2.1. Finite-directed graphs and their $C^*$-algebras

Let $E=(E^0,\,E^1,\,r,\,s)$ be a directed graph, where $E^0$ and $E^1$ are finite sets of vertices and edges, respectively, and $r,\,s:E^1\to E^0$ are range and source maps, respectively. A path $\mu$ of length $|\mu |=k\geq 1$ is a sequence $\mu =(\mu _1,\,\ldots,\,\mu _k)$ of $k$ edges $\mu _j$ such that $r(\mu _j)=s(\mu _{j+1})$ for $j=1,\,\ldots,\, k-1$. We view the vertices as paths of length $0$. The set of all paths of length $k$ is denoted $E^k$, and $E^*$ denotes the collection of all finite paths (including paths of length zero). The range and source maps naturally extend from edges $E^1$ to paths $E^k$. A sink is a vertex $v$ which emits no edges, i.e. $s^{-1}(v)=\emptyset$. A source is a vertex $w$ which receives no edges, i.e. $r^{-1}(w)=\emptyset$. By a cycle we mean a path $\mu$ of length $|\mu |\geq 1$ such that $s(\mu )=r(\mu )$. A cycle $\mu =(\mu _1,\,\ldots,\,\mu _k)$ has an exit if there is a $j$ such that $s(\mu _j)$ emits at least two distinct edges. If $\alpha$ is an initial subpath of $\beta$ then we write $\alpha \prec \beta$. Graph $E$ is transitive if for any two vertices $v,\,w$ there exists a path $\mu \in E^*$ from $v$ to $w$ of non-zero length. Thus, a transitive graph does not contain any sinks or sources. Given a graph $E$, we will denote by $A=[A(v,\,w)]_{v,w\in E^0}$ its adjacency matrix. That is, $A$ is a matrix with rows and columns indexed by the vertices of $E$, such that $A(v,\,w)$ is the number of edges with source $v$ and range $w$. If the graph $E$ is transitive then the corresponding matrix $A$ is irreducible, in the sense that for any two vertices $v,\,w$ there is a positive integer $k$ such that $A^k(v,\,w)>0$. Here $A^k$ is the $k$’th power of matrix $A$ and hence $A^k(w,\, v)$ gives the number of paths from vertex $w$ to vertex $v$.

The $C^*$-algebra $C^*(E)$ corresponding to a graph $E$ is by definition, [Reference Kumjian, Pask and Raeburn19, Reference Kumjian, Pask, Raeburn and Renault20], the universal $C^*$-algebra generated by mutually orthogonal projections $P_v$, $v\in E^0$, and partial isometries $S_e$, $e\in E^1$, subject to the following two relations:

1. (GA1) $S_e^*S_e=P_{r(e)}$,

2. (GA2) $P_v=\sum \nolimits _{s(e)=v}S_e S_e^*$ if $v\in E^0$ emits at least one edge.

For a path $\mu =(\mu _1,\,\ldots,\,\mu _k)$, we denote by $S_\mu = S_{\mu _1}\cdots S_{\mu _k}$ the corresponding partial isometry in $C^*(E)$. We agree to write $S_v=P_v$ for a $v\in E^0$. Each $S_\mu$ is non-zero with the domain projection $P_{r(\mu )}$. Then $C^*(E)$ is the closed span of $\{S_\mu S_\nu ^*:\mu,\,\nu \in E^*\}$. Note that $S_\mu S_\nu ^*$ is non-zero if and only if $r(\mu )=r(\nu )$. In that case, $S_\mu S_\nu ^*$ is a partial isometry with domain and range projections equal to $S_\nu S_\nu ^*$ and $S_\mu S_\mu ^*$, respectively.

The range projections $P_\mu =S_\mu S_\mu ^*$ of all partial isometries $S_\mu$ mutually commute, and the abelian $C^*$-subalgebra of $C^*(E)$ generated by all of them is called the diagonal subalgebra and denoted ${\mathcal {D}}_E$. We set ${\mathcal {D}}^0_E = {\rm span}\{P_v : v\in E^0 \}$ and, more generally, ${\mathcal {D}}_E^k= {\rm span}\{P_\mu : \mu \in E^k \}$ for $k\geq 0$. $C^*$-algebra ${\mathcal {D}}_E$ coincides with the norm closure of $\bigcup _{k=0}^\infty {\mathcal {D}}_E^k$. If $E$ does not contain sinks and all cycles have exits then ${\mathcal {D}}_E$ is a MASA (maximal abelian subalgebra) in $C^*(E)$ by [Reference Hopenwasser, Peters and Power15, Theorem 5.2]. Throughout this paper, we make the following

standing assumption: all graphs we consider are finite, transitive and all cycles in these graphs admit exits.

There exists a strongly continuous action $\gamma$ of the circle group $U(1)$ on $C^*(E)$, called the gauge action, such that $\gamma _z(S_e)=zS_e$ and $\gamma _z(P_v)=P_v$ for all $e\in E^1$, $v\in E^0$ and $z\in U(1)\subseteq {\mathbb {C}}$. The fixed-point algebra $C^*(E)^\gamma$ for the gauge action is an AF-algebra, denoted ${\mathcal {F}}_E$ and called the core AF-subalgebra of $C^*(E)$. ${\mathcal {F}}_E$ is the closed span of $\{S_\mu S_\nu ^*:\mu,\,\nu \in E^*,\,\;|\mu |=|\nu |\}$. For $k\in {\mathbb {N}}=\{0,\,1,\,2,\,\ldots \}$ we denote by ${\mathcal {F}}_E^k$ the linear span of $\{S_\mu S_\nu ^*:\mu,\,\nu \in E^*,\,\;|\mu |=|\nu |= k\}$. $C^*$-algebra ${\mathcal {F}}_E$ coincides with the norm closure of $\bigcup _{k=0}^\infty {\mathcal {F}}_E^k$.

We consider the usual shift on $C^*(E)$, [Reference Cuntz and Krieger10], given by

(1)\begin{equation} \varphi(x)=\displaystyle\sum_{e\in E^1} S_e x S_e^*, \quad x\in C^*(E). \end{equation}

In general, for finite graphs without sinks and sources, the shift is a unital, completely positive map. However, it is an injective $*$-homomorphism when restricted to the relative commutant $({\mathcal {D}}_E^0)'\cap C^*(E)$ of ${\mathcal {D}}_E^0$ in $C^*(E)$.

We observe that for each $v\in E^0$ projection $\varphi ^k(P_v)$ is minimal in the centre of ${\mathcal {F}}_E^k$. The $C^*$-algebra ${\mathcal {F}}_E^k\varphi ^k(P_v)$ is the linear span of partial isometries $S_\mu S_\nu ^*$ with $|\mu |=|\nu |=k$ and $r(\mu )=r(\nu )=v$. It is isomorphic to the full matrix algebra of size $\sum \nolimits _{w\in E^0} A^k(w,\, v)$. The multiplicity of ${\mathcal {F}}_E^k\varphi ^k(P_v)$ in ${\mathcal {F}}_E^{k+1}\varphi ^{k+1}(P_w)$ is $A(v,\,w)$, so the Bratteli diagram for ${\mathcal {F}}_E$ is induced from the graph $E$, see [Reference Bates, Pask, Raeburn and Szymański3, Reference Cuntz and Krieger10, Reference Kumjian, Pask, Raeburn and Renault20].

We denote

(2)\begin{equation} \mathfrak B := ({\mathcal{D}}_E^0)'\cap {\mathcal{F}}_E^1.\end{equation}

That is, $\mathfrak B$ is the linear span of elements $S_e S_f^*$, $e,\,f\in E^1$, with $s(e)=s(f)$. We note that $\mathfrak B$ is contained in the multiplicative domain of $\varphi$. We have ${\mathcal {D}}_E^1 \subseteq \mathfrak B \subseteq {\mathcal {F}}_E^1$ and

(3)\begin{equation} \varphi^k(\mathfrak B) = ({\mathcal{F}}_E^k)' \cap {\mathcal{F}}_E^{k+1} \cong \bigoplus_{v,w\in E^0}M_{A(v,w)}({\mathbb{C}}) \end{equation}

for all $k$. For $v,\,w\in E^0$, we denote

(4)\begin{equation} {}_vQ_w := \displaystyle\sum_{e\in E^1, s(e)=v, r(e)=w}P_e. \end{equation}

Each $_vQ_w$ is a minimal projection in the centre of $\mathfrak B$ and $\mathfrak B_vQ_w\cong M_{A(v,w)}({\mathbb {C}})$. We put

(5)\begin{equation} \mathfrak B_E^k := \bigvee_{j=0}^{k-1}\varphi^j(\mathfrak B), \end{equation}

for $k\geq 1$, the $C^*$-algebra generated by $\bigcup _{j=0}^{k-1}\varphi ^j(\mathfrak B)$. In general, if $A$ and $B$ are both $C^*$-subalgebras of a $C^*$-algebra $C$, then we denote by $A\vee B$ the $C^*$-subalgebra of $C$ generated by $A$ and $B$. Since for all $k$, we have

(6)\begin{equation} {\mathcal{D}}_E^k = \bigvee_{j=0}^{k-1}\varphi^j({\mathcal{D}}_E^1),\end{equation}

it is easy to see that

(7)\begin{equation} {\mathcal{D}}_E^k \subseteq \mathfrak B_E^k \subseteq {\mathcal{F}}_E^k. \end{equation}

We observe that

(8)\begin{equation} {}_vQ_w\varphi(_{v'}Q_{w'}) = \delta_{w,v'} \displaystyle\sum_{s(e)=v, r(e)=s(f)=w, r(f)=w'} P_{ef}. \end{equation}

This implies that

\begin{align*} \mathfrak B_E^k & = \bigoplus_{v_1,\ldots,v_{k+1}\in E^0} \mathfrak B _{v_1}Q_{v_2} \vee \varphi(\mathfrak B _{v_2}Q_{v_3}) \vee \ldots \vee \varphi^{k-1}(\mathfrak B_{v_k}Q_{v_{k+1}}) \\ & = \bigoplus_{v_1,\ldots,v_{k+1}\in E^0} \mathfrak B _{v_1}Q_{v_2} \otimes \varphi(\mathfrak B _{v_2}Q_{v_3}) \otimes \ldots \otimes \varphi^{k-1}(\mathfrak B_{v_k}Q_{v_{k+1}}). \end{align*}

There exist faithful conditional expectations $\varPhi _{{\mathcal {F}}}:C^*(E)\to {\mathcal {F}}_E$ and $\varPhi _{{\mathcal {D}}}:C^*(E)\to {\mathcal {D}}_E$ such that $\varPhi _{{\mathcal {F}}}(S_\mu S_\nu ^*)=0$ for $|\mu |\neq |\nu |$ and $\varPhi _{{\mathcal {D}}}(S_\mu S_\nu ^*)=0$ for $\mu \neq \nu$. We note that $\varPhi _{{\mathcal {D}}} = \varPhi _{{\mathcal {D}}}\circ \varPhi _{{\mathcal {F}}}$ and

\begin{align*} \varPhi_{{\mathcal{D}}}\circ\varphi & = \varphi\circ\varPhi_{{\mathcal{D}}} \text{ on } {\mathcal{D}}_E, \\ \varPhi_{{\mathcal{F}}}\circ\varphi & = \varphi\circ\varPhi_{{\mathcal{F}}} \text{ on } {\mathcal{F}}_E. \end{align*}

For an integer $m\in {\mathbb {Z}}$, we denote by $C^*(E)^{(m)}$ the spectral subspace of the gauge action corresponding to $m$. That is,

(9)\begin{equation} C^*(E)^{(m)}:=\{x\in C^*(E) \mid \gamma_z(x)=z^m x,\,\forall z\in U(1)\}. \end{equation}

In particular, $C^*(E)^{(0)}=C^*(E)^\gamma$. For each $m\in {\mathbb {Z}}$ there is a unital, contractive and completely bounded map $\varPhi ^m:C^*(E)\to C^*(E)^{(m)}$ given by

(10)\begin{equation} \varPhi^m(x) = \displaystyle\int_{z\in U(1)} z^{{-}m}\gamma_z(x){\rm dx}.\end{equation}

In particular, $\varPhi ^0=\varPhi _{\mathcal {F}}$. We have $\varPhi ^m(x)=x$ for all $x\in C^*(E)^{(m)}$. If $x\in C^*(E)$ and $\varPhi ^m(x)=0$ for all $m\in {\mathbb {Z}}$ then $x=0$.

2.2. The trace on the core AF-subalgebra

We recall the definition of a natural trace on the core $AF$-subalgebra ${\mathcal {F}}_E$. For relevant facts from the Perron–Frobenius theory, see for example [Reference Goodman, de la Harpe and Jones12, Reference Graham13].

Let $\beta$ be the Perron–Frobenius eigenvalue of the matrix $A$ and let $(x(v))_{v\in E^0}$ be the corresponding Perron–Frobenius eigenvector. That is, $\beta >0$, for each $v\in E^0$ we have $x(v)>0$, and

(11)\begin{equation} \displaystyle\sum_{w\in E^0} A(v,w) x(w) = \beta x(v). \end{equation}

We set $X:=\sum \nolimits _{v\in E^0} x(v)$ and define a tracial state $\tau$ on ${\mathcal {F}}_E$ so that

(12)\begin{equation} \tau(S_{\mu}S_{\nu}^*) = \delta_{\mu,\nu}\displaystyle\frac{x(r(\mu))}{X\beta^k} \end{equation}

for $\mu,\,\nu \in E^k$. We have $\tau (\varPhi _{{\mathcal {D}}}(x)) = \tau (x)$ for all $x\in {\mathcal {F}}_E$.

Remark 2.1 Trace $\tau$ defined above is not shift invariant, in general. That is, it may happen that $\tau (\varphi (x))\neq \tau (x)$ for some $x\in {\mathcal {F}}_E$. In fact, $\tau$ is $\varphi$-invariant if and only if

\[ \sum_{v\in E^0} A(v,w) = \beta \]

for each $w\in E^0$. For example, the matrix

\[ A = \left( \begin{array}{@{}cc@{}} 2 & 1 \\ 1 & 4 \end{array} \right) \]

does not satisfy this condition.

2.3. Endomorphisms determined by unitaries

Cuntz's classical approach to the study of endomorphisms of ${\mathcal {O}}_n$, [Reference Cuntz9], has been developed further in [Reference Conti and Szymański7] and extended to graph $C^*$-algebras in [Reference Avery, Johansen and Szymański1, Reference Conti, Hong and Szymański4, Reference Johansen, Sørensen and Szymański18].

We denote by ${\mathcal {U}}_E$ the collection of all those unitaries in $C^*(E)$ which commute with all vertex projections $P_v$, $v\in E^0$. That is

(13)\begin{equation} {\mathcal{U}}_E:={\mathcal{U}}(({\mathcal{D}}_E^0)'\cap C^*(E)). \end{equation}

If $u\in {\mathcal {U}}_E$ then $uS_e$, $e\in E^1$, are partial isometries in $C^*(E)$ which together with projections $P_v$, $v\in E^0$, satisfy (GA1) and (GA2). Thus, by the universality of $C^*(E)$, there exists a unital $*$-homomorphism $\lambda _u:C^*(E)\to C^*(E)$ such thatFootnote ¹

(14)\begin{equation} \lambda_u(S_e)=u S_e \text{ and } \lambda_u(P_v)=P_v, \text{ for } e\in E^1,\quad v\in E^0. \end{equation}

The mapping $u\mapsto \lambda _u$ establishes a bijective correspondence between ${\mathcal {U}}_E$ and the semigroup of those unital endomorphisms of $C^*(E)$ which fix all $P_v$, $v\in E^0$. As observed in [Reference Conti, Hong and Szymański4, Proposition 2.1], if $u\in {\mathcal {U}}_E\cap {\mathcal {F}}_E$ then $\lambda _u$ is automatically injective. We say $\lambda _u$ is invertible if $\lambda _u$ is an automorphism of $C^*(E)$. If $u$ belongs to ${\mathcal {U}}_E\cap {\mathcal {F}}_E^k$ for some $k$, then the corresponding endomorphism $\lambda _u$ is called localized, [Reference Conti, Hong and Szymański4, Reference Conti and Pinzari6].

If $u\in {\mathcal {U}}(\mathfrak B)$ then $\lambda _u$ is automatically invertible with inverse $\lambda _{u^*}$ and the map

(15)\begin{equation} {\mathcal{U}}(\mathfrak B) \ni u\mapsto \lambda_u \in\operatorname{Aut}(C^*(E)) \end{equation}

is a group homomorphism with a range inside the subgroup of quasi-free automorphisms of $C^*(E)$, see [Reference Zacharias24]. Note that this group is almost never trivial and it is non-commutative if graph $E$ contains two edges $e,\,f\in E^1$ such that $s(e)=s(f)$ and $r(e)=r(f)$.

The shift $\varphi$ globally preserves ${\mathcal {U}}_E$, ${\mathcal {F}}_E$ and ${\mathcal {D}}_E$. For $k\geq 1$, we denote

(16)\begin{equation} u_k := u\varphi(u)\cdots\varphi^{k-1}(u). \end{equation}

For each $u\in {\mathcal {U}}_E$ and all $e\in E^1$, we have $S_e u = \varphi (u) S_e$, and thus

(17)\begin{equation} \lambda_u(S_\mu S_\nu^*)=u_{|\mu|}S_\mu S_\nu^*u_{|\nu|}^* \end{equation}

for any two paths $\mu,\,\nu \in E^*$.

2.4. The Popa criterion

In the analysis of uniqueness of Cartan subalgebras of tracial von Neumann algebras, Popa's intertwining-by-bimodules technique has been extremely successful. This method goes back to [Reference Popa22], but has been polished over the years and recently even extended to type $III$ case [Reference Houdayer and Isono16]. The following result contains its essential ingredient.

This beautiful theorem is inapplicable to graph $C^*$-algebras, of course. However, the following simple fact remains valid in the $C^*$-algebraic setting.

Proof. Indeed, let $w_n\in {\mathcal {U}}(A)$ be as in the lemma and suppose $v\in {\mathcal {U}}(M)$ is such that $vAv^*\subseteq B$. Then

\[ 1=||vw_nv^*||_2=|| \varPhi_B(vw_n v^*)||_2 \underset{n\to\infty}{\longrightarrow} 0, \]

which gives a contradiction.

4.1. Out-splitting

Given a graph $E$ satisfying our standing assumption, we consider its out-split graph $E_s({\mathcal {P}})$, as defined by Bates and Pask in [Reference Bates and Pask2]. Namely, for each vertex $v\in E^0$, we partition the set of edges emitted by $v$, that is $s^{-1}(v)$, into $m(v)$ non-empty disjoint subsets $E^1_v,\, \ldots,\, E^{m(v)}_v$. Denote by ${\mathcal {P}}$ the resulting partition of $E^1$. The out-split graph $E_s({\mathcal {P}})$ has the following vertices, edges, source and range functions:

\begin{align*} E_s({\mathcal{P}})^0 & = \{ v^i \mid v\in E^0, \, 1\leq i \leq m(v) \}, \\ E_s({\mathcal{P}})^1 & = \{ e^j \mid e\in E^1, \, 1\leq j \leq m(r(e)) \}, \\ s(e^j) & = s(e)^i, \text{ and } r(e^j) = r(e)^j. \end{align*}

As shown in [Reference Bates and Pask2, Theorem 3.2], the $C^*$–algebras $C^*(E)$ and $C^*(E_s({\mathcal {P}}))$ are isomorphic by an isomorphism which maps the diagonal MASA ${\mathcal {D}}_E$ of $C^*(E)$ onto the diagonal MASA ${\mathcal {D}}_{E_s({\mathcal {P}})}$ of $C^*(E_s({\mathcal {P}}))$. However, the groups of quasi-free automorphims of $C^*(E)$ and $C^*(E_s({\mathcal {P}}))$ may be different. For example, in the following case

the groups ${\mathcal {U}}(\mathfrak B)$ in $C^*(E_s({\mathcal {P}}))$ and $C^*(E)$ are isomorphic to $U(1)\times U(1)\times U(1)\times U(1)$ and $U(2)$, respectively.

Thus, the isomorphism $C^*(E)\cong C^*(E_s({\mathcal {P}}))$ may identify a quasi-free automorphism of one of the two algebras with a non quasi-free automorphism of the other. In this way, our Theorem 3.3 leads to non-trivial examples of non quasi-free automorphisms of graph algebras that map the diagonal MASA onto another MASA that is not inner conjugate to it.

4.2. Two graphs for ${\mathcal {O}}_2$

Consider the following graph:

Then the graph algebra $C^*(E)$ is isomorphic to the Cuntz algebra ${\mathcal {O}}_2=C^*(T_1,\,T_2)$, [Reference Cuntz8]. Here $C^*(T_1,\,T_2)$ is the universal $C^*$-algebra for the relations $1 = T_1T_1^* + T_2T_2^* = T_1^*T_1 = T_2^*T_2$. That is, it is a graph algebra for the graph consisting of one vertex and two edges attached to it. The isomorphism between $C^*(E) = C^*(S_a,\,S_b,\,S_c,\,S_d,\,S_e)$ and ${\mathcal {O}}_2=C^*(T_1,\,T_2)$ is obtained by the identification

\[ S_a=T_{11}T_1^*, \quad S_b=T_{121}T_1^*,\ S_c=T_{122}T_2^*,\ S_d=T_{22}T_2^*,\ S_e=T_{21}T_1^*. \]

The inverse map is given by

\[ T_1=S_a + (S_b +S_c)(S_d + S_e)^*,\quad T_2=S_d + S_e. \]

Note that this isomorphism carries the diagonal MASA ${\mathcal {D}}_E$ of $C^*(E)$ onto the standard diagonal MASA ${\mathcal {D}}_2$ of ${\mathcal {O}}_2$. Indeed, it follows from the above definition that every product $x$ of the generators of $C^*(E)$ is mapped onto an element of the form $T_\alpha T_\beta ^*$ in ${\mathcal {O}}_2$, with $\alpha,\,\beta$ some words on the alphabet $\{1,\,2\}$. Thus, the range projection $xx^*$ of that product is mapped onto a projection in the diagonal MASA ${\mathcal {D}}_2$ of ${\mathcal {O}}_2$. Hence, the image of ${\mathcal {D}}_E$ is contained in ${\mathcal {D}}_2$. But under an isomorphism, the image of a MASA in $C^*(E)$ is a MASA in ${\mathcal {O}}_2$. Thus, the image of MASA ${\mathcal {D}}_E$ of $C^*(E)$ is the entire MASA ${\mathcal {D}}_2$ of ${\mathcal {O}}_2$, as claimed.

Now, using the isomorphism above, we may find a quasi-free automorphism of $C^*(E)$ such that the corresponding automorphism of ${\mathcal {O}}_2$ is not quasi-free and yet carries the diagonal MASA ${\mathcal {D}}_2$ of ${\mathcal {O}}_2$ onto a MASA which is not inner conjugate to ${\mathcal {D}}_2$. Indeed, let

\[ \left[ \begin{array}{@{}cc@{}} \xi_{aa} & \xi_{ab} \\ \xi_{ba} & \xi_{bb} \end{array} \right] \]

be a unitary matrix whose all entries are non-zero complex numbers. Then

\[ u=\xi_{aa}S_a S_a^* + \xi_{ab}S_a S_b^* + \xi_{ba}S_b S_a^* + \xi_{bb}S_b S_b^* + S_c S_c^* + S_d S_d^* + S_e S_e^* \]

is a unitary in ${\mathcal {F}}_E^1$ satisfying the hypothesis of Theorem 3.3. The above isomorphism transports the quasi-free automorphism $\lambda _u$ of $C^*(E)$ onto an automorphism $\lambda _U$ of ${\mathcal {O}}_2$ corresponding to the unitary

\[ U= \xi_{aa}T_{11}T_{11}^* + \xi_{ab}T_{11}T_{121}^* + \xi_{ba}T_{121}T_{11}^* + \xi_{bb}T_{121}T_{121}^* + T_{122}T_{122}^* + T_2T_2^*. \]

Clearly, $\lambda _U$ is not a quasi-free automorphism of ${\mathcal {O}}_2$, since $U$ does not belong to the linear span of $T_iT_j^*$, $i,\,j=1,\,2$. Theorem 3.3 implies that there is no unitary $w\in {\mathcal {O}}_2$ satisfying $w{\mathcal {D}}_2w^*=\lambda _U({\mathcal {D}}_2)$.

6.1. The conditional expectations

Lemma 6.1 Let $A$ and $B$ be unital $C^*$-subalgebras of a finite-dimensional $C^*$-algebra, such that $ab=ba$ for all $a\in A,$ $b\in B$. Let $D_A$ and $D_B$ be MASAs of $A$ and $B,$ respectively, so that $D:=D_A\vee D_B$ is a MASA of $A\vee B$. Let $\tau$ be a faithful tracial state on $A\vee B,$ and let $E_D,$ $E_{D_A}$ and $E_{D_B}$ be $\tau$-preserving conditional expectations from $A\vee B$ onto $D,$ $D_A$ and $D_B,$ respectively. Then we have

\[ E_D(ab) = E_{D_A}(a)E_{D_B}(b) \]

for all $a\in A,$ $b\in B$.

Proof. If $A$ is a full matrix algebra (i.e., the centre of $A$ is trivial) then $A\vee B\cong A\otimes B$ and $\tau (ab)=\tau (a)\tau (b)$ for all $a\in A$, $b\in B$. Thus, in this case, the claim obviously holds.

In the general case, let $\{p_1,\,\ldots,\,p_n\}$ be the minimal central projections in $A$. Then

\[ A\vee B = \bigoplus_{i=1}^n(A\vee B){p_i} \cong \bigoplus_{i=1}^n A{p_i}\otimes B{p_i}. \]

The $\tau$-preserving conditional expectation $E_i$ from $(A\vee B){p_i}$ onto $(D_A\vee D_B){p_i}$ satisfies

\[ E_i(ab)=E_{D_A p_i}(a)E_{D_B p_i}(b) \]

for all $a\in Ap_i$ and $b\in Bp_i$, by the preceding argument. Since

\[ E_D(x) = \sum_{i=1}^n E_i(xp_i), \]

the claim follows.

Corollary 6.2 For all $x_1,\,x_2,\,\ldots,\,x_k\in \mathfrak B$, we have

\[ \varPhi_{{\mathcal{D}}}(x_1\varphi(x_2)\cdots\varphi^{k-1}(x_k)) = \varPhi_{{\mathcal{D}}}(x_1)\varphi(\varPhi_{{\mathcal{D}}}(x_2))\cdots \varphi^{k-1}(\varPhi_{{\mathcal{D}}}(x_k)). \]

Proof. Since $\mathfrak B$, $\varphi (\mathfrak B)$, …, $\varphi ^{k-1}(\mathfrak B)$ are mutually commuting unital finite-dimensional $C^*$-algebras, by Lemma 6.1, we have

\[ \varPhi_{{\mathcal{D}}}(x_1\varphi(x_2)\cdots\varphi^{k-1}(x_k)) = \varPhi_{{\mathcal{D}}}(x_1)\varPhi_{{\mathcal{D}}}(\varphi(x_2))\cdots \varPhi_{{\mathcal{D}}}(\varphi^{k-1}(x_k)). \]

The claims follow since the conditional expectation $\varPhi _{{\mathcal {D}}}$ commutes with the shift $\varphi$.

6.2. The Perron-frobenius theory

Let $A$ be an $n\times n$ matrix with non-negative integer entries. We assume that $A$ is irreducible in the sense that for each pair of indices $(i,\,j)$ there exists a positive integer $k$ such that $A^k(i,\,j)>0$. Let $\beta$ be the Perron–Frobenius eigenvalue and let $(x(1),\,x(2),\,\ldots,\,x(n))$ be the corresponding Perron–Frobenius eigenvector. That is, $\beta >0$, $x(i)>0$ for all indices $i=1,\,\ldots,\,n$, and

\[ \sum_j A(i,j) x(j) = \beta x(i). \]

In this subsection, for a (not necessary square) matrix $B$ we write $B\geq 0$ if $B(i,\,j)\geq 0$ for all $(i,\,j)$. Likewise, we write $B> 0$ if $B(i,\,j)> 0$ for all $(i,\,j)$. For a column vector $y\geq 0$, we set

\[ \lambda(y,A) = \max\{ \lambda\geq 0 \mid Ay\geq\lambda y \}. \]

The following lemma is part of the classical Perron–Frobenius theory, hence its proof is omitted.

Lemma 6.3 For an irreducible matrix $A,$ as above, we have

\[ \beta = \max\{ \lambda(y,A) \mid y\geq 0, ||y||=1 \}. \]

Lemma 6.4 Let $\beta '>0$ be the Perron–Frobenius eigenvalue of the transpose matrix ${^t}A$. Let $\{y(v)\}_v$ be the Perron–Frobenius eigenvector of ${^t}A$. That is,

\[ \sum_v A(v,w)y(v)=\beta' y(w). \]

Set $m=\min _v y(v)$ and $M=\max _v y(v)$. For any $f\in {\mathcal {D}}_E$, we have

\[ \tau(\varphi(f))\leq \dfrac{\beta' M}{\beta m}\tau(f). \]

Hence we have

\[ ||\varphi^p(f)||_2^2\leq \Bigl(\dfrac{\beta'M} {\beta m}\Bigr)^p||f||_2^2. \]

Proof. We may assume that $f=S_\mu S_\mu ^*$. We see that

\begin{align*} \tau(\varphi(S_\mu S_\mu^*))& = \sum_{e,r(e)=s(\mu)}\tau(S_{e\mu}S_{e\mu}^*)= \sum_{v}A(v,s(\mu)) \dfrac{x(r(\mu))}{X\beta^{|\mu|+1}}\\ & \leq \sum_{v}A(v,s(\mu))\dfrac{y(v)}{m} \dfrac{x(r(\mu))}{X\beta^{|\mu|+1}}= \beta'\dfrac{y(s(\mu))}{m} \dfrac{x(r(\mu))}{X\beta^{|\mu|+1}}\\ & \leq \beta' \dfrac{M}{m} \dfrac{x(r(\mu))}{X\beta^{|\mu|+1}}= \dfrac{\beta'M}{\beta m}\tau(S_\mu S_\mu^*). \end{align*}

Lemma 6.5 For an irreducible matrix $A,$ as above, we set $X=\sum \nolimits _i x(i),$ $\alpha = \min _i x(i),$ and $\alpha '=\max _i x(i)$. Then, for every positive integer $k$, we have

\[ 0<\frac{X}{\alpha'} \leq \frac{\sum_{i,j}A^k(i,j)}{\beta^k} \leq \frac{X}{\alpha}. \]

Proof. Since $x(j)/\alpha ' \leq 1 \leq x(j)/\alpha$ for all $j$ and $\sum \nolimits _j A^k(i,\,j)x(j)=\beta ^k x(i)$ for all $i$, we have

\[ \frac{X}{\alpha'} = \frac{\sum_{i,j}A^k(i,j)x(j)}{\beta^k\alpha'} \leq \frac{\sum_{i,j}A^k(i,j)}{\beta^k} \leq \frac{\sum_{i,j}A^k(i,j)x(j)}{\beta^k\alpha} = \frac{X}{\alpha}. \]

For an irreducible matrix $A$, as above, and a fixed pair of indices $(i_1,\,j_1)$ we set

(26)\begin{equation} A_1(i,j) := \left\{ \begin{array}{@{}ll} A(i,j) & \text{if } (i,j) \neq (i_1,j_1) \\ 1 & \text{if } (i,j) = (i_1,j_1) \end{array} \right. \end{equation}

Theorem 6.6 Let $A$ be an irreducible matrix, as above. Assume that $A(i_1,\,j_1)\geq 2$. Then $A_1$ is an irreducible matrix such that $A_1\leq A$ and we have

\[ \lim_{k\to\infty} \frac{\sum_{i,j}A_1^k(i,j)}{\beta^k} =0, \]

with $\beta$ the Perron–Frobenius eigenvalue of $A$.

Proof. It is clear that $A_1$ is irreducible and $A_1\leq A$. Let $\beta _1$ be the Perron–Frobenius eigenvalue of $A_1$, with the corresponding Perron–Frobenius eigenvector $(x_1(1),\,\ldots,\,x_1(n))$. We have

\[ \frac{\sum_{i,j}A_1^k(i,j)}{\beta^k} = \frac{\sum_{i,j}A_1^k(i,j)}{\beta_1^k}\cdot\frac{\beta_1^k}{\beta^k}. \]

Thus, in view of Lemma 6.5, it suffices to show that $\beta _1<\beta$.

Now, for each pair of indices $(i,\,j)$ we can find an $l_{i,j}$ such that $A_1^{l_{i,j}}(i,\,j) < A^{l_{i,j}}(i,\,j)$. Indeed, denote by $E_1$ a graph with the adjacency matrix $A_1$. We may view $E_1$ as a subgraph of $E$. Given $(i,\,j)$ we can find a path $\mu \in E^*\setminus E_1^*$ with source in vertex $i$ and range in vertex $j$. To this end, take a path $\mu _1$ from $i$ to $i_1$, a path $\mu _2$ from $j_1$ to $j$, and an edge $e\in E^1\setminus E_1^1$ from $i_1$ to $j_1$. Then put $\mu :=\mu _1 e\mu _2$. Setting $l_{i,j}:=|\mu |$ we have $A_1^{l_{i,j}}(i,\,j) < A^{l_{i,j}}(i,\,j)$, as desired. Let $k$ be an integer such that $k > l_{i,j}$ for all $i,\,j$. Then we have

\[ \sum_{j=1}^k A_1^j < \sum_{j=1}^k A^j . \]

Now, we set $\overline {A}=\sum \nolimits _{j=1}^k A^j$, $\overline {A}_1=\sum \nolimits _{j=1}^k A^j_1$, $\overline {\beta }=\sum \nolimits _{j=1}^k \beta ^j$, and $\overline {\beta }_1=\sum \nolimits _{j=1}^k \beta ^j_1$. We have $\overline {A}x=\overline {\beta }x$ and $\overline {A}_1x_1=\overline {\beta }_1x_1$. To prove the theorem, it suffices to show that $\overline {\beta }_1 < \overline {\beta }$. Thus, without loss of generality, we may simply assume that $A_1 < A$.

Let $I$ be the $n\times n$ matrix with $I(i,\,j)=1$ for all $i,\,j$. Since $A > A_1$, we have

\[ A \geq A_1 + I. \]

With $X_1:=\sum \nolimits _j x_1(j) >0$, we see that

\[ Ax_1 \geq (A_1+I)x_1 = \beta_1x_1 + \left( \begin{array}{@{}c@{}} X_1 \\ \vdots \\ X_1 \end{array} \right). \]

We can take a small $\epsilon >0$ such that

\[ \beta_1x_1 + \left( \begin{array}{@{}c@{}} X_1 \\ \vdots \\ X_1 \end{array} \right) \geq (\beta_1+\epsilon)x_1 \]

This means that $\lambda (x_1,\,A) \geq \beta _1+\epsilon > \beta _1$. Since $\beta \geq \lambda (x_1,\,A)$, we may finally conclude that $\beta >\beta _1$.