2.1. KMS states on graph $C^{\ast }$-algebra without sink at the critical inverse temperature

A finite directed graph is a collection of finitely many edges and vertices. If we denote the edge set of a graph $\Gamma$ by $E=(e_{1},\ldots ,e_{n})$ and the vertex set of $\Gamma$ by $V=(v_{1},\ldots ,v_{m})$ then recall the maps $s,t:E\rightarrow V$ from [Reference Pask and Rennie13] and the vertex (or the adjacency) matrix $D$ which is an $m\times m$ matrix whose $ij$th entry is $k$ if there are $k$-number of edges from $v_{i}$ to $v_{j}$. We denote the space of paths by $E^{\ast }$ (see [Reference Huef, Laca, Raeburn and Sims6]). $vE^{\ast }w$ will denote the set of paths between two vertices $v$ and $w$.

Now we recall some basic facts about graph $C^{\ast }$-algebras. The reader might consult [Reference Pask and Rennie13] for details on graph $C^{\ast }$-algebras. Let $\Gamma =\{E=(e_{1},\ldots ,e_{n}),V=(v_{1},\ldots ,v_{m})\}$ be a finite, directed graph without sink. In this paper, all the graphs are finite, without sink and without any multiple edges. We assign partial isometries $S_{i}$'s to edges $e_{i}$ for all $i=1,\ldots ,n$ and projections $p_{v_{i}}$ to the vertices $v_{i}$ for all $i=1,\ldots ,m$.

Definition 2.3 The graph $C^{\ast }$-algebra $C^{\ast }(\Gamma )$ is defined as the universal $C^{\ast }$-algebra generated by partial isometries $\{S_{i}\}_{i=1,\ldots ,n}$ and mutually orthogonal projections $\{p_{v_{k}}\}_{k=1,\ldots ,m}$ satisfying the following relations:

\[ S_{i}^{{\ast}}S_{i}=p_{t(e_{i})},\quad \sum_{s(e_{j})=v_{l}}S_{j}S_{j}^{{\ast}}=p_{v_{l}}. \]

The KMS states at various inverse temperatures on a graph $C^{\ast }$-algebra are well known and we refer the reader to [Reference Huef, Laca, Raeburn and Sims6] for details. The critical inverse temperature of a graph $C^{\ast }$-algebra is given by $\textrm {ln}(\rho (D))$ where $\rho (D)$ is the spectral radius of the vertex matrix $D$ of the underlying graph. In this subsection, we mainly collect a few results on the existence of KMS states at the critical inverse temperature on graph $C^{\ast }$-algebras coming from graphs without sink. We continue to assume $\Gamma$ to be a finite, connected graph without sink and with vertex matrix $D$. We denote the spectral radius of $D$ by $\rho (D)$. With this notation, combining Proposition 4.1 and Corollary 4.2 of [Reference Huef, Laca, Raeburn and Sims6], we have the following

Proof. Suppose $\beta \in \mathbb {R}$ is another possible inverse temperature where a KMS state say $\phi$ could occur. Then since we have assumed our graph to be without sink, $e^{\beta }$ is an eigenvalue of $D$. Let us denote an eigenvector corresponding to $e^{\beta }$ by $\textbf {w}=(w_{1},\ldots ,w_{m})$ so that $w_{i}=\phi (p_{v_{i}})$. Since $\phi$ is a state, $w_{i}\geq 0$ for all $i=1,\ldots ,m$ with at least one entry being strictly positive. We have

\begin{align*} D\textbf{w}&=e^{\beta}\textbf{w}\\ &\Rightarrow \textbf{v}^{T}D\textbf{w}=\textbf{v}^{T}e^{\beta}\textbf{w}\\ &\Rightarrow (\rho(D)-e^{\beta})\textbf{v}^{T}\textbf{w}=0 \end{align*}

By assumption, all the entries of $\textbf {v}$ are strictly positive and $w_{i}\geq 0$ for all $i$ with at least one entry being strictly positive which imply that $\textbf {v}^{T}\textbf {w}\neq 0$ and hence $e^{\beta }=\rho (D)$ i.e. $\beta =\ln(\rho (D))$.

We discuss examples of two classes of graphs which are without sink such that they admit KMS states only at the critical inverse temperature. We shall use them later in this paper.

Strongly connected graphs:

We state the following two well-known results without proof.

As a corollary we have

Circulant graphs:

It is easy to see that if a graph is a circulant, then its vertex matrix is determined by its first-row vector say $(d_{0},\ldots ,d_{m-1})$. More precisely, the vertex matrix $D$ of a circulant graph is given by

\[ \begin{bmatrix} d_{0} & d_{1}\ldots & d_{m-1}\\ d_{m-1} & d_{0} \ldots & d_{m-2}\\ .\\ .\\ .\\ d_{1} & d_{2}... & d_{0} \end{bmatrix}. \]

Let $\epsilon$ be a primitive $m$th root of unity. The following is well known (see [Reference Kra and Santiago10]):

Proposition 2.13 For a circulant graph with vertex matrix as above, the eigenvalues are given by

\[ \lambda_{l}=d_{0}+\epsilon^{l}d_{1}+\cdots+\epsilon^{(m-1)l}d_{m-1},\quad l=0,\ldots,(m-1) \]

It is easy to see that $\lambda =\sum \nolimits _{i=0}^{m-1}d_{i}$ is an eigenvalue of $D$ and it has a normalized eigenvector given by $(\frac {1}{m},\ldots ,\frac {1}{m})$. Since $|\lambda _{l}|\leq \lambda$, we have

Combining the above corollary with Proposition 2.4, we have

Note that KMS states at the critical inverse temperature are not necessarily unique, since the dimension of the eigenspace of the eigenvalue $\lambda$ could be strictly larger than 1 as the example in Figure 1 illustrates. We take the graph whose vertex matrix is given by

\[ \begin{bmatrix} 1 & 0 & 1 & 0\\ 0 & 1 & 0 & 1\\ 1 & 0 & 1 & 0\\ 0 & 1 & 0 & 1 \end{bmatrix}. \]

Hence the graph is circulant with its spectral radius $2$ as an eigenvalue with multiplicity $2$. So the dimension of the corresponding eigenspace is $2$ violating the uniqueness of the KMS state at the critical inverse temperature $\textrm {ln}(2)$.

Figure 1. Circulant graph with more than one KMS state.

2.2. Quantum automorphism group of graphs as symmetry of graph $C^{\ast }$-algebra

2.2.1 Compact quantum groups and quantum automorphism groups

In this subsection, we recall the basics of compact quantum groups and their actions on $C^{\ast }$-algebras. The facts collected in this subsection are well known and we refer the readers to [Reference Maes and Van Daele12, Reference Wang15, Reference Woronowicz16] for details. All the tensor products in this paper are minimal.

Definition 2.17 A compact quantum group (CQG) $\mathbb {G}$ is a pair $(C(\mathbb {G}),\Delta _{\mathbb {G}})$ such that $C(\mathbb {G})$ is a unital $C^{\ast }$-algebra and $\Delta _{\mathbb {G}}:C(\mathbb {G})\rightarrow C(\mathbb {G})\otimes C(\mathbb {G})$ is a unital $C^{\ast }$-homomorphism satisfying

1. (i) $(\textrm {id}\otimes \Delta _{\mathbb {G}})\circ \Delta _{\mathbb {G}}=(\Delta _{\mathbb {G}}\otimes \textrm {id})\circ \Delta _{\mathbb {G}}$.

2. (ii) Span$\{\Delta _{\mathbb {G}}(C(\mathbb {G}))(1\otimes C(\mathbb {G}))\}$ and Span$\{\Delta _{\mathbb {G}}(C(\mathbb {G}))(C(\mathbb {G})\otimes 1)\}$ are dense in $C(\mathbb {G})\otimes C(\mathbb {G})$.

Remark 2.18 Strictly speaking the quantum group $\mathbb {G}$ is the dual object of the pair $(C(\mathbb {G}),\Delta _{\mathbb {G}})$. But following [Reference Maes and Van Daele12], we shall refer the dual object $(C(\mathbb {G}),\Delta _{\mathbb {G}})$ as the quantum group itself without mentioning it explicitly.

Given a CQG $\mathbb {G}$, there is a canonical dense Hopf $\ast$-algebra $C(\mathbb {G})_{0}$ in $C(\mathbb {G})$ on which an antipode $\kappa$ and counit $\epsilon$ are defined. Given two CQG's $\mathbb {G}_{1}$ and $\mathbb {G}_{2}$, a CQG morphism between them is a unital $C^{\ast }$-homomorphism $\pi :C(\mathbb {G}_{1})\rightarrow C(\mathbb {G}_{2})$ such that $(\pi \otimes \pi )\circ \Delta _{\mathbb {G}_{1}}=\Delta _{\mathbb {G}_{2}}\circ \pi$.

Definition 2.19 Given a (unital) $C^{\ast }$-algebra ${{\mathcal {C}}}$, a CQG $\mathbb {G}$ is said to act faithfully on ${{\mathcal {C}}}$ if there is a unital $C^{\ast }$-homomorphism $\alpha :{{\mathcal {C}}}\rightarrow {{\mathcal {C}}}\otimes C(\mathbb {G})$ satisfying

1. (i) $(\alpha \otimes \textrm {id})\circ \alpha =(\textrm {id}\otimes \Delta _{\mathbb {G}})\circ \alpha$.

2. (ii) Span$\{\alpha ({{\mathcal {C}}})(1\otimes C(\mathbb {G}))\}$ is dense in ${{\mathcal {C}}}\otimes C(\mathbb {G})$.

3. (iii) The $\ast$-algebra generated by the set $\{(\omega \otimes \textrm {id})\circ \alpha ({{\mathcal {C}}}): \omega \in {{\mathcal {C}}}^{\ast }\}$ is norm-dense in $C(\mathbb {G})$.

Definition 2.20 An action $\alpha :{{\mathcal {C}}}\rightarrow {{\mathcal {C}}}\otimes C(\mathbb {G})$ is said to be ergodic if $\alpha (c)=c\otimes 1$ implies $c\in \mathbb {C}1$.

Definition 2.21 Given an action $\alpha$ of a CQG $\mathbb {G}$ on a $C^{\ast }$-algebra ${{\mathcal {C}}}$, $\alpha$ is said to preserve a state $\tau$ on $C(\mathbb {G})$ if $(\tau \otimes \textrm {id})\circ \alpha (a)=\tau (a)1$ for all $a\in {{\mathcal {C}}}$.

Definition 2.22 Def 2.1 of [Reference Bichon3]

Given a unital $C^{\ast }$-algebra ${{\mathcal {C}}}$, the quantum automorphism group of ${{\mathcal {C}}}$ is a CQG $\mathbb {G}$ acting faithfully on ${{\mathcal {C}}}$ satisfying the following universal property:

If $\mathbb {B}$ is any CQG acting faithfully on ${{\mathcal {C}}}$, there is a surjective CQG morphism $\pi :C(\mathbb {G})\rightarrow C(\mathbb {B})$ such that $(\textrm {id}\otimes \pi )\circ \alpha =\beta$, where $\beta :{{\mathcal {C}}}\rightarrow {{\mathcal {C}}}\otimes C(\mathbb {B})$ is the corresponding action of $\mathbb {B}$ on ${{\mathcal {C}}}$ and $\alpha$ is the action of $\mathbb {G}$ on ${{\mathcal {C}}}$.

From now on, we shall drop the suffix of $\Delta$ whenever the quantum group is clear from the context.

Example 2.23 If we take a space of $n$ points $X_{n}$ then the quantum automorphism group of the $C^{\ast }$-algebra $C(X_{n})$ is denoted by $S_{n}^{+}$. The underlying $C^{\ast }$-algebra $C(S^{+}_{n})$ is the universal $C^{\ast }$ algebra generated by $\{u_{ij}\}_{i,j=1,\ldots ,n}$ satisfying the following relations (see Theorem 3.1 of [Reference Wang15]):

\[ u_{ij}^{2}=u_{ij},\quad u_{ij}^{{\ast}}=u_{ij}, \quad \sum_{k=1}^{n}u_{ik}=\sum_{k=1}^{n}u_{ki}=1,\ i,j=1,\ldots,n. \]

The coproduct on the generators is given by $\Delta (u_{ij})=\sum \nolimits _{k=1}^{n}u_{ik}\otimes u_{kj}$.

2.2.2 Quantum automorphism group of finite graphs and graph $C^{\ast }$-algebras

Recall the definition of finite, directed graph $\Gamma =((V=v_{1},\ldots ,v_{m}),( E= e_{1},\ldots , e_{n}))$ without any multiple edge from Subsection 2.1.

Definition 2.24 The quantum automorphism group of a graph $\Gamma$ without any multiple edge will be denoted by $\textrm {Aut}^{+}(\Gamma )$. The underlying $C^{\ast }$-algebra $C(\textrm {Aut}^{+}(\Gamma ))$ is defined to be the quotient $C(S^{+}_{n})/(AD-DA)$, where $A=((u_{ij}))_{i.j=1,\ldots ,m}$, and $D$ is the adjacency matrix for $\Gamma$. The coproduct on the generators is again given by $\Delta (u_{ij})=\sum \nolimits _{k=1}^{m}u_{ik}\otimes u_{kj}$.

For the classical automorphism group $\textrm {Aut}(\Gamma )$, the commutative $C^{\ast }$-algebra $C(\textrm {Aut}(\Gamma ))$ is generated by $u_{ij}$ where $u_{ij}$ is a function on $S_{n}$ taking value $1$ on the permutation which sends $i$th vertex to $j$th vertex and takes the value zero on other elements of the group. It is a quantum subgroup of $\textrm {Aut}^{+}(\Gamma )$. The surjective CQG morphism $\pi :C(\textrm {Aut}^{+}(\Gamma ))\rightarrow C(\textrm {Aut}(\Gamma ))$ sends the generators to generators.

Remark 2.25 Since $\textrm {Aut}^{+}(\Gamma )$ is a quantum subgroup of $S_{n}^{+}$, it is a Kac algebra and hence $\kappa (u_{ij})=u_{ji}^{\ast }=u_{ji}$. Applying $\kappa$ to the equation $AD=DA$, we get $A^{T}D=DA^{T}$, where $A^{T}=((u_{ji}))$.

With analogy of vertex-transitive action of the automorphism group of a graph, we have the following

Definition 2.26 A graph $\Gamma$ is said to be quantum vertex transitive if the generators $u_{ij}$ of $C(\textrm {Aut}^{+}(\Gamma ))$ are all non-zero.

Remark 2.27 It is easy to see that if a graph is vertex transitive, it must be quantum vertex transitive.

Proposition 2.28 Corollary 3.7 of [Reference Lupini, Mancinska and Roberson11]

The action of $\textrm {Aut}^{+}(\Gamma )$ on $C(V)$ is ergodic if and only if the action is quantum vertex transitive.

Remark 2.29 For a graph $\Gamma =(V,E)$, when we talk about ergodic action, we always take the corresponding $C^{\ast }$-algebra to be $C(V)$.

In the next proposition, we shall see that in fact for a finite, connected graph $\Gamma$ without multiple edge the CQG $\textrm {Aut}^{+}(\Gamma )$ has a $C^{\ast }$-action on the infinite-dimensional $C^{\ast }$-algebra $C^{\ast }(\Gamma )$.

Proposition 2.30 see Theorem 4.1 of [Reference Schmidt and Weber14]

Given a directed graph $\Gamma$ without multiple edge, $\textrm {Aut}^{+}(\Gamma )$ has a $C^{\ast }$-action on $C^{\ast }(\Gamma )$. The action is given by

\begin{align*} \alpha(p_{v_{i}})&=\sum_{k=1}^{m}p_{v_{k}}\otimes u_{ki}\\ \alpha(S_{j})&=\sum_{l=1}^{n}S_{l}\otimes u_{s(e_{l})s(e_{j})}u_{t(e_{l})t(e_{j})}. \end{align*}

Proposition 2.31 Suppose $\Gamma =(V=(v_{1},\ldots ,v_{m}),E=(e_{1},\ldots ,e_{n}))$ is a finite, directed graph without any multiple edge as before. For a $\textrm {KMS}_{\beta }$ state $\tau$ on the graph $C^{\ast }$-algebra $C^{\ast }(\Gamma )$, $\textrm {Aut}^{+}(\Gamma )$ preserves $\tau$ if and only if $(\tau \otimes \textrm {id})\circ \alpha (p_{v_{i}})=\tau (p_{v_{i}})1$ for all $i=1,\ldots ,m$.

We start with proving the following lemma which will be used to prove the above proposition. In the following lemma, $\Gamma =(V,E)$ is again a finite, directed graph without any multiple edge.

Lemma 2.32 Suppose $\textrm {Aut}^{+}(\Gamma )$ preserves some linear functional $\tau$ on $C(V)$. Then $\tau (p_{v_{i}})\neq \tau (p_{v_{j}})\Rightarrow u_{ij}=0$.

Proof. Let $i,j$ be such that $\tau (p_{v_{i}})\neq \tau (p_{v_{j}})$. By the assumption,

\begin{align*} (\tau\otimes\textrm{id})\circ\alpha(p_{v_{i}})&=\tau(p_{v_{i}})1\\ \Rightarrow \sum_{k}\tau(p_{v_{k}})u_{ki}&=\tau(p_{v_{i}})1. \end{align*}

Multiplying both sides of the last equation by $u_{ji}$ and using the orthogonality, we get $\tau (p_{v_{j}})u_{ji}=\tau (p_{v_{i}})u_{ji}$ i.e. $(\tau (p_{v_{j}})-\tau (p_{v_{i}}))u_{ji}=0$ and hence $u_{ji}=0$ as $\tau (p_{v_{i}})\neq \tau (p_{v_{j}})$. Applying $\kappa$, we get $u_{ij}=u_{ji}=0$.

Proof of Proposition 2.31. If $\textrm {Aut}^{+}(\Gamma )$ preserves $\tau$, then $(\tau \otimes \textrm {id})\circ \alpha (p_{v_{i}})=\tau (p_{v_{i}})1$ for all $i=1,\ldots ,m$ trivially. For the converse, given $(\tau \otimes \textrm {id})\circ \alpha (p_{v_{i}})=\tau (p_{v_{i}})1$ for all $i=1,\ldots ,m$, we need to show that $(\tau \otimes \textrm {id})\circ \alpha (S_{\mu }S_{\nu }^{\ast })=\tau (S_{\mu }S_{\nu }^{\ast })1$ for all $\mu ,\nu \in E^{\ast }$. The proof is similar to that of Theorem 3.5 of [Reference Joardar and Mandal8]. It is easy to see that for $|\mu |\neq |\nu |$, $(\tau \otimes \textrm {id})\circ \alpha (S_{\mu }S_{\nu }^{\ast })=0=\tau (S_{\mu }S_{\nu }^{\ast })$. So let $|\mu |=|\nu |$. For $\mu =\nu =e_{i_{1}}e_{i_{2}}\ldots e_{i_{p}}$, we have $S_{\mu }S_{\mu }^{\ast }=S_{i_{1}}\ldots S_{i_{p}}S_{i_{p}}^{\ast }\ldots S_{i_{1}}^{\ast }$. So

\begin{align*} (\tau\otimes\textrm{id})\circ\alpha(S_{\mu}S_{\mu}^{{\ast}})&=(\tau\otimes\textrm{id})\bigg(\sum S_{j_{1}}\cdots S_{j_{p}}S_{j_{p}}^{{\ast}}\cdots S_{j_{1}}^{{\ast}}\otimes u_{s(e_{j_{1}})s(e_{i_{1}})}u_{t(e_{j_{1}})t(e_{i_{1}})}\\ &\quad \cdots u_{s(e_{j_{p}})s(e_{i_{p}})}u_{t(e_{j_{p}})t(e_{i_{p}})}u_{s(e_{j_{p}})s(e_{i_{p}})} \cdots u_{t(e_{j_{1}})t(e_{i_{1}})}u_{s(e_{j_{1}})s(e_{i_{1}})}\bigg) \end{align*}

By the same argument as given in the proof of the Theorem 3.5 of [Reference Joardar and Mandal8], for $S_{j_{1}}\cdots S_{j_{p}}=0$, $u_{s(e_{j_{1}})s(e_{i_{1}})}u_{t(e_{j_{1}})t(e_{i_{1}})}\cdots u_{s(e_{j_{p}})s(e_{i_{p}})}u_{t(e_{j_{p}})s(t_{i_{p}})}=0$ and hence the last expression equals to

\begin{align*} &\sum e^{-\beta|\mu|}\tau(p_{t(e_{j_{p}})}) u_{s(e_{j_{1}})s(e_{i_{1}})}u_{t(e_{j_{1}})t(e_{i_{1}})}\ldots u_{s(e_{j_{p}})s(e_{i_{p}})}u_{t(e_{j_{p}})t(e_{i_{p}})}\\ &\quad u_{s(e_{j_{p}})s(e_{i_{p}})}\ldots u_{t(e_{j_{1}})t(e_{i_{1}})}u_{s(e_{j_{1}})s(e_{i_{1}})} \end{align*}

Observe that any $\textrm {KMS}_{\beta }$ state restricts to a state on $C(V)$ so that by Lemma 2.32, for $\tau (p_{t(e_{j_{p}})})\neq \tau (p_{t(e_{i_{p}})})$, $u_{t(e_{j_{p}})t(e_{i_{p}})}=0$. Using this, the last summation reduces to

\begin{align*} & e^{-\beta|\mu|}\sum\tau(p_{t(e_{i_{p}})})u_{s(e_{j_{1}})s(e_{i_{1}})}u_{t(e_{j_{1}})t(e_{i_{1}})}\cdots u_{s(e_{j_{p}})s(e_{i_{p}})}u_{t(e_{j_{p}})t(e_{i_{p}})}\\ &\quad u_{s(e_{j_{p}})s(e_{i_{p}})}\ldots u_{t(e_{j_{1}})t(e_{i_{1}})}u_{s(e_{j_{1}})s(e_{i_{1}})}. \end{align*}

Using the same arguments used in the proof of Theorem 3.13 in [Reference Joardar and Mandal7] repeatedly, it can be shown that the last summation actually equals to $e^{-\beta |\mu |}\tau (p_{t(e_{i_{p}})})=\tau (S_{\mu }S_{\mu }^{\ast })$. Hence

\[ (\tau\otimes\textrm{id})\circ\alpha(S_{\mu}S_{\mu}^{{\ast}})=\tau(S_{\mu}S_{\mu}^{{\ast}})1. \]

With similar reasoning, it can easily be verified that for $\mu \neq \nu$, $(\tau \otimes \textrm {id})\circ \alpha (S_{\mu }S_{\nu }^{\ast })=0=\tau (S_{\mu }S_{\nu }^{\ast })1$. Hence by linearity and continuity of $\tau$, for any $a\in C^{\ast }(\Gamma )$, $(\tau \otimes \textrm {id})\circ \alpha (a)=\tau (a).1$.

If we apply Proposition 2.31 to the action of classical automorphism group of a graph on the corresponding graph $C^{\ast }$-algebra, we get the following

Lemma 2.33 Given a $\textrm {KMS}_{\beta }$ state $\tau$ on $C^{\ast }(\Gamma )$, we denote the vector $(\tau (p_{v_{1}}),\ldots ,\tau (p_{v_{m}}))$ by ${{\mathcal {N}}}^{\tau }$. If we denote the permutation matrix corresponding to an element $g\in \textrm {Aut}(\Gamma )$ by $B$, then $\textrm {Aut}(\Gamma )$ preserves $\tau$ if and only if $B{{\mathcal {N}}}^{\tau }={{\mathcal {N}}}^{\tau }$.

Proof. Follows from the easy observation that $(\tau \otimes \textrm {id})\circ \alpha (p_{v_{i}})=\tau (p_{v_{i}})1$ implies $B{{\mathcal {N}}}^{\tau }={{\mathcal {N}}}^{\tau }$ for the classical automorphism group of the graph.

3.1. Strongly connected graphs

Recall the unique KMS state of $C^{\ast }(\Gamma )$ for a strongly connected graph $\Gamma$ with vertex matrix $D$. We denote the $ij$th entry of $D$ by $d_{ij}$. The unique KMS state at the critical inverse temperature $\textrm {ln}(\rho (D))$ is determined by the unique normalized Perron–Frobenius eigenvector of $D$ corresponding to the eigenvalue $\rho (D)$. If the state is denoted by $\tau$, the eigenvector is given by $((\tau (p_{v_{i}})))_{i=1,\ldots ,m}$ where $m$ is the number of vertices. Now we prove the main result of this subsection.

Proof. Note that by Proposition 2.31, it suffices to show that $(\tau \otimes \textrm {id})\circ \alpha (p_{v_{i}})=\tau (p_{v_{i}})1 \forall \ i=1,\ldots ,m$. Recall the action of $\textrm {Aut}^{+}(\Gamma )$ on $C^{\ast }(\Gamma )$. We continue to denote the matrix $((u_{ij}))$ by $A$. For a state $\phi$ of $C(\textrm {Aut}^{+}(\Gamma ))$, we denote the vector whose $i$th entry is $(\tau \otimes \phi )\circ \alpha (p_{v_{i}})$ by $\textbf {v}_{\phi }$. Then

\begin{align*} (D\textbf{v}_{\phi})_{i}&=\sum_{j}d_{ij}(\tau\otimes\phi)\circ\alpha(p_{v_{j}})\\ &= \sum_{j,k}\tau(p_{v_{k}})\phi(d_{ij}u_{kj})\\ &= \sum_{k}\tau(p_{v_{k}})\phi\bigg(\sum_{j}d_{ij}u_{kj}\bigg)\\ &= \sum_{k}\tau(p_{v_{k}})\phi\bigg(\sum_{j}u_{ji}d_{jk}\bigg) (DA^{T}=A^{T}D \text{by Remark 2.25})\\ &= \sum_{j,k}d_{jk}\tau(p_{v_{k}})\phi(u_{ji})\\ &=\rho(D)\sum_{j}\tau(p_{v_{j}})\phi(u_{ji}) (D((\tau(p_{i})))^{T}=\rho(D)((\tau(p_{v_{i}})))^{T})\\ &= \rho(D) (\tau\otimes\phi)\circ\alpha(p_{v_{i}}). \end{align*}

Hence $\textbf {v}_{\phi }$ is an eigenvector of $D$ corresponding to the eigenvalue $\rho (D)$. By the one dimensionality of the eigenspace, we have some constant $C_{\phi }$ such that $(\tau \otimes \phi )\circ \alpha (p_{v_{i}})=C_{\phi }\tau (p_{v_{i}})$ for all $i=1,\ldots ,m$. To determine the constant $C_{\phi }$, we take the summation over $i$ on both sides and get

\begin{align*} &\sum_{i}(\tau\otimes\phi)\circ\alpha(p_{v_{i}})=C_{\phi}\sum_{i}\tau(p_{v_{i}})\\ &\quad \Rightarrow \sum_{i,j}\tau(p_{v_{j}})\phi(u_{ji})=C_{\phi}\\ &\quad\Rightarrow \sum_{j}\tau(p_{v_{j}})\sum_{i}\phi(u_{ji})=C_{\phi}\\ &\quad\Rightarrow C_{\phi}=1. \end{align*}

Hence $\phi ((\tau \otimes \textrm {id})\circ \alpha (p_{v_{i}}))=\phi (\tau (p_{v_{i}})1)$ for all $i$ and for all state $\phi$ which implies that $(\tau \otimes \textrm {id})\circ \alpha (p_{v_{i}})=\tau (p_{v_{i}})1$ for all $i=1,\ldots ,m$.

We end this subsection with a proposition about the non-ergodicity of the action of $\textrm {Aut}^{+}(\Gamma )$ on $C(V)$ for a strongly connected graph $\Gamma =(V,E)$. Note that the following proposition does not deal with states on the infinite-dimensional $C^{\ast }$-algebra $C^{\ast }(\Gamma )$.

3.2. Circulant graphs

Consider a finite graph $\Gamma =(V,E)$ with $m$-vertices $(v_{1},\ldots ,v_{m})$. Recall the notation ${{\mathcal {N}}}^{\tau }$ for a $\textrm {KMS}_{\beta }$ state $\tau$ on $C^{\ast }(\Gamma )$.

Now recall from the discussion following Corollary 2.15 and Lemma 2.16 that for a circulant graph, the KMS states exist only at the critical inverse temperature, but they are not necessarily unique. We shall prove that if we further assume the invariance of such a state under the action of the automorphism group of the graph, then it is unique.

3.3. Graph of the Mermin–Peres magic square game

We start this subsection by clarifying a few notation to be used in this subsection. Given an undirected graph $\Gamma$, we make it directed by declaring that both $(i,j)$ and $(j,i)$ are in the edge set whenever there is an edge between two vertices $v_{i}$ and $v_{j}$. The vertex matrix of such a directed graph is symmetric by definition. In this subsection, we use the notation $\overrightarrow {\Gamma }$ for the directed graph coming from an undirected graph $\Gamma$ in this way. $\Gamma$ will always denote an undirected graph.

Given two graphs $\Gamma _{1}=(V_{1},E_{1}),\Gamma _{2}=(V_{2},E_{2})$, their disjoint union $\Gamma _{1}\cup \Gamma _{2}$ is defined to be the graph $\Gamma =(V,E)$ such that $V=V_{1}\cup V_{2}$. There is an edge between two vertices $v_{i}, v_{j}\in V_{1}\cup V_{2}$ if both the vertices belong to either $\Gamma _{1}$ or $\Gamma _{2}$ and they have an edge in the corresponding graph.

To prove the proposition, we require the following

Proof of Proposition 3.9. Proof of Proposition 3.9

We assume that $\overrightarrow {\Gamma _{1}}$ has $n$-vertices and $\overrightarrow {\Gamma _{2}}$ has $m$-vertices. Let us denote the vertex matrix of $\overrightarrow {\Gamma }$ by $D$. $D$ is given by the matrix

\[ \begin{bmatrix} D_{1} & 0_{n\times m}\\ 0_{m\times n} & D_{2} \end{bmatrix}. \]

Then the spectral radius is equal to $\lambda$ by Lemma 3.10. Also, it is easy to see that the spectral radius is an eigenvalue of the matrix $D$. Now $\lambda$ has a one-dimensional eigenspace for $D_{1}$ spanned by say $w_{1}$ and a one-dimensional eigenspace for $D_{2}$ spanned by say $w_{2}$ as both the graphs are connected and hence strongly connected as directed graphs. We take both the eigenvectors normalized for convenience. Then for $D$, the eigenspace corresponding to $\lambda$ is two dimensional spanned by the vectors $\textbf{w}_{\textbf{1}}=(w_{1},0_{m})$ and $\textbf{w}_{\textbf{2}}=(0_{n},w_{2})$ where $0_{k}$ is the zero $k$-tuple. So the eigenspace of $D$ corresponding to the eigenvalue $\lambda$ is given by $\{\xi \textbf {w}_{\textbf{1}}+\eta \textbf {w}_{\textbf{2}}:(\xi ,\eta )\in \mathbb {C}^{2}- ( 0,0) \}$. For any $(\xi ,\eta )\in \mathbb {C}^{2}$, $\xi \textbf {w}_{\textbf{1}}+\eta \textbf {w}_{\textbf{2}}=(\xi w_{1},\eta w_{2})$. It is easy to see that there are infinitely many choices of $\xi ,\eta$ such that corresponding eigenvector is normalized with all its entries being non-negative which in turn give rise to infinitely many KMS states. The set of normalized vectors is given by $\{(\xi w_{1},(1-\xi )w_{2}):0\leq \xi \leq 1\}$. We shall show that any $B\in \textrm {Aut}(\overrightarrow {\Gamma })$ keeps such a normalized eigenvector invariant. Let $\textbf {w}=(\xi w_{1},(1-\xi )w_{2})$ be one such choice. By Proposition 3.8, any $B\in \textrm {Aut}(\overrightarrow {\Gamma })$ can be written in the matrix form $\left [\begin {smallmatrix} B_{1} & 0_{n\times m}\\ 0_{m\times n} & B_{2}.\end {smallmatrix}\right ]$, for $B_{i}\in \textrm {Aut}(\overrightarrow {\Gamma _{i}})$ and $i=1,2$. Then $B\textbf {w}=(\xi B_{1}w_{1},(1-\xi ) B_{2}w_{2})$. Since $w_{1}, w_{2}$ are Perron–Frobenius eigenvectors of $D_{1}, D_{2}$ respectively with both the graphs $\overrightarrow {\Gamma _{1}}, \overrightarrow {\Gamma _{2}}$ strongly connected, by Proposition 3.1, $B_{i}(w_{i})=w_{i}$ for $i=1,2$. So $B\textbf {w}=(\xi w_{1},(1-\xi )w_{2})=\textbf {w}$. Hence an application of Lemma 2.33 completes the proof of the proposition.

Now we turn to the main object of study of this subsection. A linear binary constraint system (LBCS) ${{\mathcal {F}}}$ consists of a family of binary variables $x_{1},\ldots ,x_{n}$ and constraints $C_{1},\ldots ,C_{m}$, where each $C_{l}$ is a linear equation over $\mathbb {F}_{2}$ in some subset of the variables i.e. each $C_{l}$ is of the form $\sum \nolimits _{x_{i}\in S_{l}}x_{i}=b_{l}$ for some $S_{l}\subset \{x_{1},\ldots ,x_{n}\}$. Corresponding to every LBCS ${{\mathcal {F}}}$, one can associate a graph (see section 6.2 of [Reference Atserias, Mancinska, Roberson, Samal, Severini and Varvitsiotis1]) to be denoted by ${{\mathcal {G}}}({{\mathcal {F}}})$. The following is an example of an LBCS.

\begin{align*} & x_{1}+x_{2}+x_{3}=0 \quad x_{1}+x_{4}+x_{7}=0\\ & x_{4}+x_{5}+x_{6}=0 \quad x_{2}+x_{5}+x_{8}=0\\ & x_{7}+x_{8}+x_{9}=0 \quad x_{3}+x_{6}+x_{9}=1, \end{align*}

where the addition is over $\mathbb {F}_{2}$. From now on ${{\mathcal {F}}}$ will always mean the above LBCS.

In light of the Theorem 6.2, 6.3 and 6.4 of [Reference Atserias, Mancinska, Roberson, Samal, Severini and Varvitsiotis1], the corresponding graphs ${{\mathcal {G}}}({{\mathcal {F}}})$ and ${{\mathcal {G}}}({{\mathcal {F}}}_{0})$ are quantum isomorphic, but not isomorphic. The graph ${{\mathcal {G}}}({{\mathcal {F}}})$ is called the graph of Mermin–Peres magic square game.

Both the graphs ${{\mathcal {G}}}({{\mathcal {F}}})$ and ${{\mathcal {G}}}({{\mathcal {F}}}_{0})$ are vertex transitive (in fact they are Cayley as mentioned in [Reference Lupini, Mancinska and Roberson11]) and hence quantum vertex transitive by Remark 2.27. Combining the facts that ${{\mathcal {G}}}({{\mathcal {F}}})$ and ${{\mathcal {G}}}({{\mathcal {F}}}_{0})$ are quantum isomorphic and quantum vertex transitive with Lemma 4.15 of [Reference Lupini, Mancinska and Roberson11], we get

By Remark 3.7,

It can be verified that both the graphs ${{\mathcal {G}}}({{\mathcal {F}}})$ and ${{\mathcal {G}}}({{\mathcal {F}}}_{0})$ are connected with $24$ vertices each such that the vertex matrices of the graphs $\overrightarrow {{{\mathcal {G}}}({{\mathcal {F}}})}$ and $\overrightarrow {{{\mathcal {G}}}({{\mathcal {F}}})_{0}}$ have spectral radius $9$ (see Figures 2 and 3). Then since they are non-isomorphic, by Proposition 3.9, $C^{\ast }(\overrightarrow {{{\mathcal {G}}}({{\mathcal {F}}})\cup {{\mathcal {G}}}({{\mathcal {F}}}_{0})})$ has infinitely many KMS states at the critical inverse temperature $\textrm {ln}(9)$ all of which are invariant under the action of the classical automorphism group of $\overrightarrow {{{\mathcal {G}}}({{\mathcal {F}}})\cup {{\mathcal {G}}}({{\mathcal {F}}}_{0})}$. But as mentioned earlier, if we further assume that the KMS state at the critical inverse temperature is invariant under the action of $\textrm {Aut}^{+}(\overrightarrow {{{\mathcal {G}}}({{\mathcal {F}}})\cup {{\mathcal {G}}}({{\mathcal {F}}}_{0})})$, then it is necessarily unique. We prove it in the next theorem.

Theorem 3.14 For the LBCS ${{\mathcal {F}}}$, the graph $C^{\ast }$-algebra $C^{\ast }(\overrightarrow {{{\mathcal {G}}}({{\mathcal {F}}})\cup {{\mathcal {G}}}({{\mathcal {F}}}_{0})})$ has a unique $\textrm {Aut}^{+}(\overrightarrow {{{\mathcal {G}}}({{\mathcal {F}}})\cup {{\mathcal {G}}}({{\mathcal {F}}}_{0})})$-invariant KMS state $\tau$ given by

\[ \tau(S_{\mu}S_{\nu}^{{\ast}})=\delta_{\mu,\nu}\frac{1}{9^{|\mu|}48}. \]

Figure 2. Graph ${{\mathcal {G}}}({{\mathcal {F}}})$.

Figure 3. Graph ${{\mathcal {G}}}({{\mathcal {F}}}_{0})$.

Proof. By Corollary 3.13, the graph $\overrightarrow {({{\mathcal {G}}}({{\mathcal {F}}})\cup {{\mathcal {G}}}({{\mathcal {F}}}_{0})})$ is quantum vertex transitive. Hence by Lemma 2.32, for any $\textrm {Aut}^{+}(\overrightarrow {{{\mathcal {G}}}({{\mathcal {F}}})\cup {{\mathcal {G}}}({{\mathcal {F}}}_{0})})$ invariant KMS state $\tau$ on $C^{\ast }(\overrightarrow {{{\mathcal {G}}}({{\mathcal {F}}})\cup {{\mathcal {G}}}({{\mathcal {F}}}_{0})}))$ at the critical inverse temperature $\textrm {ln}(9)$, we have $\tau (p_{v_{i}})=\tau (p_{v_{j}})$ for all $i,j$. That forces $\tau (p_{v_{i}})$ to be $\frac {1}{48}$ for all $i=1,\ldots ,48$. Since $(\frac {1}{48},\ldots ,\frac {1}{48})$ is an eigenvector corresponding to the eigenvalue $9$, there is a unique KMS state on $C^{\ast }(\overrightarrow {{{\mathcal {G}}}({{\mathcal {F}}})\cup {{\mathcal {G}}}({{\mathcal {F}}}_{0})})$ at the critical inverse temperature $\textrm {ln}(9)$ satisfying

\[ \tau(S_{\mu}S_{\nu}^{{\ast}})=\delta_{\mu,\nu}\frac{1}{9^{|\mu|}48}. \]

$\textrm {Aut}^{+}(\overrightarrow {{{\mathcal {G}}}({{\mathcal {F}}})\cup {{\mathcal {G}}}({{\mathcal {F}}}_{0})})$ preserves the above KMS state by following the same line of arguments as given in Remark 3.6. To complete the proof, we need to show that the only possible inverse temperature where a KMS state could occur is the critical inverse temperature. For that first notice that the graph $\overrightarrow {{{\mathcal {G}}}({{\mathcal {F}}})\cup {{\mathcal {G}}}({{\mathcal {F}}}_{0})}$ is without sink. We have already observed that the spectral radius $9$ has an eigenvector with all entries being strictly positive (column vector with all its entries $\frac {1}{48}$). Also the vertex matrix of the graph $\overrightarrow {{{\mathcal {G}}}({{\mathcal {F}}})\cup {{\mathcal {G}}}({{\mathcal {F}}}_{0})}$ is symmetric implying that $(\frac {1}{48},\ldots ,\frac {1}{48})D=9(\frac {1}{48},\ldots ,\frac {1}{48})$. Hence an application of Lemma 2.5 finishes the proof of the theorem.