The hyperrigidity of tensor algebras of C$^*$-correspondences

TL;DR

This paper characterizes when tensor algebras of C*-correspondences are hyperrigid, providing necessary and sufficient conditions, especially for those arising from topological graphs, thus advancing understanding of their rigidity properties.

Contribution

It offers a complete characterization of hyperrigidity for tensor algebras of C*-correspondences, including those from topological graphs, filling a gap in operator algebra theory.

Findings

Necessary and sufficient conditions for hyperrigidity.

Complete characterization for topological graph correspondences.

Advances understanding of rigidity in tensor algebras.

Abstract

Given a C$^*$-correspondence $X$, we give necessary and sufficient conditions for the tensor algebra $\mathcal T_X^+$ to be hyperrigid. In the case where $X$ is coming from a topological graph we obtain a complete characterization.

Equations

$$\tau :{\mathrm{C}}_{\text{env}}^{\ast }(\mathcal{A})⟶B(\mathcal{H})$$

$$\rho (c)t(x)=t({\phi }_X(c)(x)),\text{and}$$

$$t(x{)}^{\ast }t(x^{\prime })=\rho (⟨x,x^{\prime }⟩),$$

$$t(x_1)\cdots t(x_n)t(y_1{)}^{\ast }\cdots t(y_m{)}^{\ast },x_i,y_j\in X.$$

$${\psi }_t({\theta }_{x,y})=t(x)t(y{)}^{\ast },$$

$${\mathcal{J}}_X\equiv \ker {\phi }_X^{\perp }\cap {\phi }_X^{-1}(\mathcal{K}(X)).$$

$${\psi }_t({\phi }_X(c))=\rho (c),$$

$${\phi }_X({\mathcal{J}}_X)X{\otimes }_{\sigma }\mathcal{H}\ne X{\otimes }_{\sigma }\mathcal{H}.$$

$$({\phi }_X({\mathcal{J}}_X)\otimes \mathrm{id})X^{\otimes n}{\otimes }_{\sigma }\mathcal{H}\ne X^{\otimes n}{\otimes }_{\sigma }\mathcal{H}.$$

$${\phi }_X({\mathcal{J}}_X)X{\otimes }_{\sigma }\mathcal{K}\ne X{\otimes }_{\sigma }\mathcal{K}.$$

$$\langle f\otimes_{\sigma}h,f\otimes_{\sigma}h\rangle=\langle h,\sigma((\langle f,f\rangle)h\rangle=\|\sigma\big{(}\langle f,f\rangle^{1/2}\big{)}h\|\neq 0.$$

$$0\ne f{\otimes }_{\sigma }h\in ({\phi }_X({\mathcal{J}}_X)X{\otimes }_{\sigma }\mathcal{H}{)}^{\perp }$$

$$t_{{\mid }_{\mathcal{M}}}(x{)}^{\ast }{t}_{{\mid }_{\mathcal{M}}}(y)$$

$$=pt(x{)}^{\ast }t(y)p$$

$$=p\rho (⟨x,y⟩)p={\rho }_{{\mid }_{\mathcal{M}}}(⟨x,y⟩)$$

$$t(Kx)={\psi }_t(K)t(x),x\in X.$$

$$\rho (c)t(x)h=t({\phi }_X(c)x)h={\psi }_t({\phi }_X(c))t(x)h.$$

$${\phi }_X({\mathcal{J}}_X)X{\otimes }_{\sigma }\mathcal{H}\ne X{\otimes }_{\sigma }\mathcal{H}$$

$$({\phi }_X({\mathcal{J}}_X)\otimes I){\mathcal{M}}_0=\{0\}.$$

$$\displaystyle\big{\langle}(\varphi_{X}(j)\otimes I)f\,,\,(\varphi_{X}(j)\otimes I)f\big{\rangle}=\langle f,(\varphi_{X}(j^{*}j)\otimes I)f\rangle=0$$

$$({\phi }_X(\mathcal{C})\otimes I){\mathcal{M}}_0={\mathcal{M}}_0.$$

$$({\phi }_X(\mathcal{C})\otimes I)({\phi }_X({\mathcal{J}}_X)X{\otimes }_{\sigma }\mathcal{H})={\phi }_X({\mathcal{J}}_X)X{\otimes }_{\sigma }\mathcal{H},$$

$${\rho }_0$$

$$t_0$$

$$\mathcal{M}:$$

$$=({\rho }_0⋊t_0)({\mathcal{T}}_X^{+})(0\oplus {\mathcal{M}}_0\oplus 0\oplus 0\oplus \dots )\subseteq \mathcal{F}(X){\otimes }_{\sigma }\mathcal{H},$$

$$t(X)\mathcal{M}=0\oplus 0\oplus (X\otimes {\mathcal{M}}_0)\oplus (X^{\otimes 2}\otimes {\mathcal{M}}_0)\oplus ...,$$

$$\mathcal{M}\ominus t(X)\mathcal{M}=0\oplus {\mathcal{M}}_0\oplus 0\oplus 0\oplus \dots .$$

$$t_0(x{)}^{\ast }h\in (\mathcal{C}{\otimes }_{\sigma }\mathcal{H})\oplus 0\oplus 0\oplus ...,x\in X$$

$${\psi }_t({\phi }_X(j))h\in {t_0}_{{\mid }_{\mathcal{M}}}(X)({t_0}_{{\mid }_{\mathcal{M}}})(X{)}^{\ast }h=\{0\}.$$

$$\rho (j)h\in 0\oplus ({\phi }_X({\mathcal{J}}_X)\otimes I){\mathcal{M}}_0\oplus 0\oplus 0\oplus \cdots =\{0\},$$

$$\begin{gathered} {\mathcal{M}}^{\perp }=\mathcal{C}\oplus ({\phi }_X({\mathcal{J}}_X)X{\otimes }_{\sigma } \\ H)\oplus (X\otimes {\mathcal{M}}_0{)}^{\perp }\oplus \dots \end{gathered}$$

Full text

2010 Mathematics Subject Classification. 46L07, 46L08, 46L55, 47B49, 47L40

Key words and phrases: $\mathrm{C}^{*}$-correspondence, tensor algebra, hyperrigid, topological graph, operator algebra

1. Introduction

A not necessarily unital operator algebra ${\mathcal{A}}$ is said to be hyperrigid if given any non-degenerate $*$-homomorphism

[TABLE]

then $\tau$ is the only completely positive, completely contractive extension of the restricted map $\tau_{\mid{\mathcal{A}}}$. Arveson coined the term hyperrigid in [1] but he was not the only one considering properties similar to this at the time, e.g. [4].

There are many examples of hyperrigid operator algebras such as those which are Dirichlet but the situation was not very clear in the case of tensor algebras of C∗-correspondences. It was known that the tensor algebra of a row-finite graph is hyperrigid [4], [5] and Dor-On and Salmomon [3] showed that row-finiteness completely characterizes hyperrigidity for such graph correspondences. These approaches, while successful, did not lend themselves to a more general characterization.

The authors, in a previous work [11], developed a sufficient condition for hyperrigidity in tensor algebras. In particular, if Katsura’s ideal acts non-degenerately on the left then the tensor algebra is hyperrigid. The motivation was to provide a large class of hyperrigid C∗-correspondence examples as crossed products of operator algebras behave in a very nice manner when the operator algebra is hyperrigid. This theory was in turn leveraged to provide a positive confirmation to the Hao-Ng isomorphism problem in the case of graph correspondences and arbitrary groups. For further reading on the subject please see [9, 10, 11].

In this paper, we provide a necessary condition for the hyperrigidity of a tensor algebra, that a C∗-correspondence cannot be $\sigma$-degenerate, and show that this completely characterizes the situation where the C∗-correspondence is coming from a topological graph, which generalizes both the graph correspondence case and the semicrossed product arising from a multivariable dynamical system.

1.1. Regarding hyperrigidity

The reader familiar with the literature recognizes that in our definition of hyperrigidity, we are essentially asking that the restriction on ${\mathcal{A}}$ of any non-degenerate representation of $\mathrm{C}^{*}_{\text{env}}({\mathcal{A}})$ possesses the * unique extension property* (abbr. UEP). According to [3, Proposition 2.4] a representation $\rho:{\mathcal{A}}\rightarrow B({\mathcal{H}})$, degenerate or not, has the UEP if and only if $\rho$ is a maximal representation of ${\mathcal{A}}$, i.e., whenever $\pi$ is a representation of ${\mathcal{A}}$ dilating $\rho$, then $\pi=\rho\oplus\pi^{\prime}$ for some representation $\pi^{\prime}$. Our definition of hyperrigidity is in accordance with Arveson’s nomenclature [1], our earlier work [7, 11] and the works of Dor-On and Salomon [3] and Salomon [16], who systematized quite nicely the non-unital theory.

An alternative definition of hyperrigidity for ${\mathcal{A}}$ may ask that any representation of $\mathrm{C}^{*}_{\text{env}}({\mathcal{A}})$, not just the non-degenerate ones, possesses the UEP when restricted on ${\mathcal{A}}$. It turns out that for operator algebras with a positive contractive approximate unit111which includes all operator algebras appearing in this paper, such a definition would be equivalent to ours [16, Proposition 3.6 and Theorem 3.9] . However when one moves beyond operator algebras with an approximate unit, there are examples to show that the two definitions differ. One such example is the non-unital operator algebra ${\mathcal{A}}_{V}$ generated by the unilateral forward shift $V$. It is easy to see that ${\mathcal{A}}_{V}$ is hyperrigid according to our definition and yet the zero map, as a representation on ${\mathcal{H}}={\mathbb{C}}$, does not have the UEP. (See for instance [16, Example 3.4].)

2. Main results

A $\mathrm{C}^{*}$-correspondence $(X,{\mathcal{C}},\varphi_{X})$ (often just $(X,{\mathcal{C}})$) consists of a $\mathrm{C}^{*}$-algebra ${\mathcal{C}}$, a Hilbert ${\mathcal{C}}$-module $(X,\left\langle\phantom{,},\phantom{,}\right\rangle)$ and a (non-degenerate) $*$-homomorphism $\varphi_{X}\colon{\mathcal{C}}\rightarrow{\mathcal{L}}(X)$ into the C∗-algebra of adjointable operators on $X$.

An isometric (Toeplitz) representation $(\rho,t,{\mathcal{H}})$ of a $\mathrm{C}^{*}$-correspondence $(X,{\mathcal{C}})$ consists of a non-degenerate $*$-homomorphism $\rho\colon{\mathcal{C}}\rightarrow B({\mathcal{H}})$ and a linear map $t\colon X\rightarrow B({\mathcal{H}})$, such that

[TABLE]

[TABLE]

for all $c\in{\mathcal{C}}$ and $x,x^{\prime}\in X$. These relations imply that the $\mathrm{C}^{*}$-algebra generated by this isometric representation equals the closed linear span of

[TABLE]

Moreover, there exists a $*$-homomorphism $\psi_{t}:{\mathcal{K}}(X)\rightarrow B$, such that

[TABLE]

where ${\mathcal{K}}(X)\subset{\mathcal{L}}(X)$ is the subalgebra generated by the operators $\theta_{x,y}(z)=x\langle y,z\rangle,\ x,y,x\in X$, which are called by analogy the compact operators.

The Cuntz-Pimsner-Toeplitz $\mathrm{C}^{*}$-algebra ${\mathcal{T}}_{X}$ is defined as the $\mathrm{C}^{*}$-algebra generated by the image of $(\rho_{\infty},t_{\infty})$, the universal isometric representation. This is universal in the sense that for any other isometric representation there is a $*$-homomorphism of ${\mathcal{T}}_{X}$ onto the $\mathrm{C}^{*}$-algebra generated by this representation in the most natural way.

The tensor algebra ${\mathcal{T}}_{X}^{+}$ of a $\mathrm{C}^{*}$-correspondence $(X,{\mathcal{C}})$ is the norm-closed subalgebra of ${\mathcal{T}}_{X}$ generated by $\rho_{\infty}({\mathcal{C}})$ and $t_{\infty}(X)$. See [14] for more on these constructions.

Consider Katsura’s ideal

[TABLE]

An isometric representation $(\rho,t)$ of $(X,{\mathcal{C}},\varphi_{X})$ is said to be covariant (or Cuntz-Pimsner) if and only if

[TABLE]

for all $c\in{\mathcal{J}}_{X}$. The Cuntz-Pimsner algebra ${\mathcal{O}}_{X}$ is the universal $\mathrm{C}^{*}$-algebra for all isometric covariant representations of $(X,{\mathcal{C}})$, see [13] for further details. Furthermore, the first author and Kribs [8, Lemma 3.5] showed that ${\mathcal{O}}_{X}$ contains a completely isometric copy of ${\mathcal{T}}_{X}^{+}$ and $\mathrm{C}^{*}_{\text{env}}({\mathcal{T}}_{X}^{+})\simeq{\mathcal{O}}_{X}$.

We turn now to the hyperrigidity of tensor algebras. In [11] a sufficient condition for hyperrigidity was developed, Katsura’s ideal acting non-degenerately on the left of $X$. To be clear, non-degeneracy here means that $\overline{\varphi_{X}({\mathcal{J}}_{X})X}=X$ which by Cohen’s factorization theorem implies that we actually have $\varphi_{X}({\mathcal{J}}_{X})X=X$.

Theorem 2.1** (Theorem 3.1, [11]).**

Let $(X,{\mathcal{C}})$ be a $\mathrm{C}^{*}$-correspondence with $X$ countably generated as a right Hilbert ${\mathcal{C}}$-module. If $\varphi_{X}({\mathcal{J}}_{X})$ acts non-degenerately on $X$, then ${\mathcal{T}}^{+}_{X}$ is a hyperrigid operator algebra.

The proof shows that if $\tau^{\prime}\colon{\mathcal{O}}_{X}\longrightarrow B({\mathcal{H}})$ is a completely contractive and completely positive map that agrees with a $*$-homomorphism of ${\mathcal{O}}_{X}$ on ${\mathcal{T}}^{+}_{X}$ then the multiplicative domain of $\tau^{\prime}$ must be everything. This is accomplished through the multiplicative domain arguments of [2, Proposition 1.5.7] and the fact that by $X$ being countably generated, Kasparov’s Stabilization Theorem implies the existence of a sequence $\{x_{n}\}_{n=1}^{\infty}$ in $X$ so that $\sum_{n=1}^{k}\theta_{x_{n},x_{n}}$, $k=1,2,\dots$, is an approximate unit for ${\mathcal{K}}(X)$. After quite a lot of inequality calculations one arrives at the fact that all of ${\mathcal{T}}_{X}^{+}$ is in the multiplicative domain and thus so is ${\mathcal{O}}_{X}$.

A $\mathrm{C}^{*}$-correspondence $(X,{\mathcal{C}})$ is called regular if and only if ${\mathcal{C}}$ acts faithfully on $X$ by compact operators, i.e., ${\mathcal{J}}_{X}={\mathcal{C}}$. We thus obtain the following which also appeared in [11].

Corollary 2.2**.**

The tensor algebra of a regular, countably generated $\mathrm{C}^{*}$-correspondence is necessarily hyperrigid.

We seek a converse to Theorem 2.1.

Definition 2.3**.**

Let $(X,{\mathcal{C}})$ be a $\mathrm{C}^{*}$-correspondence and let ${\mathcal{J}}_{X}$ be Katsura’s ideal. We say that $\varphi_{X}({\mathcal{J}}_{X})$ acts $\sigma$-degenerately on $X$ if there exists a representation $\sigma\colon{\mathcal{C}}\rightarrow B({\mathcal{H}})$ so that

[TABLE]

Remark 2.4**.**

In particular, if there exists $n\in{\mathbb{N}}$ so that

[TABLE]

then by considering the Hilbert space ${\mathcal{K}}:=X^{\otimes n-1}\otimes_{\sigma}{\mathcal{H}}$, we see that

[TABLE]

and so $\varphi_{X}({\mathcal{J}}_{X})$ acts $\sigma$-degenerately on $X$.

The following gives a quick example of a $\sigma$-degenerate action. Note that this is possibly stronger than having a not non-degenerate action.

Proposition 2.5**.**

Let $(X,{\mathcal{C}})$ be a $\mathrm{C}^{*}$-correspondence. If $(\varphi_{X}({\mathcal{J}}_{X})X)^{\perp}\neq\{0\}$, then $\varphi_{X}({\mathcal{J}}_{X})$ acts $\sigma$-degenerately on $X$.

Proof..

Let $0\neq f\in(\varphi_{X}({\mathcal{J}}_{X})X)^{\perp}$. Let $\sigma:{\mathcal{C}}\rightarrow B({\mathcal{H}})$ be a $*$-representation and $h\in{\mathcal{H}}$ so that \sigma\big{(}\langle f,f\rangle^{1/2}\big{)}h\neq 0. Then,

[TABLE]

A similar calculation shows that

[TABLE]

and we are done. ∎

We need the following

Lemma 2.6**.**

Let $(X,{\mathcal{C}})$ be a $\mathrm{C}^{*}$-correspondence and $(\rho,t)$ an isometric representation of $(X,{\mathcal{C}})$ on ${\mathcal{H}}$.

1. (i)

If ${\mathcal{M}}\subseteq{\mathcal{H}}$ is an invariant subspace for $(\rho\rtimes t)({\mathcal{T}}_{X}^{+})$, then the restriction $(\rho_{\mid_{{\mathcal{M}}}},t_{\mid_{{\mathcal{M}}}})$ of $(\rho,t)$ on ${\mathcal{M}}$ is an isometric representation. 2. (ii)

If $\rho(c)h=\psi_{t}(\varphi_{X}(c))h,$ for all $c\in{\mathcal{J}}_{X}$ and $h\in[t(X){\mathcal{H}}]^{\perp}$, then $(\rho,t)$ is a Cuntz-Pimsner representation.

Proof..

(i) If $p$ is the orthogonal projection on ${\mathcal{M}}$, then $p$ commutes with $\rho({\mathcal{C}})$ and so $\rho_{\mid_{{\mathcal{M}}}}(\cdot)=p\rho(\cdot)p$ is a $*$-representation of ${\mathcal{C}}$.

Furthermore, for $x,y\in X$, we have

[TABLE]

and the conclusion follows.

(ii) It is easy to see on rank-one operators and therefore by linearity and continuity on all compact operators $K\in{\mathcal{K}}(X)$ that

[TABLE]

Now if $c\in{\mathcal{J}}_{X}$, then for any $x\in X$ and $h\in{\mathcal{H}}$ we have

[TABLE]

By assumption $\rho(c)h=\psi_{t}(\varphi_{X}(c))h$, for any $h\in[t(X){\mathcal{H}}]^{\perp}$ and the conclusion follows. ∎

Theorem 2.7**.**

Let $(X,{\mathcal{C}})$ be a $\mathrm{C}^{*}$-correspondence. If Katsura’s ideal ${\mathcal{J}}_{X}$ acts $\sigma$-degenerately on $X$ then the tensor algebra ${\mathcal{T}}^{+}_{X}$ is not hyperrigid.

Proof..

Let $\sigma\colon{\mathcal{C}}\rightarrow B({\mathcal{H}})$ so that

[TABLE]

and let ${\mathcal{M}}_{0}:=(\varphi_{X}({\mathcal{J}}_{X})X\otimes_{\sigma}{\mathcal{H}})^{\perp}$.

We claim that

[TABLE]

Indeed for any $f\in{\mathcal{M}}_{0}$ and $j\in{\mathcal{J}}_{X}$ we have

[TABLE]

since $f\in(\varphi_{X}({\mathcal{J}}_{X})X\otimes_{\sigma}{\mathcal{H}})^{\perp}$. This proves the claim.

We also claim that

[TABLE]

Indeed this follows from the fact that

[TABLE]

which is easily verified.

Using the subspace ${\mathcal{M}}_{0}$ we produce a Cuntz-Pimsner representation $(\rho,t)$ of $(X,{\mathcal{C}})$ as follows. Let $(\rho_{\infty},t_{\infty})$ be the universal representation of $(X,{\mathcal{C}})$ on the Fock space ${\mathcal{F}}(X)=\oplus_{n=0}^{\infty}X^{\otimes n}$, $X^{\otimes 0}:={\mathcal{C}}$. Let

[TABLE]

Define

[TABLE]

with the second equality following from (2). Clearly, ${\mathcal{M}}$ is an invariant subspace for $(\rho_{0}\rtimes t_{0})({\mathcal{T}}_{X}^{+})$.

Let $\rho:={\rho_{0}}_{\mid_{{\mathcal{M}}}}$ and $t:={t_{0}}_{\mid_{{\mathcal{M}}}}$. By Lemma 2.6(i), $(\rho,t)$ is a representation of $(X,{\mathcal{C}})$. We claim that $(\rho,t)$ is actually Cuntz-Pimsner.

Indeed by Lemma 2.6(ii) it suffices to examine whether $\psi_{t}(\varphi_{X}(j))h=\rho(j)h$, for any $h\in{\mathcal{M}}\ominus t(X){\mathcal{M}}$. Note that since

[TABLE]

we have that

[TABLE]

From this it follows that for any $h\in{\mathcal{M}}\ominus t(X){\mathcal{M}}$ we have

[TABLE]

and so in particular for any $j\in{\mathcal{J}}_{X}$ we obtain

[TABLE]

On the other hand,

[TABLE]

because of (2). Hence $(\rho,t)$ is Cuntz-Pimsner.

At this point by restricting on ${\mathcal{T}}_{X}^{+}$, we produce the representation $\rho\rtimes t\mid_{{\mathcal{T}}_{X}^{+}}$ of ${\mathcal{T}}_{X}^{+}$ coming from a $*$-representation of its $\mathrm{C}^{*}$-envelope ${\mathcal{O}}_{X}$, which admits a dilation, namely $\rho_{0}\rtimes t_{0}\mid_{{\mathcal{T}}_{X}^{+}}$. If we show now that $\rho_{0}\rtimes t_{0}\mid_{{\mathcal{T}}_{X}^{+}}$ is a non-trivial dilation of $\rho\rtimes t\mid_{{\mathcal{T}}_{X}^{+}}$, i.e. ${\mathcal{M}}_{0}$ is not reducing for $(\rho_{0}\rtimes t_{0})({\mathcal{T}}_{X}^{+})$, then $\rho\rtimes t\mid_{{\mathcal{T}}_{X}^{+}}$ is not a maximal representation of ${\mathcal{T}}^{+}_{X}$. Proposition 2.4 [3] shows $\rho\rtimes t\mid_{{\mathcal{T}}_{X}^{+}}$ does not have the UEP and so ${\mathcal{T}}_{X}^{+}$ is not hyperrigid, as desired.

Towards this end, note that

[TABLE]

and so

[TABLE]

Therefore ${\mathcal{M}}^{\perp}$ is not an invariant subspace for $(\rho_{0}\rtimes t_{0})({\mathcal{T}}_{X}^{+})$ and so ${\mathcal{M}}$ is not a reducing subspace for $(\rho_{0}\rtimes t_{0})({\mathcal{T}}_{X}^{+})$. This completes the proof. ∎

3. Topological graphs

A broad class of $\mathrm{C}^{*}$-correspondences arises naturally from the concept of a topological graph. For us, a topological graph $G=(G^{0},G^{1},r,s)$ consists of two $\sigma$-locally compact spaces $G^{0}$, $G^{1}$, a continuous proper map $r:G^{1}\rightarrow G^{0}$ and a local homeomorphism $s:G^{1}\rightarrow G^{0}$. The set $G^{0}$ is called the base (vertex) space and $G^{1}$ the edge space. When $G^{0}$ and $G^{1}$ are both equipped with the discrete topology, we have a discrete countable graph.

With a given topological graph $G=(G^{0},G^{1},r,s)$ we associate a $\mathrm{C}^{*}$-correspondence $X_{G}$ over $C_{0}(G^{0})$. The right and left actions of $C_{0}(G^{0})$ on $C_{c}(G^{1})$ are given by

[TABLE]

for $F\in C_{c}(G^{1})$, $f,g\in C_{0}(G^{0})$ and $e\in G^{1}$. The inner product is defined for $F,H\in C_{c}(G^{1})$ by

[TABLE]

for $v\in G^{0}$. Finally, $X_{G}$ denotes the completion of $C_{c}(G^{1})$ with respect to the norm

[TABLE]

When $G^{0}$ and $G^{1}$ are both equipped with the discrete topology, then the tensor algebra ${\mathcal{T}}_{G}^{+}\equiv{\mathcal{T}}^{+}_{X_{G}}$ associated with $G$ coincides with the quiver algebra of Muhly and Solel [14]. See [15] for further reading.

Given a topological graph $G=(G^{0},G^{1},r,s)$, we can describe the ideal ${\mathcal{J}}_{X_{G}}$ as follows. Let

[TABLE]

Both sets are easily seen to be open and in [12, Proposition 1.24] Katsura shows that

[TABLE]

From the above it is easy to see that ${\mathcal{J}}_{X_{G}}=C_{0}(G^{0}_{\operatorname{reg}})$, where

[TABLE]

We need the following

Lemma 3.1**.**

Let $G=(G^{0},G^{1},r,s)$ be a topological graph. Then r^{-1}\big{(}G^{0}_{\operatorname{reg}}\big{)}=G^{1} if and only if $r:G^{1}\rightarrow G^{0}$ is a proper map satisfying r(G^{1})\subseteq\big{(}\overline{r(G^{1})}\big{)}^{\circ}.

Proof..

Notice that

[TABLE]

and so r^{-1}\big{(}G^{0}_{\operatorname{reg}}\big{)}=G^{1} is equivalent to $r^{-1}(G^{0}_{\operatorname{fin}})=r^{-1}(\overline{G^{0}_{\operatorname{sce}}})=G^{1}$

First we claim that $r^{-1}(G^{0}_{\operatorname{fin}})=G^{1}$ if and only if $r$ is a proper map. Indeed, assume that $r^{-1}(G^{0}_{\operatorname{fin}})=G^{1}$ and let $K\subseteq r(G^{1})$ compact in the relative topology. For every $x\in K$, let $V_{x}$ be a compact neighborhood of $x$ such that $r^{-1}(V_{x})$ is compact and so $r^{-1}(V_{x}\cap K))$ is also compact. By compactness, there exist $x_{1},x_{2},\dots,x_{n}\in K$ so that $K=\cup_{i=1}^{n}(V_{x_{i}}\cap K)$ and so

[TABLE]

and so $r^{-1}(K)$ is compact.

Conversely, if $r$ is proper then any compact neighborhood V of any point in $G^{0}$ is inverted by $r^{-1}$ to a compact set and so $r^{-1}(G^{0}_{\operatorname{fin}})=G^{1}$.

We now claim that $r^{-1}(\overline{G^{0}_{\operatorname{sce}}})=\emptyset$ if and only r(G^{1})\subseteq\big{(}\overline{r(G^{1})}\big{)}^{\circ}.

Indeed, $e\in r^{-1}(\overline{G^{0}_{\operatorname{sce}}})$ is equivalent to $r(e)\in\overline{(r(G^{1})^{c})^{\circ}}$ and so $r^{-1}(\overline{G^{0}_{\operatorname{sce}}})=\emptyset$ is equivalent to

[TABLE]

as desired. ∎

If $G=(G^{0},G^{1},r,s)$ is a topological graph and $S\subseteq G^{1}$, then $N(S)$ denotes the collection of continuous functions $F\in X_{G}$ with $F_{|S}=0$, i.e., vanishing at $S$. The following appears as Lemma 4.3(ii) in [6].

Lemma 3.2**.**

Let $G=(G^{0},G^{1},r,s)$ be a topological graph. If $S_{1}\subseteq G^{0}$, $S_{2}\subseteq G^{1}$ closed, then

[TABLE]

Theorem 3.3**.**

Let $G=(G^{0},G^{1},r,s)$ be a topological graph and let $X_{G}$ the $\mathrm{C}^{*}$-correspondence associated with $G$. Then the following are equivalent

• (i)

the tensor algebra ${\mathcal{T}}_{X_{G}}^{+}$ is hyperrigid

• (ii)

$\varphi({\mathcal{J}}_{X_{G}})$* acts non-degenerately on $X_{G}$*

• (iii)

$r:G^{1}\rightarrow G^{0}$* is a proper map satisfying r(G^{1})\subseteq\big{(}\overline{r(G^{1})}\big{)}^{\circ}*

Proof..

If $\varphi({\mathcal{J}}_{X_{G}})$ acts non-degenerately on $X_{G}$, then Theorem 2.1 shows that ${\mathcal{T}}_{X_{G}}^{+}$ is hyperrigid. Thus (ii) implies (i).

For the converse, assume that $\varphi({\mathcal{J}}_{X_{G}})$ acts degenerately on $X_{G}$. If we verify that $\varphi({\mathcal{J}}_{X_{G}})$ acts $\sigma$-degenerately on $X_{G}$, then Theorem 2.7 shows that ${\mathcal{T}}_{X_{G}}^{+}$ is not hyperrigid and so (i) implies (ii).

Towards this end note that ${\mathcal{J}}_{X_{G}}={\mathcal{C}}_{0}({\mathcal{U}})$ for some proper open set ${\mathcal{U}}\subseteq G^{0}$. (Actually we know that ${\mathcal{U}}=G^{0}_{\operatorname{reg}}$ but this is not really needed for this part of the proof!) Hence

[TABLE]

according to Lemma 3.2.

Since $\varphi({\mathcal{J}}_{X_{G}})$ acts degenerately on $X_{G}$, (4) shows that $r^{-1}({\mathcal{U}})^{c}\neq\emptyset$. Let $e\in r^{-1}({\mathcal{U}})^{c}$ and let $F\in C_{c}(G^{1})\subseteq X_{G}$ with $F(e)=1$ and $F(e^{\prime})=0$, for any other $e^{\prime}\in G^{1}$ with $s(e^{\prime})=s(e)$. Consider the one dimensional representation $\sigma:C_{0}(G_{0})\rightarrow{\mathbb{C}}$ coming from evaluation at $s(e)$. We claim that

[TABLE]

Indeed for any $G\in\varphi({\mathcal{J}}_{X_{G}})X_{G}=N(r^{-1}({\mathcal{U}})^{c})$ we have

[TABLE]

Furthermore,

[TABLE]

and so $0\neq F\otimes_{\sigma}1\in(\varphi_{X_{G}}({\mathcal{J}}_{X_{G}})X_{G}\otimes_{\sigma}{\mathbb{C}})^{\perp}$. This establishes the claim and finishes the proof of (i) implies (ii).

Finally we need to show that (ii) is equivalent to (iii). Notice that (4) implies that $\varphi({\mathcal{J}}_{X_{G}})$ acts degenerately on $X_{G}$ if and only if

[TABLE]

The conclusion now follows from Lemma 3.1. ∎

The statement of the previous Theorem takes its most pleasing form when $G^{0}$ is a compact space. In that case ${\mathcal{T}}^{+}_{X}$ is hyperrigid if and only is $G^{1}$ is compact and $r(G^{1})\subseteq G^{0}$ is clopen.

Acknowledgement. Both authors would like to thank MacEwan University for providing project funding to bring the first author out to Edmonton for a research visit. The first author received support for this project in the form of a Summer Research Award from the Thomas Harriott College of Arts Sciences at ECU. He also expresses his gratitude to Adam Dor-On and Guy Salomon for several discussions regarding their work on hyperrigidity. The second author was partially supported by an NSERC grant.