension, for example if one is already an isometry. In this case, we just add a unitary summand to one of them.) Let V = V_{A_{1}A_{2}}.

Note that both V_{1}V_{2} and V_{2}V_{1} are isometries of the form

[\begin{matrix}A_{1}A_{2} & 0 \\ ∗ & ∗ \end{matrix}] .

So by the uniqueness of the minimal dilation, we can write

V_{1}V_{2} ≃ V ⊕ W_{1} and V_{2}V_{1} ≃ V ⊕ W_{2},

where W_{i} is an isometry acting on a Hilbert space K_{i} (possibly of dimension 0).

Now we dilate A_{1} to

S_{1} := V_{1} ⊕ W_{1}^{(∞)} ⊕ W_{2}^{(∞)}

and dilate A_{2} to

S_{2} := V_{2} ⊕ I_{K_{1}}^{(∞)} ⊕ I_{K_{2}}^{(∞)}

on

K ⊕ K_{1}^{(∞)} ⊕ K_{2}^{(∞)}.

So

S_{1}S_{2} ≃ V ⊕ W_{1} ⊕ W_{1}^{(∞)} ⊕ W_{2}^{(∞)} ≃ V ⊕ W_{1}^{(∞)} ⊕ W_{2}^{(∞)}

and

S_{2}S_{1} ≃ V ⊕ W_{2} ⊕ W_{1}^{(∞)} ⊕ W_{2}^{(∞)} ≃ V ⊕ W_{1}^{(∞)} ⊕ W_{2}^{(∞)}.

These unitary equivalences both fix H. Therefore there is a unitary operator U that fixes H so that

S_{2}S_{1} = U^{∗}S_{1}S_{2}U.

Now define isometric dilations U^{∗}S_{1} of A_{1} and S_{2}U of A_{2}. Then

(U^{∗}S_{1})(S_{2}U) = U^{∗}S_{1}S_{2}U = S_{2}S_{1} = (S_{2}U)(U^{∗}S_{1}).

This yields a commuting isometric dilation.

Moreover, if

V_{A_{2}} = [\begin{matrix}A_{2} & 0 \\ D_{2} & J \end{matrix}] ,

then S_{2}U has the form

S_{2}U ≃ [\begin{matrix}A_{2} & 0 & 0 \\ D_{2} & J & 0 \\ 0 & 0 & I \end{matrix}] [\begin{matrix}I & 0 & 0 \\ 0 & U_{22} & U_{23} \\ 0 & U_{32} & U_{33} \end{matrix}] ≃ [\begin{matrix}A_{2} & 0 & 0 \\ D_{2} & J & 0 \\ 0 & 0 & U′ \end{matrix}]

The basic observation is that J ⊕ I is an isometry on H^{⊥} with range equal to the orthocomplement of Ran D_{2}. The same is therefore true for the lower 2 × 2 corner of S_{2}U. By the uniqueness of the minimal isometric dilation, this corner splits as a direct sum J ⊕ U′ where U′must map onto the complement of the range of D and J. So U′ is unitary as claimed.

Example 7.12. One might ask to dilate both A_{1} and A_{2} to commuting isometries of the form V_{A_{i}} ⊕U_{i} with U_{i} unitary. This is not possible, as the following example due to Orr Shalit shows. Let

A_{1} = 0 and A_{2} = S,

where S is the unilateral shift on H = ℓ^{2}. Then

V_{A_{2}} = A_{2} = S

and

V_{A_{1}} = I ⊗ S acting on H ⊗ ℓ^{2}.

Suppose that U ∈ B(H_{0}) is unitary, and that

W_{1} = U ⊕ V_{A_{1}}

commutes with

W_{2} = A_{2} ⊕ X = S ⊕ X (with H appropriately identified).

Then they can be written as

W_{1} = [\begin{matrix}U & 0 & 0 & 0 & … \\ 0 & 0 & 0 & 0 & … \\ 0 & I & 0 & 0 & … \\ 0 & 0 & I & 0 & … \\ ⋮ & ⋮ & ⋮ & ... & ... \end{matrix}] and W_{2} = [\begin{matrix}X_{00} & 0 & X_{02} & X_{03} & … \\ 0 & S & 0 & 0 & … \\ X_{20} & 0 & X_{22} & X_{23} & … \\ X_{30} & 0 & X_{32} & X_{33} & … \\ ⋮ & ⋮ & ⋮ & ... & ... \end{matrix}] .

Computing the two products yields

[\begin{matrix}UX_{00} & 0 & UX_{02} & UX_{03} & … \\ 0 & 0 & 0 & 0 & … \\ 0 & S & 0 & 0 & … \\ X_{20} & 0 & X_{22} & X_{23} & … \\ X_{30} & 0 & X_{32} & X_{33} & … \\ ⋮ & ⋮ & ⋮ & ... & ... \end{matrix}] = [\begin{matrix}X_{00}U & X_{02} & X_{03} & X_{04} & … \\ 0 & 0 & 0 & 0 & … \\ 0 & S & 0 & 0 & … \\ X_{20}U & X_{22} & X_{23} & X_{24} & … \\ X_{30}U & X_{32} & X_{33} & X_{34} & … \\ ⋮ & ⋮ & ⋮ & ... & ... \end{matrix}] .

Equating terms shows that

X_{00}U = UX_{00}, X_{jj} = S for j ≥ 2 and X_{ij} = 0 otherwise.

Thus

W_{2} ≃ X_{00} ⊕ (S ⊗ I),

which is not V_{A_{2}} direct sum a unitary.

Katsoulis and Kakariadis [32, Theorem 3.5] (in the special case of the identity automorphism) show that every tensor algebra of a C*-correspondence has the weak Ando property. As noted in Example 5.3, the Muhly-Solel Commutant Lifting Theorem [42] shows that the tensor algebra T^{+}(E) of a C*-corresp-ondence E satisfies SCLT. Thus tensor algebras have the Ando property by Theorem 7.10. This forms a large class of algebras with this property. This includes all tensor algebras of graphs. The case of the non-commutative disk algebra follows from [17]. We would like to know if all semi-Dirichlet algebras have this property.

Theorem 7.13 (Katsoulis-Kakariadis). The tensor algebra of a C*- correspondence has the Ando property.

Corollary 7.14. The tensor algebra of a directed graph has the Ando property. In particular, the non-commutative disk algebras have this property.

The proof in [32] proves more, providing a lifting for relations that intertwine an automorphism. More will be said about this later when we discuss semi-crossed products.

Another proof can be based on an Ando type theorem of Solel [56, Theorem 4.4]. He shows that any representation of a product system over ℕ^{2} coextends to an isometric representation. One can think of a product system over ℕ^{2} as a pair of C*-correspondences over ℕ together with commutation relations. Here we only need a special case, where there is one C*-correspondence E over ℕ, and take the second correspondence to be F = ℂ with the relations that F commutes with E. Then a representation ρ of T^{+}(E) which commutes with a contraction X determines a representation of the product system. Applying Solel’s Theorem yields the desired isometric lifting. This verifies ICLT and, in fact, the weak Ando property. Now Theorem 7.10 shows that T^{+}(E) has the Ando property.

We finish this section by giving the intertwining version of the Ando property.

Theorem 7.15. Suppose that A has the Ando property. Suppose also that ρ_{1} and ρ_{2} are representations of A on H_{1} and H_{2} and X is a contraction in B(H_{2}, H_{1}) satisfying

ρ_{1}(a)X = Xρ_{2}(a) for all a ∈ A.

Given a fully extremal coextension σ_{2} of ρ_{2}, there exist a fully extremal coextension σ̃_{1} of ρ_{1} on K_{1}, a maximal representation π determining a fully extremal coextension σ̃_{2} = σ_{2} ⊕ π of ρ_{2} on K_{2}, and an isometry W ∈ B(K_{2}, K_{1}) so that

P_{H_{1}}W = XP_{H_{2}},

and

σ̃_{1}(a)W = W σ̃_{2}(a) for all a ∈ A.

Proof. The proof parallels the proof of Theorem 7.10 using the intertwining versions of CLT and ICLT. So we just sketch the plan.

One starts with the fully extremal coextension σ_{2} of ρ_{2} on K_{2}. By Theorem 5.12, coextend X to Y and ρ_{1} to a fully extremal σ_{1} on K_{1} so that

σ_{1}(a)Y = Y σ_{2}(a) for all a ∈ A.

Then use Proposition 7.7 to coextend Y to an isometry V and σ_{i} to a Shilov representation τ_{i} on L_{i} so that

τ_{1}(a)V = V τ_{2}(a) for all a ∈ A.

Since σ_{i} are fully extremal, we have τ_{i} = σ_{i} ⊕ τ′_{i}. If Z = P_{L_{1}⊖K_{1}}V |_{K_{2}} is the 2, 1 entry of V with respect to the decompositions L_{i} = K_{i} ⊕ (L_{i} ⊖ K_{i}), then

τ′_{1}(a)Z = Zσ_{2}(a) for all a ∈ A.

Since τ_{i} are Shilov, so are τ′_{i}. So choose maximal representations π_{i} on P_{i} which have invariant subspaces M_{i} so that π_{i}|_{M_{i}} ≃ τ′_{i}. Set

σ̃_{1} = σ_{1} ⊕ (π_{1} ⊕ π_{2})^{(∞)} and σ̃_{2} = σ_{2} ⊕ (π_{2} ⊕ π_{1})^{(∞)}.

Then the isometry

W ∈ B(K_{2} ⊕ (P_{2} ⊕ P_{1})^{(∞)}, K_{1} ⊕ (P_{1} ⊕ P_{2})^{(∞)}

described in the proof of Theorem 7.10 is the desired intertwiner. The details are left to the reader.

8 Incidence Algebras

An incidence algebra is a subalgebra A of the n × n matrices, 𝔐_{n}, containing the diagonal algebra 𝔇_{n} with respect to a fixed orthonormal basis. Clearly, A is spanned by the matrix units E_{ij} that it contains. One can define a partial order R on {1, 2, …, n} by

i ≺ j (or (i, j) ∈ R) if E_{ij} ∈ A.

This can be identified with a directed graph, but note that generally the algebra is not the same as the tensor algebra of the graph. There is a reduced partial order obtained by identifying equivalent indices

i ≡ j if i ≺ j and j ≺ i.

The representation theory of the algebra of a partial order and its reduced partial order are related simply by multiplicity.

The algebra A∩A^{∗} is a C*-algebra containing the diagonal 𝔇_{n}, and is spanned by {E_{ij} : i ≡ j}. A representation ρ restricts to a completely contractive representation of A ∩ A^{∗}, and thus is a ∗-representation. So each diagonal matrix unit E_{ii} is sent to an orthogonal projection P_{i} = ρ(E_{i}) onto a subspace H_{i}. Since ρ is unital,

H = ∑_{1≤i≤n}⊕ H_{i}.

In general, there are contractions T_{ij} ∈ B(H_{j}, H_{i}) so that

ρ(E_{ij}) = P_{i}T_{ij}P_{j}.

When i ≡ j, T_{ij} is a unitary and T_{ji} = T_{ij}^{∗}. The homomorphism property shows that

T_{ik} = T_{ij}T_{jk} when i ≺ j ≺ k.

These relations are sufficient to determine an algebraic homomorphism.

Not all choices of contractions {T_{ij} : (i, j) ∈ R} yield a completely contractive representation in general. However this does hold in some situations. Paulsen and Power [46] establish this for nest algebras. Davidson, Paulsen and Power [22] establish this for bilateral tree algebras. These are the algebras where the reduced relation is generated as a transitive relation by a directed bilateral tree (a directed graph with no loops). Finally the class of all such algebras was determined by Davidson in [15] as the interpolating graphs.

Muhly and Solel consider unilateral tree algebras, which are the incidence algebras A which are semi-Dirichlet. They show that graphically means that the relation is generated by a directed unilateral tree (each vertex is the range of at most one edge and there are no loops). These incidence algebras actually coincide with the tensor algebra of the unilateral tree because there is always at most one edge into each vertex. For example, the algebra

A = span{E_{11}, E_{22}, E_{33}, E_{13}, E_{23}},

which is determined by the graph formed by edges from v_{3} to each of v_{1} and v_{2}, is a unilateral tree algebra. However A^{∗}, which is determined by edges from v_{1} and v_{2} into v_{3} is not a unilateral tree algebra, but it is a bilateral tree algebra. See [40, Chapter 5] for a discussion of “trees and trees”.

If ρ is a representation of an incidence algebra A as above, then a coextension σ will act on a Hilbert space

K = ∑_{1≤i≤n}⊕ K_{i},

where K_{i} ⊃ H_{i} is the range of the projection σ(E_{ii}), and is determined by coextensions V_{ij} of T_{ij} in B(K_{j}, K_{i}) of the form

V_{ij} = [\begin{matrix}T_{ij} & 0 \\ D_{ij} & S_{ij} \end{matrix}]

with respect to the decompositions K_{i} = H_{i}⊕K′_{i}. The homomorphism property requires that

V_{ij}V_{jk} = V_{ik} whenever i ≺ j ≺ k.

In general, these are complicated relations to dilate. One of the simplest examples where things get complicated is the 2k-cycle graph C_{2k} for k ≥ 2. This graph has vertices {1, 2, …, 2k} and edges

2i + 1 ≻ 2i, 2i + 1 ≻ 2i + 2 and 1 ≻ 2k.

The algebra for this graph has representations such that ρ(e_{ij}) = T_{ij} are all contractions, but ∥ρ∥_{cb} > 1 [15, Theorem 2.2]. This is related to the famous example of Parrott [44] for A(𝔻^{3}) and a similar example due to Paulsen and Power [47, Theorem 3.1] for the incidence algebra T_{2} ⊗ T_{2} ⊗ T_{2}, where T_{2} is the algebra of 2 × 2 upper triangular matrices. Also see the exposition in [14, Example 20.27].

The case of bilateral tree algebras is more condusive to analysis.

Theorem 8.1. Let A be a bilateral tree incidence algebra. Then a representation ρ is an extremal coextension if and only if each edge E_{ij} is mapped to a partial isometry V_{ij} such that V_{ij}^{∗}V_{ij} = ρ(E_{ii}).

If ρ is a representation, then a coextension σ of ρ is fully extremal if and only if it is extremal and K_{i} = H_{i} ∨ V_{ij}K_{j} for all edges of the tree.

Proof. The key observation from [22] is that every matrix unit in A factors uniquely as a product of matrix units in the tree T , corresponding to the combinatorial fact that every edge in the transitive relation corresponds to the unique path on the tree from one vertex to another. Thus if for each matrix unit in the tree, T_{ij} = ρ(E_{ij})|_{H_{j}} is coextended to V_{ij}, then we can extend this definition to every matrix unit in a unique way; and the homomorphism property guarantees that each is a coextension of T_{ij}. It is possible to coextend each T_{ij} in the tree to an isometry from K_{j} to K_{i}. If more than one edge is entering a single vertex i, then one has to ensure that K_{i} is large enough to accomodate all T_{ij}. (Of course, the ranges can overlap or even coincide.) By [22], this representation is still completely contractive. Thus to be extremal, each ρ(E_{ij}) needs to be isometric from K_{j} into K_{i}.

Conversely, if each T_{ij} = ρ(E_{ij})|_{H_{j}} is an isometry on H_{j}, consider any coextension σ of ρ. Then each σ(E_{ii}) has range K_{i} ⊃ H_{i}, and each isometry T_{ij} coextends to a contraction S_{ij} = σ(E_{ij})|_{K_{j}}. Therefore S_{ij}|_{H_{j}} = T_{ij} and S_{ij}|_{K_{j}⊖H_{j}} is a contraction with range orthogonal to H_{i}. Hence K′ = K ⊖ H is reducing. Therefore σ decomposes as a direct sum of ρ and σ′ = σ|_{K′}.

Fully extremal coextensions of ρ are more complicated. If we start with ρ, and coextend to an extremal σ, so that each T_{ij} is coextended to an isometry V_{ij}, it may be possible to dilate σ to τ ≻_{c} ρ if there is some ‘room’ left. More precisely, if for some edge E_{ij} in the tree, we have

K_{i} ≠ H_{i} ∨ V_{ij}K_{j},

then one can extend σ to τ, a coextension of ρ, to use this extra space. Pick a vector e ∈ K′_{i} = K_{i} ⊖ H_{i} which is orthogonal to the range of V_{ij}. Form

K̃_{j} = K_{j} ⊕ ℂf.

Extend V_{ij} to

Ṽ_{ij} := V_{ij} + ef^{∗}.

It is apparent that this is indeed a coextension of ρ and an extension of σ which is not obtained as a direct sum. So σ is not fully extremal.

Conversely, if for all edges i ≺ j of the tree,

K_{i} = H_{i} ∨ V_{ij}H_{j}

then for any extension τ of σ, the operators Ṽ_{ij} = τ(E_{ij}) will have to map any new summand of K̃_{j}, namely K̃_{j} ⊖K_{j}, to a space orthogonal to H_{i} because it is a coextension of ρ, and orthogonal to V_{ij}K_{j} because Ṽ_{ij} is a contraction. Hence by hypothesis, it maps into K̃_{i} ⊖ K_{i}. This makes it apparent that τ splits as a direct sum of σ and another representation. Therefore σ is fully extremal.

Corollary 8.2. Every Shilov extension of a bilateral tree algebra A is extremal.

Proof. A maximal representation π of A extends to a ∗-representation of 𝔐_{n}. In particular, each π(E_{ij}) is a unitary from H_{j} onto H_{i}. Thus any restriction σ to an invariant subspace sends each vertex to a projection onto a subspace K_{i} ⊂ H_{i} and each edge E_{ij} to an isometry of K_{j} into K_{i}. By Theorem 8.1, σ is extremal.

Remark 8.3. Since ρ(E_{ii}) can be 0, an isometry can be vacuous. So for the algebra A_{n} of Example 3.2, the coextensions σ_{2i} are all extremal. But to be fully extremal, each edge has to be mapped to an isometry with maximal range, so only σ_{n} is fully extremal.

Uniqueness. Let us explain why only the unilateral tree algebras have unique minimal fully extremal coextensions. The unilateral tree algebras are semi-Dirichlet. So this property is a consequence of Theorem 4.6.

A typical example is

A = span{E_{21}, E_{31}, 𝔇_{3}} ⊂ 𝔐_{3}.

Consider a representation ρ on H = H_{1} ⊕ H_{2} ⊕ H_{3} where ρ(E_{i1}) = T_{i} for i = 2, 3. Let

D_{i} = P_{D_{i}}(I − T_{i}^{∗}T_{i})^{1/2} ∈ B(H_{1}, D_{i}),

where

D_{i} = \overline{Ran(I − T_{i}^{∗}T_{i})}

is the closed range of (I − T_{i}^{∗}T_{i})^{1/2}. Coextend ρ to σ on K = K_{1} ⊕ K_{2} ⊕ K_{3} where

K_{1} = H_{1} and K_{i} = H_{i} ⊕ D_{i} for i = 2, 3

by setting

V_{i1} = [\begin{matrix}T_{i} \\ D_{i} \end{matrix}] .

This is easily seen to be a fully extremal coextension by the previous proposition.

Any other isometric coextension τ will act on a Hilbert space L where L_{i} = H_{i} ⊕ L′_{i}, and

τ(E_{i1}) = [\begin{matrix}T_{i} & 0 \\ U_{i}D_{i} & S_{i} \end{matrix}] for i = 2, 3.

Here U_{i} is an isometric imbedding of D_{i} into L′_{i} and S_{i} is an isometry of L′_{1} into L′_{i} with range orthogonal to the range of

[\begin{matrix}T_{i} \\ U_{i}D_{i} \end{matrix}] .

A bit of thought shows that this splits as a direct sum of a representation on

H_{1} ⊕ (H_{2} ⊕ U_{2}D_{2}) ⊕ (H_{3} ⊕ U_{3}H_{3})

which is unitarily equivalent to σ and another piece.

On the other hand, any graph which is a bilateral tree but not a unilateral tree will have two edges mapping into a common vertex. The compression to this three dimensional space yields the algebra A^{∗}. We explain why A^{∗} has non-unique minimal fully extremal coextensions.

Fix θ ∈ (0, π/4), and consider a representation ρ on

H = H_{1} ⊕ H_{2} ⊕ H_{3}

where H_{i} = ℂ^{2} given by

ρ(E_{1i}) = T = [\begin{matrix}cos θ & 0 \\ 0 & sin θ \end{matrix}] for i = 2, 3.

Let

D = (I − T^{∗}T)^{1/2} = [\begin{matrix}sin θ & 0 \\ 0 & cos θ \end{matrix}] .

Then for any unitaries U_{i} in the 2 × 2 unitary group 𝔘_{2}, we can coextend ρ to an isometric representation σ on

K = ℂ^{4} ⊕ ℂ^{2} ⊕ ℂ^{2}

by setting

V_{i} = [\begin{matrix}T_{i} \\ U_{i}D \end{matrix}] for i = 2, 3.

These are all fully extremal coextensions of ρ by Theorem 8.1. Moreover they are evidently minimal.

Consider when two such representations will be unitarily equivalent via a unitary W which is the identity on H, and thus has the form

(I_{2} ⊕ V) ⊕ I_{2} ⊕ I_{2}.

Conjugation by W carries σ to the representation which replaces U_{i}D by V U_{i}D for i = 1, 2. It is clear that one can arrange to match up the 1, 2 entry by appropriate choice of V. But that leaves no control over the 1, 3 entry. The possible minimal fully extremal coextensions of ρ are parameterized by 𝔘_{2}.

Commutant lifting. Davidson, Paulsen and Power [22] showed that bilateral tree algebras have ICLT, and hence MCLT. They do not generally have SMCLT because of failure of unique dilations. Among finite dimensional incidence algebras, these are precisely the algebras with ICLT [15, Theorem 4.6].

Muhly and Solel show that unilateral tree algebras satisfy SCLT. This now can be seen from the fact that they have ICLT, whence MCLT. So by Theorem 4.6, we can obtain unique minimal fully extremal coextensions, and that every Shilov extension is fully extremal. So this implies SCLT.

The connected graphs which are unilateral trees and have adjoints which are unilateral trees as well are evidently chains. So the incidence algebras with this property are just direct sums of finite dimensional nest algebras. Since these algebras are Dirichlet, they have many good properties from section 4.

Proposition 8.4. Bilateral tree algebras have WCLT and WCLT*, and well as the weak Ando and Ando* properties.

Proof. If A is a bilateral tree algebra, then so is A^{∗}. So it suffices to prove WCLT. Let ρ be a representation of A, and let X be a contraction commuting with ρ(A). By [22], A has ICLT. Hence by Theorem 7.3 (ii), there is a Shilov coextension σ and an isometric coextension V of X on K ⊃ H which commute. By Corollary 8.2, σ is extremal. Thus A has WCLT and the weak Ando property.

The goal now is to refine this construction to obtain fully extremal coextensions to obtain CLT and hence the Ando property. We begin by establishing the Ando property for T_{2}, the 2 × 2 upper triangular matrices. Since

T_{2} = span{E_{11}, E_{12}, E_{22}},

a representation ρ is determined by ρ(E_{ii}) = P_{i} = P_{H_{i}}, where H = H_{1} ⊕ H_{2}, and a contraction X ∈ B(H_{2}, H_{1}) where ρ(E_{12}) = P_{1}XP_{2}. A contraction A commuting with ρ(T_{2}) commutes with P_{i}, and so has the form A = A_{1} ⊕ A_{2}; plus we have A_{1}X = XA_{2}. Thus Ando’s Theorem for T_{2} can be reformulated as a commutant lifting theorem:

Lemma 8.5. Suppose that A_{i} ∈ B(H_{i}) for i = 1, 2 and X ∈ B(H_{2}, H_{1}) are contractions such that

A_{1}X = XA_{2}.

Then there are coextensions of A_{i}, i = 1, 2 and X to isometries Ã_{i} in B(K_{i}) and X̃ in B(K_{2}, K_{1}) so that

Ã_{1}X̃ = X̃Ã_{2} and K_{1} = H_{1} ∨ X̃K_{2}.

Proof. The algebra T_{2} is a tree algebra, and so has the weak Ando property by the previous proposition. Hence there are isometric coextensions B_{i} of A_{i} and Y of X so that

B_{1}Y = Y B_{2},

acting on spaces L_{i} ⊃ H_{i}. We will modify this to obtain the desired form.

Observe that the commutation relation implies that the range of Y is invariant for B_{1}. Let

L = H_{1} ∨ Y L_{2}; and B′_{1} = P_{L}B_{1}|_{L}.

Let Y′ ∈ B(L_{2}, L) be Y considered an an operator into L. Then

B′_{1}Y′ = P_{L}B_{1}P_{L}Y = P_{L}B_{1}Y = P_{L}Y B_{2} = Y′B_{2}.

Also since Y′ is an isometry,

B_{2} = Y^{′∗}B′_{1}Y′.

In particular, the commutation relations hold, and

H_{1} ∨ Y L_{2} = L.

The contraction B′_{1} may no longer be an isometry, but it is a coextension of A_{1}.

Let Ã_{1} be the minimal isometric dilation of B′_{1} on K_{1} ⊃ L. Write K′_{1} = K_{1} ⊖L. With respect to K_{1} = L ⊕ K′_{1}, we have

Ã_{1} = [\begin{matrix}B′_{1} & 0 \\ C & D \end{matrix}] .

Define

K_{2} = L_{2} ⊕ K′_{1} and X̃ = Y′ ⊕ I ∈ B(K_{2}, K_{1}).

Set

Ã_{2} = X̃^{∗}Ã_{1}X̃

= [\begin{matrix}Y^{′∗} & 0 \\ 0 & I \end{matrix}] [\begin{matrix}B′_{1} & 0 \\ C & D \end{matrix}] [\begin{matrix}Y′ & 0 \\ 0 & I \end{matrix}]

= [\begin{matrix}Y^{′∗}B′_{1}Y′ & 0 \\ CY′ & D \end{matrix}]

= [\begin{matrix}B_{2} & 0 \\ CY′ & D \end{matrix}]

Thus A_{2} is a coextension of B_{2}, and hence of A_{2}. It is easy to verify that

Ã_{1}X̃ = X̃Ã_{2}.

Since Ã_{1} and X̃ are isometries, so is Ã_{2}. Moreover, we now have

H_{1} ∨ X̃K_{2} = (H_{1} ∨ Y′L_{2}) ⊕ K′_{1} = K_{1}.

Theorem 8.6. Bilateral tree algebras have the Ando and Ando* properties.

Proof. Again it suffices to establish the Ando property. We first assume that the A ∩ A^{∗} = 𝔇_{n}, so that the relation and reduced relation coincide.

Before proceeding, we make a few observations and set some notation. Suppose that the tree T has vertices v_{i} for 1 ≤ i ≤ n. Let ρ be a representation of the algebra A commuting with a contraction A. Then since

ρ(v_{i}) = P_{i} = P_{H_{i}}

are pairwise orthogonal projections summing to the identity, we see that

A = ∑_{1≤i≤n}⊕ A_{i}

where

A_{i} = A|_{H_{i}} ∈ B(H_{i}).

If e_{ij} is an edge of the tree, let

T_{ij} = ρ(e_{ij})|_{H_{j}} ∈ B(H_{j}, H_{i}).

We have

A_{i}T_{ij} = T_{ij}A_{j},

and conversely any A with these two properties commutes with A.

A finite bilateral tree has an elimination scheme, in the sense that every bilateral tree has a vertex v which has at most one edge e such that either s(e) = v or r(v) = v. This allows a proof by induction. So proceed by induction on the number of vertices.

If the graph has a single vertex, then it has no edges and A = ℂ. It is trivial to verify Ando’s property in this case.

Now suppose that the result holds for every bilateral tree on fewer than n vertices, and let T be a bilateral tree on n vertices. Let ρ and A be as above. We may assume that T is connected; for otherwise we may dilate each component by the induction hypothesis. So every vertex has an edge. Let v_{i_{0}} be a vertex with one edge e. Restrict the representation to T \{v_{i_{0}}, e} acting on H_{i_{0}}^{⊥}, called ρ′, commuting with A′ = A|_{H_{i_{0}}^{⊥}}. Use the induction hypothesis to coextend ρ′ to a fully extremal coextension σ′ commuting with an isometric coextension B′ of A′. Let

Ran σ′(E_{ii}) =: L′_{i} for i ≠ i_{0}.

Then

B′ = ∑_{i≠i_{0}} ⊕ B′_{i}.

There are two cases: either s(e) = v_{i_{0}} and r(e) = v_{j_{0}} or r(e) = v_{i_{0}} and s(e) = v_{j_{0}}. Assume the former. Let

ρ(e) = X ∈ B(H_{i_{0}}, H_{j_{0}}).

Then

A_{j_{0}}X = XA_{i_{0}}.

Use Lemma 8.5 to coextend A_{i_{0}}, A_{j_{0}} and X to isometries Ã_{i_{0}}, Ã_{j_{0}} and X̃ so that

Ã_{j_{0}} X̃ = X̃Ã_{i_{0}} and K_{j_{0}} = H_{j_{0}} ∨ X̃K_{i_{0}}.

We can decompose

Ã_{j_{0}} = V_{j_{0}} ⊕ W_{j_{0}},

where V_{j_{0}} is the minimal isometric coextension of A_{j_{0}}, with respect to

K_{j_{0}} = K_{j_{0}}^{0} ⊕ K_{j_{0}}^{1}.

Similarly, decompose the isometry

B′_{j_{0}} = V_{j_{0}} ⊕ W′_{j_{0}}

with respect to

L′_{j_{0}} ≃ K_{j_{0}}^{0} ⊕ L_{j_{0}}^{′1}.

Define the coextension σ of ρ and isometric coextension B of A as follows. Set

L_{i} = L′_{i} ⊕ K_{j_{0}}^{1} for i ≠ i_{0} and L_{i_{0}} = K_{i_{0}} ⊕ L_{j_{0}}^{′1}.

Define

σ(e_{ij}) = σ′(e_{ij}) ⊕ (E_{ij} ⊗ I_{K_{j_{0}}^{1}} ) for i ≠ i_{0} and σ(e) = X̃ ⊕ I_{L_{j_{0}}^{′1}}
B_{i} = B′_{i} ⊕ W_{i_{0}} for i ≠ i_{0}  and B_{i_{0}} = Ã_{i_{0}} ⊕ W_{j_{0}}.

Here E_{ij}⊗I_{K_{j_{0}}^{1}} is interpreted as the unitary that maps the copy of K_{j_{0}}^{1} contained in L_{j} to the corresponding copy in L_{i}. One needs only verify that each σ(e_{ij}) intertwines B_{j} with B_{i} and

H_{i} ∨ σ(e_{ij})L_{j} = L_{i}.

Both of these facts are routine. Thus a fully extremal coextension of ρ is produced that commutes with an isometric coextension of A. This verifies Ando’s property.

The second case, in which the edge e maps v_{j_{0}} to v_{i_{0}} is handled similarly by first dilating the graph on n − 1 vertices, and producing a dilation of the one edge e using Lemma 8.5. Then as above, split the two isometries over the vertex v_{j_{0}} into the minimal isometric coextension direct summed with another isometry; and then define an explicit coextension with the desired properties.

The following consequence is immediate.

Corollary 8.7. Bilateral tree algebras have CLT and CLT*.

9 The Fuglede Property

We introduce another property of an abstract unital operator algebra reminiscent of the classical Fuglede Theorem that the commutant of a normal operator is self-adjoint.

Definition 9.1. Let A be a unital operator algebra with C*-envelope C_{e}^{∗}(A). Say that A has the Fuglede Property (FP) if for every faithful unital ∗- representation π of C_{e}^{∗}(A), one has

π(A)′ = π(C_{e}^{∗}(A))′.

It is easy to characterize this property among abelian algebras.

Proposition 9.2. If A is an abelian operator algebra, then the following are equivalent:

(i) A has the Fuglede property. (ii) A is a function algebra. (iii) C_{e}^{∗}(A) is abelian.

Proof. If (1) holds, then for every a ∈ A, π(a) lies in π(A)′ and hence in π(C_{e}^{∗}(A))′. Thus π(C_{e}^{∗}(A)) is abelian. So (3) holds. If (3) holds, then A is a function algebra since A separates points by the definition of the C*-envelope. Finally if (2) holds, then π(A) is contained in π(C(X)) which is an algebra of commuting normal operators. So the FP property follows from the usual Fuglede Theorem.

The following is a useful class of operator algebras which has the FP property.

Proposition 9.3. Suppose that there is a family {U_{k} = [a_{ij}^{(k)}]} of unitary matrices U_{k} ∈ 𝔐_{m_{k},n_{k}}(A) such that the set of matrix coefficients {a_{ij}^{(k)}} generate A as an operator algebra. Then A has F P.

Proof. If B commutes with π(A), then

B^{(m)}π(U_{k}) = π(U_{k})B^{(n)}.

By the Fuglede–Putnam Theorem, we obtain

B^{∗(m)}U_{k} = U_{k}B^{∗(n)}.

Therefore B^{∗} commutes with each a_{ij}^{(k)}. As these generate A, we deduce that π(A)′ is self-adjoint.

Example 9.4. The non-commutative disk algebras of Popescu, 𝔄_{n}, are generated by a row isometry S = [S_{1} … S_{n}]. The C*-envelope is the Cuntz algebra O_{n}. As an element of 𝔐_{1,n}(O_{n}), S is a unitary operator. Thus 𝔄_{n} has the FP property. This property has been explicitly observed in [19, Proposition 2.10].

Example 9.5. The algebra 𝔄_{∞} generated by a countable family of isometries with pairwise orthogonal ranges does not have the Fuglede property. This is because ∗-representations of the C*-envelope, O_{∞}, are determined by any countably infinite family of isometries with pairwise orthogonal ranges, and does not force the sum of these ranges to be the whole space. In the left regular representation, the commutant of 𝔄_{∞} is the wot-closed algebra generated by the right regular representation, which is not self-adjoint.

Example 9.6. The algebra A_{d} of continuous multipliers on the Drury-Arveson space H_{d}^{2} is abelian, but the norm is not the sup norm. So this is not a function algebra. Arveson [8] identifies the C*-envelope, which contains the compact operators; so it is clearly not abelian.

Since A_{d} is a quotient of 𝔄_{d}, one sees that FP does not pass to quotients. One can also see this by noting that there are quotients of functions algebras which are not themselves functions algebras.

Example 9.7. The tensor algebra of any finite graph has FP. It does not follow immediately from Proposition 9.3, but does follow by a simple modification. A finite graph G = (V, E, r, s) consists of a finite set V of vertices, a finite set E of edges, and range and source maps r, s : E → V. The graph C*-algebra C^{∗}(G) is the universal C*-algebra generated by pairwise orthogonal projections p_{v} for v ∈ V and partial isometries u_{e} for e ∈ E such that

∑_{v∈V} p_{v} = 1, u_{e}^{∗}u_{e} = p_{s(e)} and ∑_{r(e)=v} u_{e}u_{e}^{∗} = p_{v}

unless v is a source, meaning that there are no edges with range v. The tensor algebra of the graph T^{+}(G) is the non-self-adjoint subalgebra of C^{∗}(G) generated by

{p_{v}, u_{e} : v ∈ V, e ∈ E}.

Then C_{e}^{∗}(T (G)) = C^{∗}(G) [29, 33].

Suppose that π is a ∗-representation of C^{∗}(G) and T ∈ π(T (G))′. Then T commutes with π(p_{v}) =: P_{v}, and thus T = ⊕ ∑_{v∈V} T_{v} where T_{v} ∈ B(P_{v}H). If there are edges with r(e) = v, say e_{1}, …, e_{k}, then let s(e_{i}) = v_{i} and consider

U = [π(u_{e_{1}}) … π(u_{e_{k}})] ∈ B(P_{v_{1}}H ⊕ … P_{v_{k}}H, P_{v}H).

This is a unitary element, and

T_{v}U = U(T_{v_{1}} ⊕ ⋯ ⊕ T_{v_{k}}).

By the Fuglede-Putnam Theorem, we also obtain

T_{v}^{∗}U = U(T_{v_{1}}^{∗} ⊕ ⋯ ⊕ T_{v_{k}}^{∗}).

If there are no edges with range v, there is nothing to check. We deduce as in Proposition 9.3 that T^{+}(G) has FP.

Example 9.8. The algebra of any finite k-graph has FP. This is established as in the case of a 1-graph.

Example 9.9. Let N be a nest (a complete chain of closed subspaces of a Hilbert space). Set

A = T (N) ∩ 𝔎^{+},

where T (N) is the nest algebra [14] and 𝔎^{+} = ℂI + 𝔎 is unitization of the space of compact operators. By the Erdos Density Theorem, T (N) ∩ 𝔎 contains a norm 1 approximate identity; and thus A is weak-∗ dense in T (N). Therefore its commutant is trivial, and coincides with the commutant of 𝔎^{+}, the enveloping C*-algebra. Moreover, 𝔎^{+} is the C*-envelope because 𝔎 is the only ideal, and the quotient q of 𝔎^{+} onto ℂ is clearly not isometric on A. A ∗-representation of 𝔎^{+} has the form

π(A) = q(A)I_{K_{0}} ⊕ A ⊗ I_{K_{1}}

on a Hilbert space K = K_{0} ⊕ (H ⊗ K_{1}). By the earlier remark, the commutant of π(A) is seen to be

B(K_{0}) ⊕ (ℂI_{H} ⊗ B(K_{1})),

which is the commutant of π(𝔎^{+}). So A has FP.

In particular, any finite dimensional nest algebra T ⊂ 𝔐_{n} has FP.

Example 9.10. More generally, let L be a completely distributive commutative subspace lattice (see [14]); and let Alg(L) be the corresponding CSL algebra. Let 𝔐 be a masa containing (the projections onto) L. Also let

𝔑 = Alg(L ∩ L^{⊥}).

Observe that L ∩ L^{⊥} is a completely distributive Boolean algebra, and thus is atomic. Therefore

𝔑 = ⊕ ∑_{i} B(H_{i})

is an ℓ^{∞} direct sum with respect to the decomposition H = ⊕ ∑_{i} H_{i}, where H_{i} are the ranges of the atoms of L ∩ L^{⊥}.

Also by complete distributivity, Alg(L)∩𝔎 has a norm one approximate identity. So again A = Alg(L) ∩ 𝔎^{+} is weak-∗ dense in Alg(L). It is straightforward to see that

C_{e}^{∗}(A) = {λI + ⊕ ∑_{i} A_{i} : A_{i} ∈ 𝔎^{+}(H_{i}) and lim_{i} A_{i} = 0}.

The irreducible representations of C_{e}^{∗}(A) are unitarily equivalent to compression to some H_{i} and the character that evaluates λ. So every representation is a direct sum of these irreducible ones with multiplicity. As in the nest case, it is straightforward to show that

π(A)′ = π(C_{e}^{∗}(A))′.

So A has FP.

We have one minor result relating FP with commutant lifting.

Proposition 9.11. If an operator algebra A has FP and MCLT, then it has ICLT.

Proof. Suppose that ρ is a representation of A on H and X ∈ B(H) is a contraction commuting with ρ(A). Then by MCLT, there is a maximal dilation π of ρ and a contractive dilation Y of X which commutes with π(A) and has H as a common semi-invariant subspace. Since π is maximal, it extends to a ∗-representation of C_{e}^{∗}(A) which we also denote by π. By the Fuglede property, Y commutes with π(C_{e}^{∗}(A)). Hence C^{∗}(Y) is contained in π(C_{e}^{∗}(A))′.

The standard Schaeffer dilation of Y to a unitary U on K^{(∞)} has coefficients in C^{∗}(Y). So U commutes with π^{(∞)}. This establishes ICLT.

Corollary 9.12. If an operator algebra A has properties FP, WCLT and WCLT*, then it has the weak Ando property.

If an operator algebra A has properties FP, CLT and CLT*, then it has the Ando property.

Proof. Theorem 5.14 shows that WCLT and WCLT* imply MCLT. So by the preceding proposition, we obtain ICLT. By Proposition 7.9 and Theorem 7.10, WCLT and ICLT imply the weak Ando property and CLT and ICLT imply the Ando property.

10 Completely Isometric Endomorphisms

In the category of operator algebras, the natural morphisms are completely bounded maps. Among the endomorphisms, those that work best for semicrossed products are the completely isometric ones. These are the analogue of the faithful ∗-endomorphisms of C*-algebras. In this section, we investigate lifting completely isometric endomorphisms of operator algebras to ∗- endomorphisms of some C*-cover.

Let Aut(A) denote the completely isometric automorphisms of an operator algebra A. If 𝔄 is a C*-algebra, then (completely) isometric automorphisms are automatically ∗-automorphisms. If A ⊂ 𝔄, let

Aut_{A}(𝔄) = {α ∈ Aut(𝔄) : α(A) = A}.

Similarly, let End(A) denote the completely isometric unital endomorphisms of A. Again, for a C*-algebra, these are faithful unital ∗-endomorphisms. When A ⊂ 𝔄, we let

End_{A}(𝔄) = {α ∈ End(𝔄) : α(A) ⊂ A}.

First we begin with an easy result.

Proposition 10.1. Let A be a unital operator algebra. Then every completely isometric automorphism of A lifts to a ∗-automorphism of C_{e}^{∗}(A) which fixes A (as a set). Thus

Aut_{A}(C_{e}^{∗}(A)) ≃ Aut(A)

by restriction to A.

Proof. Let α ∈ Aut(A). Consider A as a subalgebra of C_{e}^{∗}(A). The map α takes A completely isometrically and isomorphically onto itself, and the image generates C_{e}^{∗}(A) as a C*-algebra. By the basic property of C*-envelopes, α extends to a ∗-homomorphism α̃ of C_{e}^{∗}(A) onto itself. The kernel of α̃ is a boundary ideal because the map is completely isometric on A, and hence is {0}. Thus α̃ is an automorphism which fixes A as a set. The converse is evident.

The restriction of α ∈ Aut_{A}(C_{e}^{∗}(A)) to A is injective since A generates C_{e}^{∗}(A) as a C*-algebra. Thus the restriction map is an isomorphism.

Example 10.2. It is not true that every α ∈ End(A) lifts to an endomorphism of C_{e}^{∗}(A). Take A = A(𝔻) and let τ ∈ A(𝔻) be the composition of a conformal map of 𝔻 onto the rectangle

{x + iy : −1 < x < 0 and |y| ≤ 10}

followed by the exponential map. Thus τ maps 𝔻 onto the annulus

A := {z : e^{−1} < |z| < 1}.

It follows that

α(f) = f ∘ τ

is a completely isometric endomorphism. However, since τ maps parts of 𝕋 into the interior of 𝔻, this map does not extend to an endomorphism of C(𝕋).

Observe that α lifts to an endomorphism of C(\overline{𝔻}) by

α̃(f) = f ∘ τ.

Unfortunately this map is not faithful, as its kernel is

ker α̃ = I(A) = \{f ∈ C(\overline{𝔻}) : f|_{A} = 0\}.

The remedy is a bit subtle. Let

X = ⋂_{n≥0} τ^{n}(\overline{𝔻}).

This is a connected compact set with two key properties:

τ(X) = X and 𝕋 ⊂ X.

The latter holds because τ(𝕋) contains 𝕋. Now consider A(𝔻) as a subalgebra of the C*-algebra C(X). The embedding is isometric because 𝕋 ⊂ X. This is a C*-cover by the Stone-Weierstrass Theorem. Here α extends to

ᾱ(f) = f ∘ τ.

This is a faithful ∗-endomorphism because τ is surjective on X.

Example 10.3. Here is a different example. Take

A = A(𝔻) ⊕ (c ⊗ T )

where c is the space of convergent sequences and T = C^{∗}(T_{z}) is the Toeplitz algebra generated by the shift T_{z} on H^{2}. It is easy to see that

C_{e}^{∗}(A) = C(𝕋) ⊕ (c ⊗ T ).

Write an element of A as f ⊕ (T_{n})_{n≥1}, where lim_{n→∞} T_{n} =: T_{0} exists. Fix λ ∈ \overline{𝔻} and consider the map

α(f ⊕ (T_{n})_{n≥1}) = f(λ)I ⊕ (T_{f}, T_{n−1})_{n≥2}.

This is evidently a completely isometric unital endomorphism. However one can restrict α to a map β which takes A(𝔻) to a subalgebra of T by β(f) = T_{f}. The range of β generates T as a C*-algebra, which is non-abelian. Therefore there is no extension of β to a homomorphism of C(𝕋) into T . Thus α does not lift to a ∗-endomorphism of C_{e}^{∗}(A).

If |λ| = 1, we can embed A into

𝔄 = T ⊕ (c ⊗ T )

in the natural way and extend α to the endomorphism

α̃(T ⊕ (T_{n})_{n≥1}) = qT (λ)I ⊕ (T, T_{n−1})_{n≥2} where q is the quotient map of T onto C(𝕋).

However if |λ| < 1, evaluation at λ is not multiplicative on T , so α does not lift to an endomorphism of 𝔄. We can instead let

𝔅 = ℂ ⊕ T ⊕ (c ⊗ T )

and imbed A by

j(f ⊕ (T_{n})_{n≥1}) = f(λ) ⊕ T_{f} ⊕ (T_{n})_{n≥1}.

Clearly j is completely isometric. The C*-algebra generated by j(A(𝔻)) is all of 𝔅 because

j(1) − j(z)^{∗}j(z) = (1 − |λ|^{2}) ⊕ 0

shows that ℂ ⊕ 0 is contained in this algebra.

Observe that evaluation at λ is now a character of 𝔅. Moreover

j(α(f ⊕ (T_{n})_{n≥1})) = j(f(λ)I ⊕ (T_{f}, T_{n−1})_{n≥2})
= f(λ) ⊕ f(λ)I ⊕ (T_{f}, T_{n−1})_{n≥2}.

So we may extend α to α̃ ∈ End(𝔅) by

α̃(w ⊕ T ⊕ (T_{n})_{n≥1}) = w ⊕ wI ⊕ (T, T_{n−1})_{n≥2}.

A modification of Peters’ argument [48, Prop.I.8] shows that we can extend completely isometric endomorphisms to automorphisms of a larger algebra.

Proposition 10.4. If A is a unital operator algebra and α ∈ End(A), then there is a unital operator algebra B containing A as a unital subalgebra and β ∈ Aut(B) such that β|_{A} = α. Moreover, B is the closure of ⋃_{n≥0} β^{−n}(A).

Proof. First observe that the standard orbit representation makes sense for A. Let σ be a completely isometric representation of A on a Hilbert space H so that C^{∗}(σ(A)) ≃ C_{e}^{∗}(A). Form the Hilbert space H̃ = H ⊗ ℓ^{2} and define

π(a) = ∑_{n=0}^{∞} ⊕σ(α^{n}(a)) and V = I ⊗ S,

where S is the unilateral shift. Then it is evident that (π, V) is an isometric covariant representation of (A, α). In particular, π(A) is completely isometric to A. Define the corresponding endomorphims α̃ on π(A) by

α̃(π(a)) = α̃( ∑_{n=0}^{∞} ⊕σ(α^{n}(a)) = ∑_{n=0}^{∞} ⊕σ(α^{n+1}(a)) = π(α(a))

for a ∈ A.

Form the injective system

\xymatrix{π(A) \ar[r]^{α̃} \ar[d]^{α̃} & π(A) \ar[r]^{α̃} \ar[d]^{α̃} & π(A) \ar[r]^{α̃} \ar[d]^{α̃} & … \ar[r] & B \ar[d]^{β} \\ π(A) \ar[r]^{α̃} & π(A) \ar[r]^{α̃} & π(A) \ar[r]^{α̃} & … \ar[r] & B}

Then B is a unital operator algebra containing A as a unital subalgebra and β is a completely isometric endomorphism. However it is evident that β is surjective, so β is an automorphism.

Finally, observe that B is the closure of ⋃_{n≥0} β^{−n}(A).

Now we can use this to lift endomorphisms.

Theorem 10.5. If A is a unital operator algebra and α ∈ End(A), then there is a C*-cover 𝔄 of A and an endomorphisms α̃ of 𝔄 such that α̃|_{A} = α.

Proof. Use Proposition 10.4 to lift α to an automorphism β of a larger algebra B. Then apply Proposition 10.1 to lift it again to an automorphism β̃ of the C*-algebra 𝔅 = C_{e}^{∗}(B). Let 𝔄 be the C*-subalgebra of 𝔅 generated by A. Since β̃|_{A} = α, we see that A is invariant under β̃. Hence so is A^{∗}. Since 𝔄 is generated by A and A^{∗}, it is also invariant under β̃. So α̃ = β̃|_{𝔄} is the desired ∗-endomorphism.

While not all endomorphisms of A lift to C_{e}^{∗}(A), those that do lift behave well when lifted to larger algebras. This simplifies the hypotheses in some of the results in [32] as explained in the next section.

Proposition 10.6. Let A be a unital operator algebra and let 𝔄 be a C*-cover. Suppose that α ∈ End_{A}(C_{e}^{∗}(A)) and that β ∈ End_{A}(𝔄) such that β|_{A} = α|_{A}. Then αq = qβ, and hence β fixes the Shilov ideal 𝔍_{A}, where q is the canonical quotient map of 𝔄 onto C_{e}^{∗}(A).

Proof. Observe that αq and qβ are ∗-homomorphisms of 𝔄 into C_{e}^{∗}(A) which agree on the generating set A. Thus they are equal. Hence

𝔍_{A} = ker αq = ker qβ.

It follows that

𝔍_{A} = {a ∈ 𝔄 : β(a) ∈ 𝔍_{A}} = β^{−1}(𝔍_{A}).

In particular,

β(𝔍_{A}) ⊂ 𝔍_{A}.

Extremal and fully extremal coextensions behave well under automorphisms, but not for endomorphisms.

Proposition 10.7. Let ρ be a representation of a unital operator algebra A with (fully) extremal coextension σ. If α ∈ Aut(A), then σ ∘ α is a (fully) extremal coextension of ρ ∘ α.

Proof. Let ρ act on H and σ act on K ⊃ H. Suppose first that σ is extremal. Since H is coinvariant for σ(A), it is also coinvariant for σ(α(A)). Thus σ ∘ α is a coextension of ρ∘α. Suppose that τ is a coextension of σ ∘α. Then τ ∘α^{−1} is a coextension of σ. Hence it splits as

τ ∘ α^{−1} = σ ⊕ φ.

Thus

τ = σ ∘ α ⊕ φ ∘ α.

So σ ∘ α is extremal.

A similar proof works for fully extremal coextensions of ρ ∘ α.

Example 10.8. Let

A = A(𝔻) ⊕ (c ⊗ C(\overline{𝔻}))

with elements (f, (g_{n})_{n≥1}) where f ∈ A(𝔻), g_{n} ∈ C(\overline{𝔻}) for n ≥ 1 and lim_{n→∞} g_{n} =: g_{0} exists. Define

α(f, (g_{n})_{n≥1}) = (f(0), (f, g_{n−1})_{n≥2}).

This is a completely isometric endomorphism.

A representation of A has the form

ρ(f, (g_{n})_{n≥1}) = f(T) ⊕ ∑_{n≥1}⊕ ρ_{n}(g_{n})

where T is a contraction and ρ_{n} are ∗-representations of C(\overline{𝔻}). It is straightforward to show that a representation of A is an extremal coextension if and only if T is an isometry.

Taking T to be an isometry and ρ_{n} all vacuous (the zero representation on zero dimensional space), we have an extremal coextension ρ such that

ρ ∘ α(f, (g_{n})_{n≥1}) = f(0)I.

Since 0 is not an isometry, this is not extremal.

11 A Review of Semicrossed Products

Nonself-adjoint crossed products were introduced by Arveson [4, 11] as certain concretely represented operator algebras that encoded the action of a subsemigroup of a group acting on a measure space, and later a topological space.

McAsey, Muhly and Sato [37, 36] further analyzed such algebras, again relying on a concrete representation to define the algebra. In [48], Peters gave a more abstract and universal definition of the semicrossed product of a C*-algebra by a single endomorphism. Actually he provides a concrete definition, but then proves that it has the universal property which has become the de facto definition of a semicrossed product.

One can readily extend Peter’s definition of the semicrossed product of a C*- algebra by a ∗-endomorphism to unital operator algebras and unital completely isometric endomorphisms.

Definition 11.1. Let A be a unital operator algebra and α ∈ End(A). A covariant representation of (A, α) is a pair (ρ, T) consisting of a completely contractive representation ρ : A → B(H) and a contraction T ∈ B(H) such that

ρ(a)T = T ρ(α(a)) for all a ∈ A.

The semicrossed product A×_{α}ℤ_{+} is the universal operator algebra for covariant representations. This is the closure of the algebra P(A, 𝔱) of formal polynomials of the form p = ∑_{i=0}^{n} 𝔱^{i}a_{i}, where a_{i} ∈ A, with multiplication determined by

a𝔱 = 𝔱α(a)

and the norm

∥p∥ = sup_{(ρ,T) covariant} ∥ ∑_{i=0}^{n} T^{i}ρ(a_{i})∥.

This supremum is clearly dominated by ∑_{i=0}^{n} ∥a_{i}∥; so this norm is well defined. Since this is the supremum of operator algebra norms, it is also easily seen to be an operator algebra norm. By construction, for each covariant representation (ρ, T), there is a unique completely contractive representation ρ×T of A×_{α} ℤ_{+} into B(H) given by

ρ × T (p) = ∑_{i=0}^{n} T^{i}ρ(a_{i}).

This is the defining property of the semicrossed product.

When 𝔄 is a C*-algebra, the completely isometric endomorphisms are the faithful ∗-endomorphisms. Peters [48] shows that when α is a faithful ∗- endomorphism of 𝔄, the norm of 𝔄 ×_{α} ℤ_{+} can be computed by using orbit representations. Let σ be a faithful ∗-representation of 𝔄 on H. Form the ∗-representation π on H ⊗ ℓ^{2} by

π(a) = ∑_{n=0}^{∞} ⊕ σ(α^{n}(a))

and let V = I ⊗S, where S is the unilateral shift on ℓ^{2}. It is evident that (π, V) is a covariant representation of (𝔄, α). The corresponding representation π ×V of 𝔄 ×_{α} ℤ_{+} is known as the orbit representation of σ.

Theorem 11.2 (Peters). If α is a faithful ∗-endomorphism of a C*-algebra 𝔄, and σ is a faithful ∗-representation of 𝔄, then the orbit representation σ × V provides a completely isometric representation of 𝔄 ×_{α} ℤ_{+}.

Moreover, Peters [48, Prop.I.8] establishes Proposition 10.4 in the case where A is a C*-algebra. It follows [48, Prop.II.4] that 𝔄 ×_{α} ℤ_{+} is completely isometrically isomorphic to the subalgebra of the crossed product algebra 𝔅 ×_{β} ℤ generated as a non-self-adjoint algebra by j(𝔄) and the unitary 𝔲 implementing β in the crossed product. Kakariadis and the second author [32, Thm.2.5] show that this crossed product is the C*-envelope:

Theorem 11.3 (Kakariadis-Katsoulis). Let α be a faithful ∗-endomorphism of a C*-algebra 𝔄 and let (𝔅, β) be the lifting of α to an automorphism described above. Then

C_{e}^{∗}(𝔄 ×_{α} ℤ_{+}) = 𝔅 ×_{β} ℤ.

Since dilation theorems fail in many classical cases, such as commuting triples of contractions [60, 44], one can circumvent the issue by considering only isometric covariant relations. This semicrossed product was introduced by Kakariadis and the second author [32]. The results there have immediate application for us.

Definition 11.4. Let A be an operator algebra and let α ∈ End(A). A covariant representation (ρ, T) of (A, α) is called isometric if ρ is a complete isometry and T is an isometry. The isometric semicrossed product A ×_{α}^{is} ℤ_{+} is the universal operator algebra for isometric covariant representations. This is the closure of the algebra P(A, 𝔱) of formal polynomials of the form p = ∑_{i=0}^{n} 𝔱^{i}a_{i}, where a_{i} ∈ A, under the norm

∥p∥ = sup_{(ρ,T) isometric covariant} ∥ ∑_{i=0}^{n} T^{i}ρ(a_{i})∥.

Theorem 11.5 (Kakariadis-Katsoulis). If A is a unital operator algebra and α ∈ End_{A}(C_{e}^{∗}(A)), then A ×_{α}^{is} ℤ_{+} is (completely isometrically isomorphic to) a subalgebra of C_{e}^{∗}(A) ×_{α} ℤ_{+}. Moreover,

C_{e}^{∗}(A ×_{α}^{is} ℤ_{+}) = C_{e}^{∗}(C_{e}^{∗}(A) ×_{α} ℤ_{+}).

More generally, they consider an arbitrary C*-cover 𝔄 of A. Let J_{A} denote the Shilov boundary, i.e. the kernel of the unique homomorphism of 𝔄 onto C_{e}^{∗}(A) which is the identity on A. Suppose that α ∈ End_{A}(𝔄) also leaves J_{A} invariant. Then it is easy to see that this induces an endomorphism α̇ ∈ End_{A}(C_{e}^{∗}(A)). Hence A ×is ; α̇ ℤ_{+} is (canonically completely isometrically isomorphic to) a subalgebra of C_{e}^{∗}(A) ×α̇ ℤ_{+}. They show that the same norm is also induced on A ×_{α} ℤ_{+} as a subalgebra of 𝔄 ×_{α} ℤ_{+}. This result should be considered in conjunction with Proposition 10.6.

Theorem 11.6 (Kakariadis-Katsoulis). If A is a unital operator algebra with C*-cover 𝔄 and α ∈ End_{A}(𝔄) fixes J_{A}, then A ×_{α}^{is} ℤ_{+} is also (canonically completely isometrically isomorphic to) a subalgebra of 𝔄 ×_{α} ℤ_{+}.

In conjunction with Proposition 10.6, we obtain:

Corollary 11.7. If α ∈ End_{A}(C_{e}^{∗}(A)), 𝔄 is a C*-cover of A and β ∈ End_{A}(𝔄) such that β|_{A} = α|_{A}, then A ×_{β} ℤ_{+} is completely isometrically isomorphic to A ×_{α} ℤ_{+}.

12 Dilating Covariance Relations

We consider the following problem: suppose that α ∈ End(A) lifts to a ∗- endomorphism of C_{e}^{∗}(A). When is A×_{α}ℤ_{+} canonically completely isometrically isomorphic to the subalgebra of C_{e}^{∗}(A) ×_{α} ℤ_{+} generated by A and the element 𝔱 inducing the action α. To simplify statements, we will just say, in this case, that A ×_{α} ℤ_{+} is a subalgebra of 𝔄 ×_{α} ℤ_{+}. We are seeking general properties of A which make this true.

Commutant lifting properties can be seen as special cases of dilation theorems for semicrossed products in the case of the identity automorphism. The goal of this section is to provide several theorems which allow one to obtain results about general semicrossed products from the various commutant lifting properties.

The literature contains a number of dilation theorems for endomorphisms of operator algebras. Ling and Muhly [35] establish an automorphic version of Ando’s theorem, which is a lifting theorem for an action of ℤ_{+}^{2}. Peters [48] and Muhly and Solel [38, 39] dilate actions of ℤ_{+} on C*-algebras. Our first result in this section uses SCLT and models our dilation theorem for the noncommutative disk algebras [17]. We wish to contrast the explicit dilation obtained here with the more inferential one obtained in Theorem 12.3 which requires only CLT.

Theorem 12.1. Suppose that A is a unital operator algebra satisfying SCLT and ICLT. Let α ∈ Aut(A). Then every covariant representation (ρ, T) of (A, α) has a coextension (σ, V) such that σ is a fully extremal coextension of ρ and V is an isometry.

Proof. Let σ be a fully extremal coextension of ρ. Then by Proposition 10.7, σ ∘ α is also fully extremal. By Corollary 3.9,

σ ⊕ (σ ∘ α)

is a fully extremal coextension of ρ ⊕ (ρ ∘ α). Now the covariance relation implies that for a ∈ A,

[\begin{matrix}ρ(a) & 0 \\ 0 & ρ(α(a)) \end{matrix}] [\begin{matrix}0 & T \\ 0 & 0 \end{matrix}] = [\begin{matrix}0 & ρ(a)T \\ 0 & 0 \end{matrix} ] = [\begin{matrix}0 & T \\ 0 & 0 \end{matrix}] [\begin{matrix}ρ(a) & 0 \\ 0 & ρ(α(a)) \end{matrix}]

Using SCLT, we obtain a contractive coextension of [ \begin{matrix}0 & T \\ 0 & 0 \end{matrix} ] which commutes with σ ⊕ (σ ∘ α)(A). The 1, 2 entry is a contractive coextension T_{1} of T such that

σ(a)T_{1} = T_{1}σ(α(a)) for all a ∈ A.

So

[\begin{matrix}0 & T_{1} \\ 0 & 0 \end{matrix}]

also commutes with σ ⊕ (σ ∘ α)(A).

Now use ICLT to coextend this operator to an isometry V commuting with a Shilov coextension τ of σ ⊕ (σ ∘ α). As σ ⊕ (σ ∘ α) is an extremal coextension, τ decomposes as

τ = σ ⊕ (σ ∘ α) ⊕ τ_{1}.

With respect to this decomposition K ⊕ K ⊕ P, we can write

V = [\begin{matrix}0 & T_{1} & 0 \\ 0 & 0 & 0 \\ ∗ & D & ∗ \end{matrix}] .

In particular, the commutation relation shows that

τ_{1}(a)D = D σ(α(a)) for all a ∈ A.

A direct summand of a Shilov extension is Shilov, so τ_{1} is Shilov. Let π_{1} be a maximal representation on L ⊃ P so that P is invariant, and π_{1}|_{P} = τ_{1}. Then consider P_{P}D as an operator in B(K, L), and note that

π_{1}(a)P_{P}D = P_{P}D σ(α(a)) for all a ∈ A.

Define a coextension of ρ by

π = σ ⊕ ∑_{n≥0}^{⊕}π_{1} ∘ α^{n}

acting on M = K ⊕ L^{(∞)}. Since σ is a fully extremal coextension of ρ and each π_{1} ∘ α^{n} is maximal, it follows that π is a fully extremal coextension of ρ.

Now define an isometry W on M by

W = [\begin{matrix}T_{1} & 0 & 0 & 0 & … \\ P_{P}D & 0 & 0 & 0 & … \\ 0 & I & 0 & 0 & … \\ 0 & 0 & I & 0 & … \\ ⋮ & & & ... & ... \end{matrix}] .

Then one readily verifies that

π(a)W = W π(α(a)) for all a ∈ A.

This is the desired isometric coextension of the covariance relations.

Observe that Theorem 12.1 shows that if A has SCLT and ICLT, then

A ×_{α} ℤ_{+} = A ×_{α}^{is} ℤ_{+}.

Thus by applying Theorem 11.5 to see that it imbeds into C_{e}^{∗}(A) ×_{α} ℤ_{+}. Since α is an automorphism, the C*-envelope of this is just the full crossed product C_{e}^{∗}(A) ×_{α} ℤ.

Theorem 12.2. Suppose that A has properties SCLT and ICLT, and α ∈ Aut(A). Then A ×_{α} ℤ_{+} is (canonically completely isometrically isomorphic to) a subalgebra of C_{e}^{∗}(A) ×_{α} ℤ_{+}. Moreover,

C_{e}^{∗}(A ×_{α} ℤ_{+}) = C_{e}^{∗}(A) ×_{α} ℤ.

We wish to improve this theorem so that we require only CLT, not SCLT. This requires a different approach, and does not yield a direct construction of the isometric coextension of a covariant representation.

Theorem 12.3. Suppose that an operator algebra A has the Ando property, and α ∈ Aut(A). Then

A ×_{α} ℤ_{+} = A ×_{α}^{is} ℤ_{+}.

Hence A ×_{α} ℤ_{+} is (canonically completely isometrically isomorphic to)a subalgebra of C_{e}^{∗}(A) ×_{α} ℤ_{+}. Moreover,

C_{e}^{∗}(A ×_{α} ℤ_{+}) = C_{e}^{∗}(A) ×_{α} ℤ.

Proof. Suppose that (ρ, T) is a covariant representation of (A, α). Let σ_{0} be a fully extremal coextension of ρ on a Hilbert space K_{0}. Then by Proposition 10.7, σ_{0} ∘ α is also fully extremal. Theorem 7.15 yields a fully extremal coextension σ_{1} of ρ_{1} on K_{1}, an isometry V_{0} and a maximal dilation π_{0} so that

σ_{1}(a)V_{0} = V_{0}((σ_{0} ∘ α) ⊕ π_{0})(a).

Recursively, we obtain fully extremal coextension σ_{n+1} of ρ_{1} on K_{n}, an isometry V_{n} and a maximal dilation π_{n} so that

σ_{n+1}(a)V_{n} = V_{n}((σ_{n} ∘ α) ⊕ π_{n})(a).

Let

π = ∑_{n≥0} ∑_{k∈ℤ} ⊕(π_{n} ∘ α^{k})^{(∞)}.

Then set σ̃_{n} = σ_{n}⊕π acting on K̃_{n}. Extending V_{n} appropriately to an isometry Ṽ_{n}, we obtain

σ̃_{n+1}(a)Ṽ_{n} = Ṽ_{n}(σ̃_{n} ∘ α)(a).

Now define a representation on

K̃ = ∑_{n≥0} ⊕K̃_{n}

by

σ̃(a) = [\begin{matrix}σ̃_{0}(a) & 0 & 0 & 0 & … \\ 0 & σ̃_{1}(a) & 0 & 0 & … \\ 0 & 0 & σ̃_{2}(a) & 0 & … \\ 0 & 0 & 0 & σ̃_{3}(a) & ... \\ ⋮ & ⋮ & ... & ... & ... \end{matrix}]

and

Ṽ = [\begin{matrix}0 & 0 & 0 & 0 & … \\ Ṽ_{0} & 0 & 0 & 0 & … \\ 0 & Ṽ_{1} & 0 & 0 & … \\ 0 & 0 & Ṽ_{2} & 0 & ... \\ ⋮ & ⋮ & ... & ... & ... \end{matrix}] .

This is an isometric covariant representation which coextends the contractive covariant representation on H ⊗ ℓ^{2}:

ρ^{(∞)}(a) = [\begin{matrix}ρ(a) & 0 & 0 & 0 & … \\ 0 & ρ(a) & 0 & 0 & … \\ 0 & 0 & ρ(a) & 0 & … \\ 0 & 0 & 0 & ρ(a) & ... \\ ⋮ & ⋮ & ... & ... & ... \end{matrix}]

and

T ⊗ S = [\begin{matrix}0 & 0 & 0 & 0 & … \\ T & 0 & 0 & 0 & … \\ 0 & T & 0 & 0 & … \\ 0 & 0 & T & 0 & ... \\ ⋮ & ⋮ & ... & ... & ... \end{matrix}] .

This latter representation induces the same seminorm on A ×_{α} ℤ_{+} as the covariant pair (ρ^{(∞)}, T ⊗ U) on H ⊗ ℓ^{2}(ℤ), where U is the bilateral shift, because this representation is an inductive limit of copies of (ρ^{(∞)}, T ⊗ S). However (ρ^{(∞)}, T ⊗ U) has a seminorm which clearly dominates the seminorm arising from (ρ, T).

It follows that

A ×_{α} ℤ_{+} = A ×_{α}^{is} ℤ_{+}.

The rest follows as in Theorem 12.2.

In particular, one gets a dilation theorem that we cannot find by a direct construction. Indeed, even the isometric dilation is not explicitly obtained, unlike the proof of Theorem 12.1. This result considerably expands the class of operator algebras for which we can obtain these results.

Corollary 12.4. Suppose that an operator algebra A has the Ando property, and α ∈ Aut(A). Then every covariant representation (ρ, T) of (A, α) dilates to a covariant representation (π, U) of (C_{e}^{∗}(A), α) where π is a ∗-representation of C_{e}^{∗}(A) and U is unitary.

The following result applies to endomorphisms, not just automorphisms. This result was only recently established for the disk algebra [18]. The hypotheses are quite strong.

Theorem 12.5. Suppose that A is a unital operator algebra with SMCLT and FP. Let α ∈ End_{A}(C_{e}^{∗}(A)). Then A ×_{α} ℤ_{+} is (canonically completely isometrically isomorphic to) a subalgebra of C_{e}^{∗}(A) ×_{α} ℤ_{+}.

Proof. To establish that A ×_{α} ℤ_{+} is completely isometric to a subalgebra of C_{e}^{∗}(A) ×_{α} ℤ_{+}, it suffice to show that if (ρ, T) is a covariant representation of (A, α), then ρ has a ∗-dilation π of C_{e}^{∗}(A) on a Hilbert space K ⊃ H and a contraction S dilating T to K such that (π, S) is a covariant representation of (C_{e}^{∗}(A), α) with H as a coinvariant subspace. For this then shows that

∥ρ × T (p)∥ ≤ ∥π × S(p)∥ ≤ ∥p∥_{C_{e}^{∗}(A)×_{α}ℤ_{+}} .

The reverse inequality is evident. Hence the norm on A ×_{α} ℤ_{+} will coincide with the norm as a subalgebra of C_{e}^{∗}(A) ×_{α} ℤ_{+}.

First dilate ρ to a maximal dilation π. This extends to a ∗-representation of C_{e}^{∗}(A) which we also denote by π. We may write:

π(a) = [\begin{matrix}∗ & 0 & 0 \\ ∗ & ρ(a) & 0 \\ ∗ & ∗ & ∗ \end{matrix}] for a ∈ A.

Observe that the covariance condition is equivalent to

[\begin{matrix}ρ(a) & 0 \\ 0 & ρ(α(a)) \end{matrix}] [\begin{matrix}0 & T \\ 0 & 0 \end{matrix}] = [\begin{matrix}0 & T \\ 0 & 0 \end{matrix}] [\begin{matrix}ρ(a) & 0 \\ 0 & ρ(α(a)) \end{matrix}] .

Now ρ ⊕ (ρ ∘ α) dilates to a ∗-dilation

π ⊕ (π ∘ α).

So by SMCLT, there is a contraction dilating

[\begin{matrix}0 & T \\ 0 & 0 \end{matrix}]

which commutes with (π ⊕ (π ∘ α))(A) of the form

S̃ = [\begin{matrix}∗ & S \\ ∗ & ∗ \end{matrix}]

so that H ⊕ H is simultaneously semi-invariant for (π ⊕ π ∘ α)(A) and S̃. There is no loss in assuming that the ∗ entries are all 0. Commutation again means that (π, S) is a covariant representation of A.

Now by the Fuglede property, the adjoint

[\begin{matrix}0 & 0 \\ S^{∗} & 0 \end{matrix}]

also commutes with (π ⊕ (π ∘ α))(A). That means that

π(α(a))S^{∗} = S^{∗}π(a) for all a ∈ A.

Equivalently, since π is a ∗-representation,

π(a^{∗})S = Sπ(α(a^{∗})) for all a ∈ A.

But the set of operators in C_{e}^{∗}(A) satisfying the covariance relations is a closed algebra, and contains A and A^{∗}, whence it is all of C_{e}^{∗}(A). Thus we have obtained the desired dilation to covariance relations for (C_{e}^{∗}(A), α). Hence A ×_{α} ℤ_{+} is (canonically completely isometrically isomorphic to) a subalgebra of C_{e}^{∗}(A) ×_{α} ℤ_{+}.

The following is immediate from the dilation theory for C_{e}^{∗}(A) ×_{α} ℤ_{+} and Theorem 11.5. Combining this with Theorem 11.3, one obtains an explicit description of this C*-envelope.

Corollary 12.6. Suppose that a unital operator algebra A has FP and SM-CLT, and α ∈ End_{A}(C_{e}^{∗}(A)). Then every covariant representation (ρ, T) of (A, α) dilates to a covariant representation (π, U) of (C_{e}^{∗}(A), α) where π is a ∗-representation of C_{e}^{∗}(A) and U is unitary. Moreover,

C_{e}^{∗}(A ×_{α} ℤ_{+}) = C_{e}^{∗}(C_{e}^{∗}(A) ×_{α} ℤ_{+}).

13 Further Examples

The disk algebra. The first application yields a recent result about semicrossed products by completely isometric endomorphisms for the disk algebra [18]. We note that endomorphisms which are not completely isometric are also treated there, but our results do not apply in that case.

The C*-envelope of the disk algebra A(𝔻) is C(𝕋), which is generated by the unitary element z. The classical Fuglede Theorem shows that A(𝔻) has FP. Also the classical Sz.Nagy–Foiaş Commutant Lifting Theorem yields the properties SCLT and SMCLT. Ando’s property is Corollary 7.11, which was a strengthening of Ando’s theorem. As A(𝔻) is Dirichlet, we have uniqueness of extremal coextensions and of extremal extensions, which are also consequences of the original Sz.Nagy theory. As A(𝔻) ≃ A(𝔻)^{∗}, we have SCLT* as well. Indeed, A(𝔻) has all of the properties studied in this paper.

Suppose that α ∈ End_{A(𝔻)} (C(𝕋)). Then b = α(z) ∈ A(𝔻); and has spectrum

σ_{A(𝔻)} (b) = σ_{A(𝔻)} (z) = \overline{𝔻} and σ_{C(𝕋)} (b) = σ_{C(𝕋)} (z) = 𝕋.

Thus Ran(b) = \overline{𝔻} and Ran(b|_{𝕋} ) = 𝕋. It follows that b is a non-constant finite Blaschke product. We have α(f) = f ∘ b for all f ∈ C(𝕋).

Theorem 13.1. Let b be a non-constant finite Blaschke product, and let α(f) = f ∘ b in End_{A(𝔻)} (C(𝕋)). Then A(𝔻) ×_{α} ℤ_{+} is (canonically completely isometrically isomorphic to) a subalgebra of C(𝕋) ×_{α} ℤ_{+}; and

C_{e}^{∗}(A(𝔻) ×_{α} ℤ_{+}) = C_{e}^{∗}(C(𝕋) ×_{α} ℤ_{+}).

This is explicitly described as C(S_{α}) ×_{β} ℤ where S_{α} is the solenoid

S_{α} = {(z_{n})_{n≥1} : z_{n} = b(z_{n+1}), z_{n} ∈ 𝕋, n ≥ 1}

and β is the backword shift on S_{α}.

Proof. The first statement follows from Theorems 12.5. The detailed description of the C*-envelope comes from the Kakariadis-Katsoulis Theorem 11.3.

It is worth restating this theorem as a dilation result.

Corollary 13.2. Let b be a non-constant finite Blaschke product and suppose that S and T are contractions satisfying ST = T b(S). Then there exist unitary operators U and V, dilating S and T respectively, so that UV = V b(U).

The non-commutative disk algebras. For n ≥ 2 finite, the noncommutative disk algebra 𝔄_{n} has the Cuntz algebra O_{n} as its C*-envelope. The Frazho-Bunce-Popescu dilation theorem [30, 13, 50] shows that the minimal extemal coextension of a representation is unique. This also follows because 𝔄_{n} is semi-Dirichlet. Popescu [52] proves the SCLT property in a similar manner to the original proof of Sz.Nagy and Foiaş. The FP property follows from Proposition 9.3.

There are many distinct ways to extend the left regular representation to a maximal representation (see [23, §3]). In particular, the minimal fully extremal extensions are not unique. Nevertheless, 𝔄_{n} has ICLT and MCLT. This follows from our paper [17] specialized to the identity automorphism.

The completely isometric automorphisms of 𝔄_{n} are the analogues of the conformal automorphisms of the ball 𝔹_{n} of ℂ^{n}. These were first described by Voiculescu [61] as ∗-automorphisms of O_{n} which fix the analytic part. These are the only such automorphisms of 𝔄_{n} [24]. See also [53]. Thus we recover our results on semicrossed products of 𝔄_{n} in [17] as a consequence of Theorem 12.2.

Theorem 13.3 (Davidson-Katsoulis). If α ∈ Aut(𝔄_{n}) = Aut_{𝔄_{n}}(O_{n}), then

C_{e}^{∗}(𝔄_{n} ×_{α} ℤ_{+}) = O_{n} ×_{α} ℤ.

It is also easy to determine End(O_{n}). Every n-tuple of isometries 𝔱_{i} ∈ O_{n} such that ∑_{i=1}^{n} 𝔱_{i}𝔱_{i}^{∗} = 1 determines an endomorphism with α(𝔰_{i}) = 𝔱_{i} by the universal property of the Cuntz algebra. For the endomorphism α to leave 𝔄_{n} invariant, it is then necessary and sufficient that 𝔱_{i} belong to 𝔄_{n}. Given that End(O_{n}) is so rich, the following result seems surprising.

Theorem 13.4. For n ≥ 2 finite,

End_{𝔄_{n}}(O_{n}) = Aut_{𝔄_{n}}(O_{n}).

Proof. We represent 𝔄_{n} on ℓ^{2}(𝔽_{n}^{+}) by the left regular representation λ with generators L_{i} = λ(𝔰_{i}), where

L_{i}ξ_{w} = ξ_{iw}.

Note that

C^{∗}(λ(𝔄_{n})) = E_{n}

is the Cuntz-Toeplitz algebra, and that

q : E_{n} → E_{n}/𝔎 = O_{n}

is the quotient by the compact operators. Let ℜ_{n} denote the wot-closed right regular representation algebra generated by R_{i}, 1 ≤ i ≤ n, where

R_{i}ξ_{w} = ξ_{wi}.

Then λ(𝔄_{n})′ = ℜ_{n} [3, 23]. We use the notation R_{v}ξ_{w} = ξ_{wv} for words v ∈ 𝔽_{n}^{+}.

Suppose that α ∈ End_{𝔄_{n}}(O_{n}). Then 𝔱_{i} = α(𝔰_{i}) are isometries in 𝔄_{n} such that

∑_{i=1}^{n} 𝔱_{i}𝔱_{i}^{∗} = 1.

We want to clarify when this is possible. Let

T_{i} = λ(𝔱_{i}) and T = [T_{1} … T_{n}].

Then q(T_{i}) = 𝔱_{i}, and thus T is an essential isometry, as are each T_{i}. However since q is a complete isometry on 𝔄_{n}, we have ∥T ∥ = 1.

We claim that each T_{i} is an isometry. Indeed, if ζ ∈ ℓ^{2}(𝔽_{n}^{+}) with ∥ζ∥ = 1 and ∥T_{i}ζ∥ ≠ 1, then

∥T_{i}(R_{v}ζ)∥ = ∥R_{v}T_{i}ζ∥ = ∥T_{i}ζ∥.

Since R_{1}^{n}ζ tends to 0 weakly, we see that T_{i} is not an essential isometry, contrary to fact. Since ∥T ∥ = 1, the T_{1}, …, T_{n} are isometries in λ(𝔄_{n}) with pairwise orthogonal range. So T is a row isometry. Since

q(∑_{i=1}^{n} T_{i}T_{i}^{∗}) = 1,

we deduce that

P = I − ∑_{i=1}^{n} T_{i}T_{i}^{∗}

is a finite rank projection.

The range of each T_{i} is a cyclic invariant subspace for ℜ_{n}, with cyclic vector ζ_{i} = T_{i}ξ_{∅}. Let N = Ran P. Then N^{⊥} is the sum of the ranges of the T_{i}, and so it is invariant for ℜ_{n} with wandering space

W = span{ζ_{i} : 1 ≤ i ≤ n}.

Thus N is coinvariant. Let

A_{i} = P_{N}R_{i}|_{N} and A = [A_{1} … A_{n}].

Then A is a row contraction with a row isometric dilation

R = [R_{1} … R_{n}].

The minimal row isometric dilation is unique [50], and any other is the direct sum of the minimal dilation with another row isometry. Since R is irreducible, this is the minimal dilation of A.

By [20], the wandering space W of N^{⊥} is given by

W = (N + ∑_{i=1}^{n} R_{i}N) ⊖ N.

Note that

I_{N} − ∑_{i=1}^{n} A_{i}A_{i}^{∗} = P_{N}(I − ∑_{i=1}^{n} R_{i}R_{i}^{∗})|_{N} = (P_{N}ξ_{∅})(P_{N}ξ_{∅})^{∗}.

This is non-zero because if N were orthogonal to ξ_{∅}, then ξ_{∅} would also be orthogonal to the invariant subspace it generates, which is the whole space. Thus N is not contained in ∑_{i=1}^{n} R_{i}N because this space is orthogonal to ξ_{∅}. So now we compute

n = dim W = dim(N + ∑_{i=1}^{n} R_{i}N) − dim N
≥ (1 + n dim N) − dim N
= 1 + (n − 1) dim N.

Therefore dim N ≤ 1; whence dim N = 1 because no n-tuple of isometries T_{i} in 𝔏_{n} is of Cuntz type.

The only coinvariant subspaces of dimension one are ℂν_{λ}, where ν_{λ} is an eigenvector of ℜ_{n}^{∗} [3, 23]. These are indexed by points λ in the open unit ball 𝔹_{n} of ℂ^{n}. It now follows from the analysis in [24] that α is an automorphism. Briefly, one can compose α with an automorphism θ_{λ} so that λ = 0 and so N = ℂξ_{∅}. Then

W = span{ξ_{i} : 1 ≤ i ≤ n}.

So {ζ_{i}} form an orthonormal basis for W. The unitary U ∈ U_{n} which takes ξ_{i} to ζ_{i} induces a gauge unitary Ũ which takes each L_{i} to T_{i}, as this is the unique element of 𝔄_{n} with T_{i}ξ_{∅} = ζ_{i}. Hence θ_{λ}α = ad Ũ is an automorphism; whence so is α.

Finite dimensional nest algebras. A finite dimensional nest algebra can be described as the block upper triangular matrices with respect to a decomposition H = H_{1} ⊕ ⋯ ⊕ H_{k} of a finite dimensional Hilbert space into a direct sum of subspaces. These are the incidence algebras which are Dirichlet. They have SCLT, SCLT*, ICLT, ICLT*, MCLT and the Ando property. By Example 9.9, finite dimensional nest algebras have FP.

The only isometric endomorphisms are isometric automorphisms. These are unitarily implemented, and the unitary preserves the nest. (Ringrose [54] characterizes the isomorphisms between nest algebras in infinite dimensions, and includes the more elementary finite dimensional case. See [14].) Hence the unitary has the form U = U_{1} ⊕ ⋯ ⊕ U_{k} with respect to the decomposition of H. Clearly ad U extends to a ∗-automorphism of the C*-envelope B(H) ≃ 𝔐_{n}, where n = dim H.

Graph Algebras and Tensor algebras of C*-correspondences. The tensor algebra T^{+}(E) of a C*-correspondence E over a C*-algebra 𝔄 is semi-Dirichlet. Thus every Shilov coextension of a representation ρ, and in particular every extremal coextension of ρ, is fully extremal; and the minimal extremal coextension of ρ unique. So in particular, the minimal fully extremal coextension is a cyclic coextension. Muhly and Solel [42] show that the tensor algebra of a C*-correspondence has SCLT. The C*-envelope is the Cuntz-Pimsner algebra O(E) [42, 29, 34]. Kakariadis and Katsoulis [32] establish that for every α ∈ Aut_{T ^{+}(E)}(O(E)) such that α|_{𝔄} = id, the semi-crossed product T^{+}(E) ⊗_{α} ℤ_{+} imbeds canonically, completely isometically as a subalgebra of O(E) ⊗_{α} ℤ; and this is its C*-envelope. In particular, taking α = id, one obtains the Ando property, so it has ICLT and SCLT.

Thus, by circular reasoning, Theorems 12.1 and 12.2 apply. The point however is that the dilation theorems for automorphisms follow immediately once one has the appropriate commutant lifting theorems, which basically deal with the identity automorphism. In principle, and often in practice, this is much easier.

An important special case of a tensor algebra is the tensor algebra T^{+}(G) of a directed graph G. Some of the properties are somewhat easier to see here. In addition, by Example 9.7, finite graph algebras have FP.

Bilateral Tree Algebras. In the case of a bilateral tree algebra A, one readily sees that C_{e}^{∗}(A) is a direct sum of full matrix algebras corresponding to the connected components of the graph. The automorphisms of finite dimensional C*-algebras are well understood. Modulo inner automorphisms, one can only permute subalgebras of the same size. Automorphisms of the tree algebra are more restrictive, and modulo those inner automorphisms from unitaries in A ∩ A^{∗}, they come from automorphisms of the associated directed graph.

Bilateral tree incidence algebras have the Ando property by Theorem 8.6. Hence by Theorem 12.3, we obtain:

Corollary 13.5. Let A be a bilateral tree algebra, and let α ∈ Aut(A). Then

C_{e}^{∗}(A ×_{α} ℤ_{+}) = C_{e}^{∗}(C_{e}^{∗}(A) ×_{α} ℤ_{+}).

Hence if (ρ, T) is a covariant representation of (A, α), there is a maximal dilation π of ρ and a unitary dilation U of T so that (π, U) is a covariant representation of (C_{e}^{∗}(A), α).

It is an interesting question to look at the infinite dimensional WOT-closed versions. A commutative subspace lattice (CSL) is a strongly closed lattice of commuting projections. A CSL algebra is a reflexive algebra whose invariant subspace lattice is a CSL. Since every CSL is contained in a masa, one can instead define a CSL algebra to be a reflexive algebra containing a masa. The seminal paper, which provides a detailed structure theory for these algebras, is due to Arveson [7]. See also [14].

When dealing with weak-∗ closed operator algebras, the class of all representations is generally too large. Instead one restrict attention to weak-∗ continuous (completely contractive) representations. To apply the results from this paper, a weak-∗ version needs to be developed.

A CSL algebra is a bilateral tree algebra if the lattice satisfies an measure theoretic version of the discrete bilateral tree property. We will not define this precisely here, but refer the reader to [22] for the full story. The approximation results from [22] show that every bilateral tree algebra can be approximated in two very strong ways by a sequence of finite dimensional subalgebras which are completely isometrically isomorphic to bilateral tree incidence algebras. These results should be a crucial step towards deducing similar dilation results for semicrossed products of these infinite dimensional bilateral tree algebras by weak-∗ continuous endomorphisms.

References

[1] J. Agler, An abstract approach to model theory, in Surveys of some recent results in operator theory, Vol. II, pp. 1–23, Longman Sci. Tech., Harlow, 1988.

[2] T. Ando, On a pair of commuting contractions, Acta Sci. Math. (Szeged) 24 (1963), 88–90.

[3] A. Arias and G. Popescu, Factorization and reflexivity on Fock spaces, Integral Equations and Operator Theory 23 (1995), 268–286.

[4] W. Arveson, Operator algebras and measure preserving automorphisms, Acta Math. 118, (1967), 95–109.

[5] W. Arveson, Subalgebras of C*-algebras, Acta Math. 123 (1969), 141–224.

[6] W. Arveson, Subalgebras of C*-algebras II, Acta Math. 128 (1972), 271– 308.

[7] W. Arveson, Operator algebras and invariant subspaces, Ann. Math. 100 (1974), 433–532.

[8] W. Arveson, Subalgebras of C*-algebras III, Acta Math. 181 (1998), 159– 228.

[9] W. Arveson, The curvature invariant of a Hilbert module over ℂ[z_{1}, …, z_{d}], J. Reine Angew. Math. 522 (2000), 173–236.

[10] W. Arveson, The noncommutative Choquet boundary, J. Amer. Math. Soc. 21 (2008), 1065–1084.

[11] W. Arveson and K. Josephson, Operator algebras and measure preserving automorphisms II, J. Funct. Anal. 4 (1969), 100–134.

[12] D. Blecher, Z.-J. Ruan and A. Sinclair, A characterization of operator algebras, J. Funct. Anal. 89 (1990), 188–201.

[13] J. Bunce, Models for n-tuples of non-commuting operators, J. Funct. Anal. 57 (1984), 21–30.

[14] K.R. Davidson, Nest Algebras, Pitman Research Notes in Mathematics Series 191, Longman Scientific and Technical Pub. Co., London, New York, 1988.

[15] K.R. Davidson, When locally contractive representations are completely contractive, J. Funct. Anal. 128 (1995), 186–225.

[16] K.R. Davidson and E. Katsoulis, Operator algebras for multivariable dynamics, Memoirs Amer. Math. Soc. 209 (2011), no. 982, 53+viii pages.

[17] K.R. Davidson and E. Katsoulis, Dilating covariant representations of the non-commutative disc algebras, J. Funct. Anal. 259 (2010), Pages 817–831.

[18] K.R. Davidson and E. Katsoulis, Semicrossed products of the disk algebra, Proc. Amer. Math. Soc., to appear (arXiv:1104.1398).

[19] K.R. Davidson, E. Katsoulis and D.R. Pitts, The structure of free semigroup algebras, J. reine angew. Math. 533 (2001), 99–125.

[20] K.R. Davidson, D.W. Kribs and M.E. Shpigel, Isometric Dilations of noncommuting finite rank n-tuples, Canad. J. Math. 53 (2001), 506–545.

[21] K.R. Davidson and Trieu Le, Commutant lifting for commuting row contractions, Bull. London Math. Soc. 42 (2010), 506–516.

[22] K.R. Davidson, V.I. Paulsen, and S.C. Power, Tree algebras, semidiscreteness, and dilation theory, Proc. London Math. Soc.(3) 68 (1994), 178–202.

[23] K.R. Davidson and D.R. Pitts, Invariant subspaces and hyper-reflexivity for free semi-group algebras, Proc. London Math. Soc. 78 (1999), 401–430.

[24] K.R. Davidson and D.R. Pitts, The algebraic structure of non-commutative analytic Toeplitz algebras, Math Ann. 311 (1998), 275–303.

[25] K.R. Davidson and J. Roydor, C*-envelopes of tensor algebras for multivariable dynamics, Proc. Edinburgh Math. Soc. 53 (2010), 333–351.

[26] R.G. Douglas and V.I. Paulsen, Hilbert modules over function algebras, Pitman Research Notes in Math. 217, Longman Scientific & Technical, Harlow, 1989.

[27] M. Dritschel and S. McCullough, Boundary representations for families of representations of operator algebras and spaces, J. Operator Theory 53 (2005), 159–167

[28] S. Drury, A generalization of von Neumann’s inequality to the complex ball, Proc. Amer. Math. Soc. 68 (1978), 300–304.

[29] N. Fowler, P. Muhly and I. Raeburn, Representations of Cuntz-Pimsner algebras, Indiana Univ. Math. J. 52 (2003), 569–605.

[30] A. Frazho, Models for non-commuting operators, J. Funct. Anal. 48 (1982), 1–11.

[31] M. Hamana, Injective envelopes of operator systems, Publ. Res. Inst. Math. Sci. 15 (1979), 773-785.

[32] E. Kakariadis and E. Katsoulis, Semicrossed products of operator algebras and their C*-envelopes, preprint (arXiv:1101.1482).

[33] E. Katsoulis and D. Kribs, The C*-envelope of the tensor algebra of a directed graph, Integral Equations Operator Theory 56 (2006), 401–414.

[34] E. Katsoulis and D. Kribs, Tensor algebras of C*-correspondences and their C*-envelopes, J. Funct. Anal. 234 (2006), 226–233.

[35] K. Linag and P. Muhly, An automorphic form of Ando’s theorem, Integral Equations Operator Theory 12 (1989), 424–434.

[36] M. McAsey and P. Muhly, Representations of nonself-adjoint crossed products, Proc. London Math. Soc. 47(1983), 128–144.

[37] M. McAsey,P. Muhly and K. Sato, Nonself-adjoint crossed products (invariant subspaces and maximality), Trans. Amer. Math. Soc. 248(1979), 381–409.

[38] P. Muhly and B. Solel, A lifting theorem for analytic crossed products, Houston J. Math. 18 (1992), 153–166.

[39] P. Muhly and B. Solel, Extensions and dilations for C?-dynamical systems, Operator theory, operator algebras, and applications, 375–381, Contemp. Math. 414, Amer. Math. Soc., Providence, RI, 2006.

[40] P. Muhly and B. Solel, Hilbert modules over operator algebras, Mem. Amer. Math. Soc. 117 (1995), no. 559.

[41] P. Muhly and B. Solel, An algebraic characterization of boundary representations, Nonselfadjoint operator algebras, operator theory, and related topics, 189–196, Oper. Theory Adv. Appl. 104, Birkhäuser, Basel, 1998.

[42] P. Muhly and B. Solel, Tensor algebras over C*-correspondences (Representations, dilations and C*-envelopes), J. Funct. Anal. 158 (1998), 389– 457.

[43] V. Müller and F. Vasilescu, Standard models for some commuting multioperators, Proc. Amer. Math. Soc. 117 (1993), 979–989.

[44] S. Parrott, Unitary dilations for commuting contractions, Pacific J. Math. 34 (1970), 481–490.

[45] V. Paulsen, Completely bounded maps and operator algebras, Cambridge Studies in Advanced Mathematics 78, Cambridge University Press, Cambridge, (2002), xii+300 pp.

[46] V. Paulsen and S. Power, Lifting theorems for nest algebras, J. Operator Theory 20 (1988), 311–327.

[47] V. Paulsen and S. Power, Tensor products of non-self-adjoint operator algebras, Rocky Mountain J. Math. 20 (1990), 331–350.

[48] J. Peters, Semicrossed products of C*-algebras, J. Funct. Anal. 59 (1984), 498–534.

[49] J. Peters, The C*-envelope of a semicrossed product and nest representations, in Operator Structures and Dynamical Systems, M. de Jeu, S. Silvestrov, C. Skau, and J. Tomiyama, eds., Contemporary Math. 503 (2009), pp. 197–215.

[50] G. Popescu, Isometric dilations for infinite sequences of noncommuting operators, Trans. Amer. Math. Soc. 316 (1989), 523–536.

[51] G. Popescu, von Neumann inequality for (B(H)^{n})_{1}, Math. Scand. 68 (1991), 292–304.

[52] G. Popescu, On intertwining dilations for sequences of noncommuting operators, J. Math. Anal. Appl. 167 (1992), 382–402.

[53] G. Popescu, Free holomorphic functions on the unit ball of 𝔹_{n}, J. Funct. Anal. 241 (2006), 268–333.

[54] J. Ringrose, On some algebras of operators II, Proc. London Math. Soc. (3) 16 (1966), 385–402.

[55] D. Sarason, Invariant subspaces and unstarred operator algebras, Pacific J. Math. 17 (1966), 511–517.

[56] B. Solel, Representations of product systems over semigroups and dilations of commuting CP maps, J. Funct. Anal., 235(2006). 593–618.

[57] Sz. Nagy, B., Sur les contractions de l’espace de Hilbert, Acta Sci. Math. (Szeged) 15 (1953), 87–92.

[58] B. Sz. Nagy and C. Foiaş, Harmonic analysis of operators on Hilbert space, North Holland Pub. Co., London, 1970.

[59] B. Sz. Nagy and C. Foiaş, Dilatation des commutants d’operateurs, Comptes Rendus Paris 266 (1968), A493–495.

[60] N. Varopoulos, On an inequality of von Neumann and an application of the metric theory of tensor products to operators theory, J. Funct. Anal. 16 (1974), 83–100.

[61] D. Voiculescu, Symmetries of some reduced free product C*-algebras, Operator algebras and their connections with topology and ergodic theory (Busteni, 1983), Lecture Notes in Math. 1132 (1985), 556–588, Springer Verlag, New York, 1985.

Kenneth R. Davidson

Pure Mathematics Dept.

University of Waterloo

Waterloo, ON N2L–3G1

CANADA

krdavids@uwaterloo.ca Elias G. Katsoulis

Department of Mathematics

University of Athens

15784 Athens

GREECE

Alternate address:

Department of Mathematics

East Carolina University

Greenville, NC 27858

USA

katsoulise@ecu.edu