Maximality of correspondence representations

A dilation of an operator $T$ on a Hilbert space $H$ is an operator $S$ on a larger Hilbert space $K\supset H$ such that $T$ is a corner of $S$, often represented in block-diagonal form as:

The general paradigm of dilation theory is to study properties of operators by finding a dilation that exhibits features that are easier to work with. This concept is best illustrated by classical results such as Sz.-Nagy’s unitary dilation of a contraction [50], Andô’s dilation theorem [3], which shows that a pair of commuting contractions can be dilated to a pair of commuting unitaries, and Stinespring’s dilation [49] of a unital completely positive map on a C*-algebra to a *-homomorphism. The theory is now flourishing [8, 12, 18, 48, 22, 19, 53, 44] and has many applications. For a comprehensive introduction and overview, see [47].

This paper studies dilation theory for operator systems associated with C*-correspondences. We first define what operator systems and dilations mean in this context to set the stage.

An operator system $\mathcal{S}$ is a self-adjoint unital subspace of a C*-algebra $B$. We additionally require that $\mathcal{S}$ generates $B$ as a C*-algebra. A unital linear map $\varphi\colon\mathcal{S}\to\mathcal{B}(H)$ is called unital completely positive (u.c.p. map) if it sends positive $n$ by $n$ matrices with entries in $\mathcal{S}$ to positive operators on $H^{\oplus n}$. A dilation of $\varphi$ is a u.c.p. map $\psi\colon\mathcal{S}\to\mathcal{B}(K)$ together with an isometric embedding $V\colon H\hookrightarrow K$ such that

The dilation is called trivial if $\psi=\varphi\oplus\psi^{\prime}$ under the identification of $H$ with $VH$. A u.c.p. map $\varphi$ is said to be maximal if it admits no non-trivial dilations.

Building on the ideas from Agler [1] and Muhly and Solel [37], Dritschel and McCullough demonstrated that maximal u.c.p. maps can be characterized by the unique extension property (UEP) defined by Arveson [7, 4]. By definition, a u.c.p. map $\varphi\colon\mathcal{S}\to\mathcal{B}(H)$ satisfies the UEP if there is a *-representation $\rho\colon B\to\mathcal{B}(H)$ such that $\varphi=\rho|_{\mathcal{S}}$ and $\rho$ is the unique u.c.p. extension of $\varphi$. We say that a representation $\rho$ of $B$ is maximal (with respect to $\mathcal{S}$) if $\rho|_{\mathcal{S}}$ is a maximal u.c.p. map. By the above, there is a bijective correspondence between maximal u.c.p. maps and maximal $B$-representations.

According to the theorem of Dritschel and McCullough [24, Theorem 2.1], every u.c.p. map has a maximal dilation. Dritschel and McCullough used maximality to provide the first dilation-theoretic proof of the existence of the C*-envelope $C^{*}_{env}(\mathcal{S})$, which is the smallest quotient of $B$ such that the quotient map is completely isometric on $\mathcal{S}$. As they showed in [24, Theorem 4.1], maximal representations factor through $C^{*}_{env}(\mathcal{S})$, and there is a maximal representation that is faithful on $C^{*}_{env}(\mathcal{S})$.

However, not every representation of $C^{*}_{env}(\mathcal{S})$ is maximal. To better understand this phenomenon, Arveson [5] defined the noncommutative Choquet boundary $\partial_{C}\mathcal{S}\subset\widehat{C^{*}_{env}(\mathcal{S})}$ as the subset of unitary equivalence classes of maximal irreducible representations. Elements of $\partial_{C}\mathcal{S}$ are called boundary representations. Arveson [5] in the separable case and Davidson-Kennedy [21] in general proved that every operator system has sufficiently many boundary representations.

In [6] Arveson introduced the notion of hyperrigidity of operator systems and gave several equivalent definitions. The most relevant definition for us is that an operator system is hyperrigid if all representations of $C^{*}_{env}(\mathcal{S})$ are maximal. In an attempt to connect this notion with the noncommutative Choquet boundary, Arveson conjectured [6, Conjecture 4.3] that if $\partial_{C}\mathcal{S}=\widehat{C^{*}_{env}(\mathcal{S})}$, i.e., all irreducible representations of the C*-envelope are boundary representations, then the operator system is hyperrigid. This has come to be known as Arveson’s hyperrigidity conjecture, and it has received significant attention in the literature [21, 33, 14, 13, 15, 17, 20, 27, 34, 30, 51, 42, 28]. Recently, an elementary Type I counterexample was found by the author and Dor-On [9]. However, it is still interesting to determine for which classes of operator systems the conjecture holds. For example, the conjecture remains open for function systems in commutative C*-algebras.

In the current paper, we study the topics described above in the context of a C*-correspondence $X$ over a C*-algebra $A$. There are two natural operator systems associated with $X$ in the Cuntz-Toeplitz algebra $\mathcal{T}_{X}$. The first one is $\mathcal{T}_{X}^{+}+(\mathcal{T}_{X}^{+})^{*}$ for the non-self-adjoint tensor algebra $\mathcal{T}_{X}^{+}$ generated by $A$ and $X$. This operator algebra and the associated dilation theory were studied by Muhly and Solel [39] and by Katsoulis and Ramsey [30, 31]. Another operator system is $\mathcal{S}_{X}=X^{*}+A+X\subset\mathcal{T}_{X}$ which was studied by Kim [34]. Moreover, following Muhly and Solel [39], we give a definition of dilations and maximality for representations of $X$ without any reference to operator systems in Definition 2.6. We thus have, a priori, three notions of maximality for isometric representations of $X$. Fortunately, they all coincide (Proposition 2.8).

We study dilation theory for correspondences in two steps. First, we analyze the case when $X$ is a W*-correspondence over a von Neumann algebra $\mathfrak{M}$. It turns out that the dilations in this case have a much more amenable structure. Maximality is then detected by a spatial condition involving the support projection $P_{X}\in\mathfrak{M}$ of $X$ as follows: a representation $t\colon X\to\mathcal{B}(H)$ is maximal if and only if $\overline{t(X)H}=\rho(P_{X})H$. We call this property the full Cuntz-Pimsner covariance since it generalizes the notion of full Cuntz-Krieger families by Dor-On and Salomon [23] in the context of graph operator algebras.

After dealing with W*-correspondences, we reduce the problem for C*-correspondences to the W*-case by using second dual techniques (see Section 3). If $X$ is a C*-correspondence over a C*-algebra $A$, then ${A}^{**}$ is the universal von Neumann algebra of $A$, and ${X}^{**}$ is a W*-correspondence over it (see [11, Section 8.5]). We prove that the representations of $X$ are in canonical bijection with the normal representations of ${X}^{**}$ and that this bijection respects dilations (Proposition 3.3). Therefore, the notions of maximality for $X$ and ${X}^{**}$ coincide and thus the maximality criterion directly follows from the one for W*-correspondences. We define full Cuntz-Pimsner covariance for W*- and C*-correspondences simultaneously in Definition 4.3 and prove that it is equivalent to maximality in Theorem 4.4, which is the main result of the paper. A similar idea to detect maximality by a projection in the second dual algebra was investigated by Clouâtre and Saikia [16]. This shows that the reduction to von Neumann algebras via second duals is a powerful tool for studying non-commutative Choquet boundary theory.

Of course, the second dual ${A}^{**}$ is a very big algebra which is hard to describe even for the most basic C*-algebras. Therefore, we also prove several maximality criteria which do not refer to the second dual. We describe maximality for an almost proper correspondence in terms of ideals in Proposition 4.10. Furthermore, we interpret full Cuntz-Pimsner covariance for topological graphs in terms of receiving vertices in Theorem 6.5.

Using the maximality criterion, we are able to recover several earlier results. The characterization of maximal representations of graph correspondences by Dor-On and Salomon [23] is an immediate consequence, and we describe it in Example 4.5 (and Theorem 6.5 is a generalization to topological graphs). In Corollary 4.8, we prove the Katsoulis-Kribs theorem, which states that the Cuntz-Pimsner algebra $\mathcal{O}_{X}$ is the C*-envelope of $\mathcal{T}^{+}_{X}$.

Kim [34] established a general criterion for the hyperrigidity of a correspondence, refining the result of Katsoulis and Ramsey [30], who provided a sufficient condition for hyperrigidity when the correspondence is countably generated. In Corollary 4.9, we derive Kim’s hyperrigidity criterion: a correspondence $X$ is hyperrigid if and only if Katsura’s ideal acts non-degenerately on $X$.

In Section 5, we study the noncommutative Choquet boundary and Arveson’s hyperrigidity conjecture. We provide an abstract characterization of the Choquet boundary in Theorem 5.3 via the so-called atomic spectrum $\sigma_{a}(X)\subset\widehat{A}$ (see Definition 5.2). For Type I and abelian C*-algebras, we give a more concrete description of the atomic spectrum, and thus of the Choquet boundary, in Proposition 5.9.

We analyze the hyperrigidity conjecture for correspondences. A whole class of counterexamples to the hyperrigidity conjecture is presented in Proposition 5.4. We also derive a measure-theoretic criterion for a correspondence associated with a topological quiver to be a counterexample in Corollary 6.2. On the other hand, we show that the conjecture holds for proper correspondences (Proposition 5.7) and for topological graphs (Corollary 6.2). This demonstrates that correspondences provide a natural framework for studying the connections between the Choquet boundary and hyperrigidity.

Throughout the section, $A$, $B$ will denote C*-algebras and $\mathfrak{M}$, $\mathfrak{N}$ will denote von Neumann algebras.

If $X$ is an $A$-$B$-correspondence and $Y$ is a $B$-$C$-correspondence, then there is an $A$-$C$-correspondence $X\otimes_{B}Y$ which is a completion of the algebraic tensor product with respect to the tensor inner product defined by

for all $\xi,\xi^{\prime}\in X$ and $\eta,\eta^{\prime}\in Y$. In particular, if $H$ is a $B$-representation, then we can view it as a $B$-$\mathbb{C}$-correspondence, so that $X\otimes_{B}H$ is an $A$-representation. The tensor product construction is defined analogously for W*-correspondences. If X is a $\mathfrak{M}$-$\mathfrak{N}$-W*-correspondence, then $X\otimes_{\mathfrak{N}}H$ is a normal representation of $\mathfrak{M}$ if $H$ is a normal representation of $\mathfrak{N}$.

For a C*-correspondence, the Cuntz-Toeplitz algebra $\mathcal{T}_{X}$ was defined by Pimsner [43] as the universal C*-algebra generated by images of $A$ and $X$ in representations of $X$. There are a $*$-homomorphism $\boldsymbol{\rho}\colon A\to\mathcal{T}_{X}$ and a linear map $\boldsymbol{t}\colon X\to\mathcal{T}_{X}$ with the following universal property. For all representations $(H,t)$ of $X$, there is a unique representation $\rho_{H}\rtimes t\colon\mathcal{T}_{X}\to\mathcal{B}(H)$ such that $\rho_{H}=\rho_{H}\rtimes t\circ\boldsymbol{\rho}$ and $t=\rho_{H}\rtimes t\circ\boldsymbol{t}$. We usually identify $A$ and $X$ with their images via $\boldsymbol{\rho}$ and $\boldsymbol{t}$. We also identify $\mathcal{K}(X)$ with its copy inside $\mathcal{T}_{X}$ given by $\overline{\boldsymbol{t}(X)\boldsymbol{t}(X)^{*}}$.

There is a gauge action $\gamma$ of the unit circle $\mathbb{T}=\{z\in\mathbb{C}\colon|z|=1\}$ on $\mathcal{T}_{X}$. The action leaves $A$ invariant and acts on $X$ by $\gamma_{z}(\xi)=z\xi$ for all $\xi\in X$ and $z\in\mathbb{T}$

Let $I\subset\varphi_{X}^{-1}(\mathcal{K}(X))\subset A$ be an ideal. Consider the ideal $\mathcal{J}_{I}\subset\mathcal{T}_{X}$ generated by elements $\varphi_{X}(a)-\boldsymbol{\rho}(a)$ for $a\in I$. The quotient $\mathcal{O}_{X,I}=\mathcal{T}_{X}/\mathcal{J}_{I}$ is called the $I$-relative Cuntz-Pimsner algebra. An $X$-representation $(H,t)$ for which $\rho_{H}\rtimes t$ factors through $\mathcal{O}_{X,I}$ is called $I$-relative Cuntz-Pimsner covariant.

The ideal $\mathcal{I}_{X}=\varphi_{X}^{-1}(\mathcal{K}(X))\cap{(\ker\varphi_{X})}^{\perp}$ is called Katsura’s ideal. Katsura [32] showed that the ideal $\mathcal{J}_{\mathcal{I}_{X}}$ is the largest gauge-invariant ideal of $\mathcal{T}_{X}$ trivially intersecting $\boldsymbol{\rho}(A)$ and the algebra $\mathcal{O}_{X}=\mathcal{O}_{X,\mathcal{I}_{X}}$ is called just the Cuntz-Pimsner algebra of $X$. Consequently, $X$-representations for which $\rho_{H}\rtimes t$ factors through $\mathcal{O}_{X}$ are called Cuntz-Pimsner covariant.

A representation $(H,t)$ of $X$ induces an $A$-linear isometry $t_{*}\colon X\otimes_{A}H\to H$ by the formula $t_{*}(\xi\otimes v)=t(\xi)v$ for all $\xi\in X$ and $v\in H$. Conversely, given an $A$-linear isometry $t_{*}\colon X\otimes_{A}H\to H$, we can define a representation $t\colon X\to\mathcal{B}(H)$ by the formula $t(\xi)v=t(\xi\otimes v)$ for all $\xi\in X$ and $v\in H$.

For a C*-algebra $A$, its second dual ${A}^{**}$ is naturally endowed with the structure of a von Neumann algebra (see [10, III.5.2.8]). It is universal in the following sense, if $\psi\colon A\to\mathfrak{M}$ is a $*$-homomorphism into a von Neumann algebra $\mathfrak{M}$, then $\psi$ extends uniquely to a normal homomorphism $\tilde{\psi}\colon{A}^{**}\to\mathfrak{M}$. For example, if $\rho\colon A\to\mathcal{B}(H)$ is a representation, then $\tilde{\rho}\colon{A}^{**}\to\mathcal{B}(H)$ is a normal representation. Therefore, there is a one-to-one correspondence between representations of $A$ and normal representations of ${A}^{**}$.

The multiplier algebra $\mathcal{M}(A)$ can be naturally embedded into ${A}^{**}$ (see [2]). Consequently, if $B$ is another C*-algebra and $\phi\colon A\to\mathcal{M}(B)\subset{B}^{**}$ is a $*$-homomorphism, then there is a unique normal homomorphism $\tilde{\phi}\colon{A}^{**}\to{B}^{**}$, which is unital if and only if $\phi(A)B=B$.

If $\psi\colon\mathfrak{M}\to\mathfrak{N}$ is a normal $*$-homomorphism, then there is a unique central projection $P_{\psi}\in Z(\mathfrak{M})$ called the support projection of $\psi$ such that $\ker\psi=\mathfrak{M}(1-P_{\psi})$ and $\psi(\mathfrak{M})\cong\mathfrak{M}P_{\psi}$. For a $*$-homomorphism $\phi\colon A\to\mathcal{M}(B)$, we define the support projection of $\phi$ to be $P_{\phi}=P_{\tilde{\phi}}\in{A}^{**}$. For an ideal $\iota_{I}\colon I\hookrightarrow B$, we use the notation $P_{I}=\tilde{\iota}_{I}(1)\in{B}^{**}$. It is the support projection of the associated map $B\to\mathcal{M}(I)$.

If $X$ is an $A$-$B$-correspondence, then by [11, Proposition 8.5.17], ${X}^{**}$ is a $W^{*}$-module over ${B}^{**}$ such that $\mathcal{L}({X}^{**})=\mathcal{K}(X)^{**}$. The map $\varphi_{X}\colon A\to\mathcal{L}(X)=\mathcal{M}(\mathcal{K}(X))$ lifts uniquely to the normal map $\varphi_{{X}^{**}}=\tilde{\varphi}_{X}\colon{A}^{**}\to\mathcal{L}({X}^{**})$ which endows ${X}^{**}$ with a structure of a W*-correspondence. We define the support projection of $X$ to be $P_{X}=P_{\varphi_{X}}\in{A}^{**}$.

Thanks to the universal property of the second dual, there is a bijection between $*$-representations $\rho\colon A\to B(H)$ and normal $*$-representations $\tilde{\rho}\colon{A}^{**}\to\mathcal{B}(H)$. This can be extended to correspondences as follows. Since $X$ is w*-dense in ${X}^{**}$, every linear map $t\colon X\to\mathcal{B}(H)$ extends uniquely to a w*-continuous map $\tilde{t}\colon{X}^{**}\to\mathcal{B}(H)$. Because the left and right actions of ${A}^{**}$ on ${X}^{**}$ and the inner product are w*-continuous, the maps $\tilde{\rho}_{H}$ and $\tilde{t}$ define a representation of ${X}^{**}$ on $H$. On the other hand, we can recover $\rho_{H}$ and $t$ by restricting to w*-dense subspaces $A$ and $X$. Thus, we have proved the following.

The following result reduces the study of dilations for C*-correspondences to W*-correspondences. This is the key result to prove Theorem 4.4, which is the main theorem.

In this section we develop the general theory of dilations for correspondences. We first deal with W*-correspondences and then proceed to the case of C*-correspondences using the second dual technique.

Consider a W*-correspondence $X$ over a W*-algebra $\mathfrak{M}$. We assume all representations of $\mathfrak{M}$ to be normal. Let $(H,t)$ be a representation of a correspondence $X$. Denote $H_{0}=H\ominus\overline{t(X)H}$ which is an $\mathfrak{M}$-reducing subspace. The map $t_{*}\colon X\otimes_{\mathfrak{M}}H\to\overline{t(X)H}\oplus H_{0}=H$ can be now written as a column block-matrix

where $Q$ is the orthogonal projection $H\twoheadrightarrow\overline{t(X)H}$. Dilations of the representation $(H,t)$ correspond to dilations of the map $t_{*}$ and the above matrix form helps to classify those dilations.

Recall that for a W*-correspondence $X$ over $\mathfrak{M}$ we define the support projection $P_{X}\in Z(\mathfrak{M})$ as the unique central projection satisfying $\ker\varphi_{X}=\mathfrak{M}(1-P_{X})$.

Since $\varphi(P_{X})X=X$ for a W*-correspondence $X$, we have $\rho(P_{X})t(X)=t(X)$ for any representation $(H,t)$ and thus $\overline{t(X)H}\subset\rho(P_{X})H$. Therefore, the $X$-representation is fully Cuntz-Pimsner covariant if and only if $\overline{t(X)H}=\rho(P_{X})H$.

If $X$ is a C*-correspondence and $(H,t)$ is an $X$-representation, then by w*-continuity $\overline{t(X)H}=\overline{\tilde{t}({X}^{**})H}$. In particular, the representation is fully Cuntz-Pimsner covariant if and only if $\overline{t(X)H}=\tilde{\rho}(P_{X})H$.

Let $\mathcal{A}$ be a non-self-adjoint subalgebra of a C*-algebra $B$. In [7], Arveson defined the Shilov ideal $J$ of $\mathcal{A}$ to be the largest ideal of $B$ such that the quotient $B\twoheadrightarrow B/J$ is completely isometric on $\mathcal{A}$. The quotient $C^{*}_{env}(\mathcal{A})\coloneqq B/J$ is called the C*-envelope of $\mathcal{A}$. The existence of the Shilov ideal and of the C*-envelope was established by Hamana [26].