KMS states, conformal measures and ends in digraphs

Abstract

The paper develops a series of tools for the study of KMS-weights on graph $C^*$-algebras and KMS-states on their corners. The approach adopts methods and ideas from graph theory, random walks and dynamical systems.

Appendices

Appendix

We justify in this section the statements concerning existence or absence of almost $A(\beta )$-harmonic vectors made in Sect. 2 and supply the arguments for Proposition 1.

Proposition 15

Assume that $C^*(\varGamma )$ is simple and that $A(\beta )$ is recurrent, i.e. (2) holds for some $v \in \varGamma _V$. There is an almost $A(\beta )$-harmonic vector if and only if $\limsup _n (A(\beta )^n_{v,v})^{\frac{1}{n}} = 1$. When it exists it is $A(\beta )$-harmonic and unique up to multiplication by scalars.

Proof

Let $NW_{\varGamma }$ denote the set of non-wandering vertexes in $\varGamma$. The recurrence condition implies that $NW_{\varGamma } \ne \emptyset$. The stated conclusions follow then from Proposition 4.9 and Theorem 4.14 in [43]. $\square$

Proposition 16

Assume that $\varGamma$ is cofinal and row-finite without sinks and that (4) holds. There is an $e^{\beta F}$-conformal measure on $P(\varGamma )$ if and only if ${\text {Wan}}(\varGamma ) \ne \emptyset$.

Proof

If ${\text {Wan}}(\varGamma )$ is empty it follows from Lemma 11 that there are no $e^{\beta F}$-conformal measure on $P(\varGamma )$. Assume ${\text {Wan}}(\varGamma )$ is not empty. Observe first that it follows from Proposition 4.3 in [43] that in a cofinal digraph with $NW_{\varGamma } \ne \emptyset$ all infinite paths end up in the subgraph whose vertexes are the elements of $NW_{\varGamma }$. Therefore, if $NW_{\varGamma }$ is a finite set, no infinite path can be wandering. Since we assume that ${\text {Wan}}(\varGamma ) \ne \emptyset$ it follows that $NW_{\varGamma }$ is either empty or an infinite set. When $NW_{\varGamma } = \emptyset$ the existence of an $A(\beta )$-harmonic vector follows from Theorem 4.8 in [43]. When $NW_{\varGamma }$ is infinite the existence follows by combining Proposition 2 above with the Corollary in [33] and Lemma 2.3 in [42]. $\square$

We consider cofinal digraphs of type A,B,C,D and E as explained by the diagram following Theorem 3 in Sect. 2, and we assume that $A(\beta )$ is transient in the sense that (4) holds.

1.1 Type A

Since there are no infinite emitters and no sink all the KMS measures considered in Section 3 of [43] are harmonic, and hence Theorem 2.7 and Theorem 3.8 of [43] imply that there are no proper almost $A(\beta )$-harmonic vectors. The existence of an $A(\beta )$-harmonic vector follows from Propositions 16 and 2.

1.2 Type B

It follows as for type A that there are no proper almost $A(\beta )$-harmonic vectors. That there also are no $A(\beta )$-harmonic vectors follows from Propositions 2 and 16.

1.3 Type C

Since there are no sinks it follows from Corollary 3.5 of [43] that there is a bijective correspondence between the set of infinite emitters in $\varGamma$ and the set of extremal boundary $\beta$-KMS measures on the space $\varOmega _{\varGamma }$; see [43] for the definitions. By Theorem 2.7 and Theorem 3.8 in [43] the latter set is in bijective correspondence with the extremal rays of proper almost $A(\beta )$-harmonic vectors. The graphs constructed in Section 8.1 in [43] are examples of graphs of type C for which there are $A(\beta )$-harmonic vectors, also in the transient case, and Example 8 below gives an example of a strongly connected digraph of type C for which there are no $A(\beta )$-harmonic vectors when $A(\beta )$ is transient.

1.4 Type D

For the same reason as for type C there is a bijective correspondence between extremal rays of proper almost $A(\beta )$-harmonic vectors and the infinite emitters in $\varGamma _V$. As for graphs of type B it follows from Proposition 2 and Lemma 11 of the present paper that there are no $A(\beta )$-harmonic vectors.

1.5 Type E

By (3) in Corollary 4.2 of [43] there are no wandering paths in this case. The absence of $A(\beta )$-harmonic vectors follows therefore again from Proposition 2 and Lemma 11. That there is an essentially unique proper almost $A(\beta )$-harmonic vector follows from (a) of Theorem 4.8 in [43].

Example 8

Consider the following graph $\varGamma ^0$ and equip it with the constant potential $F =1$.

Let A be the adjacency matrix of $\varGamma ^0$. An $e^{-\beta }A$-harmonic vector $\psi$ must satisfy the following conditions:

• $\psi _{v_k} = e^{(k-1)\beta } \psi _{v_1}, \ k \ge 2$, and

• $\psi _{v_0} = e^{-\beta }\psi _{v_1} + \sum _{k=2}^{\infty } e^{-\beta }\psi _{v_k}$.

No non-zero non-negative vector $\psi$ meets these conditions when $\beta > 0$, i.e. $H^{v_0}_{\beta }(\varGamma ^0) = \emptyset$ when $\beta > 0$. Since $\varGamma ^0$ is cofinal and $e^{-\beta }A$ is transient for all $\beta \in {\mathbb {R}}$ this gives the example required above concerning graphs of type C. A strongly connected example can be obtained by adding return paths to $v_0$. Specifically, $\varGamma ^0$ satisfies the three conditions in Corollary 4 and we can therefore add return paths to $v_0$ to obtain a strongly connected graph $\varGamma$ for which the Gurevich entropy $h(\varGamma )$ can be any positive number $h > 0$. Let B be the adjacency matrix of $\varGamma$. For $\beta > h(\varGamma )$ the matrix $e^{-\beta }B$ is transient, but it follows from Proposition 5 that $H^{v_0}_{\beta }(\varGamma ) = \emptyset$. However, there will be a unique ray of proper almost $e^{-\beta }B$-harmonic vectors, and hence a unique ray of $\beta$-KMS weights for the gauge action on $C^*(\varGamma )$ for all $\beta \ge h(\varGamma )$; all resulting from the presence of the infinite emitter $v_0$.

1.6 The set of inverse temperatures

In this section we prove Proposition 1 from Sect. 2. The proposition consists of four items where the fourth and last item follows directly from Theorem 4.8 in [43]. It remains therefore only to consider the first three cases in which the set of non-wandering vertexes $NW_{\varGamma }$ is non-empty. By Proposition 4.9 in [43] we may therefore assume that $\varGamma$ is strongly connected.

Let $\varGamma$ be a strongly connected digraph with $C^*(\varGamma )$ simple and $F : \varGamma _{Ar} \rightarrow {\mathbb {R}}$ a potential. For all $\beta \in {\mathbb {R}}$, set

$$\begin{aligned} \rho (A(\beta )) = \limsup _n \left( A(\beta )^n_{v,v}\right) ^{\frac{1}{n}}, \end{aligned}$$

for some $v \in \varGamma _V$. Since $\varGamma$ is strongly connected this number in $[0,\infty ]$ is independent of the vertex v, and

$$\begin{aligned} {\mathbb {P}}(-\beta F) = \log \rho (A(\beta )) \end{aligned}$$

is the pressure function which was used in [43].

Lemma 47

Let $\varGamma$ be a strongly connected digraph with $C^*(\varGamma )$ simple. Assume that either

• (a) there is a loop $\mu$ in $\varGamma$ with $F(\mu ) = 0$ or

• (b) there are loops $\mu _1$ and $\mu _2$ in $\varGamma$ such that $F(\mu _1)< 0 < F(\mu _2)$.

Then $\rho (A(\beta )) > 1$ for all $\beta \in {\mathbb {R}}$, and there are no $\beta$-KMS weights for $\alpha ^F$.

Proof

Assume (a). Set $v = s(\mu )$. Since $C^*(\varGamma )$ is simple all loops have an exit by Theorem 1. Since $\varGamma$ is strongly connected this implies that there is a path $\nu$ such that $|\nu | = m|\mu |$ for some $m \in {\mathbb {N}}$, $s(\nu ) = r(\nu ) = s(\mu ) = v$ and $\nu$ is not the concatenation of m copies of $\mu$. Since $F(\mu ) =0$ it follows that

$$\begin{aligned} A(\beta )^{nm|\mu |}_{v,v} \ge \left( A(\beta )^{m|\mu |}_{v,v}\right) ^n \ge (e^{-\beta m F(\mu )} + e^{-\beta F(\nu )})^n =( 1+ e^{-\beta F(\nu )})^n \end{aligned}$$

for all $n \in {\mathbb {N}}$, showing that

$$\begin{aligned} \rho \left( A(\beta )\right) \ge \left( 1 + e^{-\beta F(\nu )}\right) ^{\frac{1}{m|\mu |}} > 1 \end{aligned}$$

for all $\beta \in {\mathbb {R}}$.

Assume (b). Let $v \in \varGamma _V$. The assumptions imply that there are loops $\nu _1,\nu _2$ in $\varGamma$ such that $s(\nu _1)= s(\nu _2) =v$ and $F(\nu _1)< 0 < F(\nu _2)$. Then

$$\begin{aligned} A(\beta )^{n|\nu _1||\nu _2|}_{v,v} \ge \max \left\{ e^{-\beta n |\nu _2| F(\nu _1)}, e^{-\beta n |\nu _1| F(\nu _2)} \right\} \end{aligned}$$

for all $n \in {\mathbb {N}}$, proving that

$$\begin{aligned} \rho \left( A(\beta )\right) \ge \max \left\{ e^{-\beta \frac{F(\nu _1)}{|\nu _1|}}, e^{-\beta \frac{F(\nu _2)}{|\nu _2|}} \right\} > 1 \end{aligned}$$

for all $\beta \ne 0$. When $\beta = 0$ we can use the arguments from (a) to deduce that for any loop $\mu$ in $\varGamma$ there is an $m \in {\mathbb {N}}$ such that

$$\begin{aligned} \rho (A(0)) \ \ge \ 2^{\frac{1}{m|\mu |}} \ > \ 1. \end{aligned}$$

(83)

Since $\rho (A(\beta )) > 1$ for all $\beta \in {\mathbb {R}}$, it follows from Lemma 4.13 in [43] that there are no $\beta$-KMS weights for $\alpha ^F$. $\square$

Lemma 48

Let $\varGamma$ be a strongly connected digraph. Then $0 < \rho \left( A(\beta )\right) \le \infty$ for all $\beta \in {\mathbb {R}}$ and the function

$$\begin{aligned} {\mathbb {R}} \ni \beta \mapsto \rho \left( A(\beta )\right) \end{aligned}$$

is lower semi-continuous.

Proof

Let $\mu$ be a loop in $\varGamma$ and set $v = s(\mu )$. Then

$$\begin{aligned} A(\beta )^{|\mu |n}_{v,v} \ge e^{-\beta n F(\mu )} \end{aligned}$$

for all n, and hence

$$\begin{aligned} \rho (A(\beta )) \ge \left( e^{-\beta F(\mu )}\right) ^{\frac{1}{|\mu |}} \ > \ 0. \end{aligned}$$

It follows from Lemma 3.10 in [20] that we can choose an increasing sequence $H_1 \subseteq H_2 \subseteq H_3 \subseteq \cdots$ of strongly connected finite subgraphs of $\varGamma$ such that

$$\begin{aligned} \lim _{n \rightarrow \infty } \rho \left( A(\beta )|_{H_n}\right) = \sup _n \rho \left( A(\beta )|_{H_n}\right) = \rho \left( A(\beta )\right) \end{aligned}$$

for all $\beta \in {\mathbb {R}}$. Since $\beta \mapsto \rho ( A(\beta )|_{H_n})$ is continuous for each n by Lemma 4.1 in [12], it follows that $\rho (A(\beta ))$ is lower semi-continuous.

$\square$

Proposition 17

Let $\varGamma$ be a strongly connected digraph with $C^*(\varGamma )$ simple and let $F : \varGamma _{Ar} \rightarrow {\mathbb {R}}$ be a potential. Assume that $\varGamma$ is infinite. There is a $\beta$-KMS weight for $\alpha ^F$ if and only if $\rho (A(\beta )) \le 1$.

Proof

When $\varGamma _V$ is infinite the assertion follows from Proposition 4.19 in [43]. Assume $\varGamma _V$ is finite. There are then infinite emitters since we assume that $\varGamma$ is infinite. If there exists a $\beta$-KMS weight it follows from Lemma 4.13 in [43] that $\rho (A(\beta )) \le 1$. Conversely, if $\rho (A(\beta )) \le 1$ and $A(\beta )$ is recurrent it follows that $\rho (A(\beta )) = 1$ and the existence of a $\beta$-KMS weight is guaranteed by Theorem 4.14 in [43]. When $\rho (A(\beta )) \le 1$ and the matrix $A(\beta )$ is transient the existence of a $\beta$-KMS weight follows from Proposition 4.16 in [43]. $\square$

Lemma 49

Let $\varGamma$ be a strongly connected infinite digraph such that $C^*(\varGamma )$ is simple, and let $F : \varGamma _{Ar} \rightarrow {\mathbb {R}}$ be a potential. If there exists a KMS weight for $\alpha ^F$ the set

$$\begin{aligned}\beta (F) = \left\{ \beta \in {\mathbb {R}}: \ \hbox {There is a} \ \beta \hbox {-KMS weight for } \alpha ^F \right\} \end{aligned}$$

is either an interval of the form $[\beta _0, \infty )$ for some $\beta _0 > 0$ or an interval of the form $(-\infty , \beta _0]$ for some $\beta _0 < 0$. The first case occurs when there is a loop $\mu$ in $\varGamma$ such that $F(\mu ) > 0$, and the second when there is a loop $\mu$ in $\varGamma$ such that $F(\mu ) < 0$.

Proof

Assume that there exists a $\beta$-KMS weight for some $\beta$. By Proposition 17 and Lemma 47 this implies that neither (a) nor (b) from Lemma 47 holds. Assume that $F(\mu ) > 0$ for all loops $\mu$ in $\varGamma$. Fix a vertex $v \in \varGamma _V$ and let $\mathcal L_n$ be the set of loops of length n based at v. If $\beta < \beta '$ we find that

$$\begin{aligned} A(\beta )^n_{v,v} = \sum _{\mu \in {\mathcal {L}}_n} e^{-\beta F(\mu )} \ \ge \ \sum _{\mu \in {\mathcal {L}}_n} e^{-\beta ' F(\mu )} = A(\beta ')^n_{v,v}. \end{aligned}$$

Hence $\beta \mapsto \rho (A(\beta ))$ is non-increasing. By assumption and Proposition 17 the set

$$\begin{aligned} \left\{ \beta \in {\mathbb {R}} : \rho (A(\beta )) \le 1 \right\} \end{aligned}$$

is not empty. Since $\rho (A(0)) > 1$ by (83) it follows that

$$\begin{aligned} \beta _0 = \inf \left\{ \beta \in {\mathbb {R}} : \rho (A(\beta )) \le 1 \right\} \ \end{aligned}$$

is not negative. In fact, since $\rho (A(0)) > 1$ and $\beta \mapsto \rho (A(\beta ))$ is lower semi-continuous by Lemma 48, $\beta _0 > 0$ and $\rho (A(\beta _0)) \le 1$. Since $\beta \mapsto \rho (A(\beta ))$ is non-increasing, $\rho (A(\beta )) \le 1$ for all $\beta \ge \beta _0$ which by Proposition 17 implies that $\beta (F)$ is the set $[\beta _0,\infty )$. The case when $F(\mu ) < 0$ for all loops is handled the same way, leading to the conclusion that there is then a $\beta _0 < 0$ such that $\beta (F)$ is the interval $(-\infty , \beta _0]$. $\square$

Since we can assume that $\varGamma$ is strongly connected the first and third item of Proposition 1 now follow from Lemma 49. Finally, the second item of Proposition 1 where we may assume that $\varGamma$ is both finite and strongly connected follows from [12]; in particular, from Lemma 4.2 and Theorem 4.10 in [12].

It remains to justify the remark made at the end of Sect. 2; ’that when the set $\beta (F)$ of Proposition 1 is an infinite interval, the recurrent case occurs only when $\beta$ is equal to $\beta _0$, the endpoint of the interval and sometimes not even then.’

Proposition 18

In the setting of Lemma 49, assume $\beta \in \beta (F)$ and that $\sum _{n=0}^{\infty } A(\beta )^n_{v,v} = \infty$ for some $v \in \varGamma _V$. Then $\beta = \beta _0$.

Proof

By Lemma 47 we have either that $F(\mu ) > 0$ for all loops $\mu$ or $F(\mu ) < 0$ for all loops $\mu$. Assume that $F(\mu ) > 0$ for all loops and for a contradiction that $\beta > \beta _0$. For each $n \in {\mathbb {N}}$ and $\beta ' \in \{\beta , \beta _0\}$, set

$$\begin{aligned} l^n(\beta ')_{v,v} = \sum _{\nu } e^{-\beta ' F(\nu )}, \end{aligned}$$

where we sum over all loops $\nu$ in $\varGamma$ of length $|\nu | = n$ with $s(\nu ) = r(\nu ) = v$ and the property that v only occurs in the start of $\nu$ and the end of $\nu$. Then

$$\begin{aligned} \sum _{n=0}^{\infty } A(\beta ')^n_{v,v} = \sum _{k=0}^{\infty } \left( \sum _{j=1}^{\infty } l^j(\beta ')_{v,v}\right) ^k \end{aligned}$$

(84)

by Lemma 4.11 in [43]. Furthermore, since $\beta ' \in \beta (F)$ it follows from Theorem 2.7 in [43] that there is a non-zero vector $\psi \in [0,\infty )^{\varGamma _V}$ such that $\sum _{w \in \varGamma _V} A(\beta ')_{u,w}\psi _w \le \psi _u$ for all $u,w \in \varGamma _V$. Note that $\psi _v > 0$ since $\varGamma$ is strongly connected. By a result of Vere-Jones, restated in Lemma 3.6 in [42], it follows that

$$\begin{aligned} \sum _{j=1}^{\infty } l^j(\beta ')_{v,v} \le 1. \end{aligned}$$

(85)

In particular, $\sum _{j=1}^{\infty } l^j(\beta _0)_{v,v} \le 1$ which implies that $\sum _{j=1}^{\infty } l^j(\beta )_{v,v} <1$ since $e^{-\beta F(\mu )} < e^{-\beta _0 F(\mu )}$ for every loop $\mu$. Then (84) implies that $\sum _{n=0}^{\infty } A(\beta )^n_{v,v}< \infty$, contradicting the assumption. The argument which handles the case when $F(\mu ) < 0$ for all loops is completely analogous. $\square$

Note that in the setting of Lemma 49 there will be a recurrent $\beta _0$-KMS weight if and only if

$$\begin{aligned} \sum _{j=1}^{\infty } l^j(\beta _0)_{v,v} = 1, \end{aligned}$$

which very often is not the case.

Appendix

In this appendix we obtain an integral representation of $e^{\beta F}$-conformal measures and $A(\beta )$-harmonic vectors similar to the Poisson-Martin integral representation from the theory of countable state Markov chains. The key tool is a selection theorem of Burgess, cf. [9] and [10].

Let $\varGamma$ be a countable digraph with a vertex $v_0$ such that (12) holds. Recall that $X_{\beta }$ is the set of infinite paths $p \in P(\varGamma )$ with the property that $\psi _v = \lim _{k \rightarrow \infty } K_{\beta }(v,s(p_k))$ exists for all $v \in \varGamma _V$ and the resulting $v_0$-normalized $A(\beta )$-harmonic vector $\psi$ is extremal. Set

$$\begin{aligned} C_{\beta } = X_{\beta } \cap {\text {Ray}}(\varGamma ) \cap Z(v_0); \end{aligned}$$

a Borel subset of $Z(v_0)$. Define $K : C_{\beta } \rightarrow \prod _{v \in \varGamma _V} \left[ 0,b_v\right]$ such that

$$\begin{aligned} K(x)_v = \lim _{k \rightarrow \infty } K_{\beta }(v,s(x_k)). \end{aligned}$$

Then K is a Borel map and it defines an equivalence relation $x \ \sim _K \ y$ on $C_{\beta }$ such that $x \ \sim _K \ y$ iff $K(x) = K(y)$.

Lemma 50

There is a topology $\tau$ on $C_{\beta }$ which is finer than the relative topology inherited from $P(\varGamma )$ such that

• $C_{\beta }$ is a Polish space in the $\tau$-topology,

• the Borel $\sigma$-algebra generated by $\tau$ is the same as the Borel $\sigma$-algebra inherited from $P(\varGamma )$ and

• $K : C_{\beta } \rightarrow \prod _{v \in \varGamma _V} \left[ 0,b_v\right]$ is continuous with respect to the $\tau$-topology.

Proof

This follows from standard results on the Borel structure of Polish spaces, cf. eg. Theorem 3.2.4 and Corollary 3.2.6 in [38]. $\square$

Thanks to Lemma 50 we can apply a result of Burgess, [9, 10], stated in the corollary to Proposition I of [10]. Recall that an analytic subset of a Polish space is the image of a Polish space under a continuous map.

Proposition 19

Consider $C_{\beta }$ as a Polish space in the topology from Lemma 50. There is a map $T: C_{\beta } \rightarrow C_{\beta }$ which is measurable with respect to the $\sigma$-algebra generated by the analytic subsets such that

• $K(T(p)) = K(p)$ for all $p \in C_{\beta }$,

• $K(p) = K(q) \ \Leftrightarrow \ T(p) = T(q)$ for all $p,q \in C_{\beta }$ and

• $T(C_{\beta })$ is a co-analytic subset of $C_{\beta }$, i.e. $C_{\beta } \backslash T(C_{\beta })$ is analytic.

The universally measurable subsets of $P(\varGamma )$ are the subsets of $P(\varGamma )$ that are $\mu$-measurable for every $\sigma$-finite Borel measure $\mu$ on $P(\varGamma )$, cf. page 280 in [15]. Thus $A \subseteq P(\varGamma )$ is universally measurable iff the following holds: For every $\sigma$-finite Borel measure $\mu$ on $P(\varGamma )$ there are Borel sets $B_1,B_2$ in $P(\varGamma )$ such that $B_1 \subseteq A \subseteq B_2$ and $\mu (B_2 \backslash B_1) = 0$. By Corollary 8.4.3 in [15] every analytic subset of a Polish space is universally measurable. Note also that the set of universally measurable sets constitute a $\sigma$-algebra containing the Borel sets. Therefore Proposition 19 has the following

Corollary 10

There is a map $T: C_{\beta } \rightarrow C_{\beta }$ which is measurable with respect to the $\sigma$-algebra of universally measurable sets in $C_{\beta }$ such that

• $K(T(p)) = K(p)$ for all $p \in C_{\beta }$,

• $K(p) = K(q) \ \Leftrightarrow \ T(p) = T(q)$ for all $p,q \in C_{\beta }$ and

• $T(C_{\beta })$ is universally measurable.

For every element $y \in C_{\beta }$ we let $m_y$ be the unique normalized $e^{\beta F}$-conformal measure on $P(\varGamma )$ such that $m_y(Z(v)) = \lim _{k \rightarrow \infty } K_{\beta }(v,s(y_k))$ for all $v \in \varGamma _V$.

Lemma 51

$m_y$ is an extremal $v_0$-normalized $e^{\beta F}$-conformal measure concentrated on

$$\begin{aligned} \left\{ p \in {\text {Wan}}(\varGamma ) : \ \lim _{k \rightarrow \infty } K_{\beta }(v,s(p_k)) = \lim _{k \rightarrow \infty } K_{\beta }(v,s(y_k)) \ \text {for all} \ v \in \varGamma _V \right\} . \end{aligned}$$

(86)

Proof

By definition of $C_{\beta }$ the vector

$$\begin{aligned} \psi _v = \lim _{k \rightarrow \infty } K_{\beta }(v,s(y_k)), \ \ v \in \varGamma _V, \end{aligned}$$

is in $\partial H^{v_0}_{\beta F}(\varGamma )$. Since the bijection of Proposition 2 is affine, this implies that $m_y \in \partial M^{v_0}_{\beta F}(\varGamma )$. It follows then from Theorem 4 and Lemma 11 that $m_y$ is concentrated on (86). $\square$

Lemma 52

$\partial M^{v_0}_{\beta F}(\varGamma ) = \left\{ m_y: \ y \in C_{\beta } \right\}$.

Proof

By Lemma 51 it remains only to prove the inclusion $\subseteq$. Let $m \in \partial M^{v_0}_{\beta F}(\varGamma )$. It follows from Theorem 4 and Lemma 11 that there is a $y' \in Z(v_0) \cap {\text {Wan}}(\varGamma )$ such that $m(Z(v)) = \lim _{k \rightarrow \infty } K_{\beta }(v,s(y'_k))$ for all $v \in \varGamma _V$. Set $y = R(y')$ where $R : {\text {Wan}}(\varGamma ) \rightarrow {\text {Ray}}(\varGamma )$ is the retraction (28). Then $y \in C_{\beta }$ and it follows from Proposition 2 that $m=m_y$. $\square$

Set

$$\begin{aligned} Y_{\beta } = T(C_{\beta }), \end{aligned}$$

where $T : C_{\beta } \rightarrow C_{\beta }$ is the universally measurable map from Corollary 10. It follows from the third item of Corollary 10 that $Y_{\beta }$ is universally measurable, and from the first item of Corollary 10 and Proposition 2 that $m_y = m_{T(y)}$ for all $y \in C_{\beta }$. By Lemma 52 we can therefore conclude that

$$\begin{aligned} \partial M^{v_0}_{\beta F}(\varGamma ) = \left\{ m_y: \ y \in Y_{\beta } \right\} . \end{aligned}$$

If $y,y' \in Y_{\beta }$ are such that $m_y = m_{y'}$ it follows from Lemma 51 that $K(y) = K(y')$ and then from the first two items of Corollary 10 that $y = y'$. Thus the map $y \mapsto m_y$ is injective on $Y_{\beta }$ and hence is a bijection from $Y_{\beta }$ onto $\partial M^{v_0}_{\beta F}(\varGamma )$.

Let $\nu$ be a Borel probability measure on $Y_{\beta }$. That is, $\nu$ is a Borel probability measure on $P(\varGamma )$ such that there is a Borel subset $B_{\nu } \subseteq Y_{\beta }$ with $\nu (B_{\nu }) =1$. An application of Lemma 2 shows that the map $C_{\beta } \ni y \mapsto m_y(B)$ is a Borel function for all Borel subsets $B \subseteq P(\varGamma )$, and the integral

$$\begin{aligned} m(B) = \int _{Y_{\beta }} m_y(B) \ d\nu (y) \end{aligned}$$

is therefore defined. The resulting measure

$$\begin{aligned} m = \int _{Y_{\beta }} m_y \ \mathrm {d}\nu (y) \end{aligned}$$

(87)

is a $v_0$-normalized $e^{\beta }$-conformal measure since each $m_y$ is.

Lemma 53

The retraction $R: {\text {Wan}}(\varGamma ) \rightarrow {\text {Ray}}(\varGamma )$ from (28) is a Borel map.

Proof

Let $\mu = e_1e_2\cdots e_m$ be a finite path in $\varGamma$. It suffices to show that $R^{-1}\left( Z(\mu ) \cap {\text {Ray}}(\varGamma )\right)$ is a Borel set in ${\text {Wan}}(\varGamma )$. Let M be the vertexes occurring in $\mu$. If they are not distinct, $Z(\mu ) \cap {\text {Ray}}(\varGamma ) = \emptyset$ and we are done. So assume that the vertexes in M are distinct. Let

$$\begin{aligned} A = \left\{ p \in Z(r(\mu )) \cap {\text {Wan}}(\varGamma ) : \ s(p_i) \notin M \ \forall i \right\} , \end{aligned}$$

which is the intersection of ${\text {Wan}}(\varGamma )$ with a closed subset of $P(\varGamma )$ and hence a Borel set. Note that

$$\begin{aligned} \begin{aligned}&R^{-1}\left( Z(\mu ) \cap {\text {Ray}}(\varGamma )\right) \\&\quad = \bigcup _{\nu _1,\nu _2,\ldots , \nu _{m}} Z\left( \nu _1e_1\nu _2e_2 \cdots e_{n-1}\nu _{m}e_m\right) A, \end{aligned} \end{aligned}$$

where the $\nu _i$’s are finite paths in $\varGamma$ such that $s(\nu _i) = r(\nu _i) = s(e_i)$ and $\nu _i$ does not contain any of the vertexes $s(e_j), \ j < i$. This exhibits $R^{-1}\left( Z(\mu ) \cap {\text {Ray}}(\varGamma )\right)$ as a Borel set. $\square$

Theorem 20

The map $\nu \mapsto m$ defined by (87) is an affine bijection between the set of Borel probability measures $\nu$ on $Y_{\beta }$ and the set $M^{v_0}_{\beta F}(\varGamma )$ of normalized $e^{\beta F}$-conformal measures m on $P(\varGamma )$.

Proof

To show that the map is surjective, let $m \in M^{v_0}_{\beta F}(\varGamma )$. Set

$$\begin{aligned} B_{\beta } = Z(v_0) \cap {\text {Wan}}(\varGamma ) \cap X_{\beta }, \end{aligned}$$

which is a Borel set such that $R(B_{\beta } ) = C_{\beta }$, where R is the retraction (28). Furthermore, $m(B_{\beta }) = m(Z(v_0)) = 1$ by Corollary 2 and Lemma 11. R is a Borel map by Lemma 53 and hence the composition $W = T \circ R$ is a universally measurable map

$$\begin{aligned} W : \ B_{\beta } \rightarrow Y_{\beta }. \end{aligned}$$

When we also let m denote its own completion we obtain a Borel measure $\nu$ on $P(\varGamma )$ defined such that

$$\begin{aligned} \nu (B) = m \circ W^{-1}\left( B \cap Y_{\beta }\right) . \end{aligned}$$

Since $Y_{\beta }$ is universally measurable and $m\left( W^{-1}(Y_{\beta })\right) = m(B_{\beta }) = 1$, we see that $\nu$ is a Borel probability measure on $Y_{\beta }$. Using the first item in Corollary 10 as well as Corollary 2 we find that

$$\begin{aligned} \begin{aligned}&\int _{Y_{\beta }} m_y(Z(v)) \ \mathrm {d}\nu (y) = \int _{B_{\beta }} m_{W(p)}(Z(v)) \ \mathrm {d} m(p) \\&\quad = \int _{B_{\beta }} \lim _{k \rightarrow \infty } K_{\beta }(v,s(W(p)_k)) \ \mathrm {d} m(p) \\&\quad = \int _{Z(v_0)} \lim _{k \rightarrow \infty } K_{\beta }(v,s(p_k)) \ \mathrm {d} m(p) = m(Z(v)) \end{aligned} \end{aligned}$$

for all $v \in \varGamma _V$; i.e. $m = \int _{Y_{\beta }} m_y \ \mathrm {d}\nu (y)$ by Proposition 2.

To show that the map is injective, assume that (87) holds. Then

$$\begin{aligned} m(U) = \int _{Y_{\beta }} m_y(U) \ \mathrm {d} \nu (y) \end{aligned}$$

for all universally measurable subsets $U \subseteq P(\varGamma )$. Indeed, there are Borel sets $B_1 \subseteq U \subseteq B_2$ such that $m(B_2 \backslash B_1) = 0$ and hence

$$\begin{aligned} 0 = \int _{Y_{\beta }} m_y(B_2 \backslash B_1) \ \mathrm {d} \nu (y), \end{aligned}$$

which implies that $m_y(U) = m_y(B_2)$ for all $\nu$-almost all y. In particular, it follows that for all universally measurable subsets $U \subseteq Y_{\beta }$,

$$\begin{aligned} m\left( W^{-1}\left( U\right) \right) = \int _{Y_{\beta }} m_y(W^{-1}(U)) \ \mathrm {d} \nu (y). \end{aligned}$$

(88)

Consider an element $y \in Y_{\beta }$. Then $y = T(y')$ for some $y' \in C_{\beta }$ and by using the first two items in Corollary 10 we find that

$$\begin{aligned} \begin{aligned} T^{-1}(y)&= \left\{ z \in C_{\beta } : \ T(z) = T(y') \right\} = \left\{ z \in C_{\beta } : K(z) = K(y') \right\} \\&= \left\{ z \in C_{\beta } : K(z) = K(y) \right\} \\&= \left\{ p \in C_{\beta }: \ \lim _{k \rightarrow \infty } K_{\beta }(v,s(p_k)) = \lim _{k \rightarrow \infty } K_{\beta }(v,s(y_k)) \ \forall v \in \varGamma _V \right\} , \end{aligned} \end{aligned}$$

and hence

$$\begin{aligned} \begin{aligned}&W^{-1}(y) = \\&\quad \left\{ p \in Z(v_0) \cap {\text {Wan}}(\varGamma ): \ \lim _{k \rightarrow \infty } K_{\beta }(v,s(p_k)) = \lim _{k \rightarrow \infty } K_{\beta }(v,s(y_k)) \ \forall v \in \varGamma _V \right\} . \end{aligned} \end{aligned}$$

Therefore, if $y \notin U$, the set $W^{-1}(U)$ will be disjoint from the set

$$\begin{aligned} \left\{ p \in X_{\beta }: \ \lim _{k \rightarrow \infty } K_{\beta }(v,s(p_k)) = \lim _{k \rightarrow \infty } K_{\beta }(v,s(y_k)) \ \forall v \in \varGamma _V \right\} \ \end{aligned}$$

where $m_y$ is concentrated by Lemma 51. Hence $m_y(W^{-1}(U)) = 0$ when $y \notin U$. When $y \in U$ it follows for the same reasons that

$$\begin{aligned} m_y(W^{-1}(U)) = m_y(W^{-1}(y)) = m_y(Z(v_0)) = 1. \end{aligned}$$

Inserted into (88) it follows that

$$\begin{aligned} m\left( W^{-1}\left( U\right) \right) = \nu (U), \end{aligned}$$

showing that m determines $\nu$. $\square$

In view of Proposition 2 we have the following

Corollary 11

Let $\psi \in H^{v_0}_{\beta F}(\varGamma )$. There is a unique Borel probability measure $\nu$ on $Y_{\beta }$ such that

$$\begin{aligned} \psi _v = \int _{Y_{\beta }} \lim _{k \rightarrow \infty } K_{\beta }(v, s(y_k)) \ \mathrm {d}\nu (y) \end{aligned}$$

(89)

for all $v \in \varGamma _V$. Conversely, for every Borel probability measure $\nu$ on $Y_{\beta }$ the Eq. (89) defines an element $\psi \in H^{v_0}_{\beta F}(\varGamma )$.

Remark 13

By taking $A(\beta )$ to be a stochastic matrix, Corollary 11 gives an integral representation of the harmonic functions of the associated countable state Markov chain, very similar to the Martin representation, cf. Theorem 4.1 in [37]. With the approach presented here it is not necessary to introduce the Martin boundary. The subset $Y_{\beta }$ of $P(\varGamma )$ plays the role of the minimal Martin boundary, but no substitute of the Martin boundary itself is needed for the integral representation.

The fact that the set of $e^{\beta F}$-conformal measures on $P(\varGamma )$ in the transient case can be identified with the bounded Borel measures on a set of rays should not lead one to think that $e^{\beta F}$-conformal measures are concentrated on the set of rays. In fact, the $e^{\beta F}$-conformal measures very often annihilate the set of rays. The following gives a simple example of this.

Example 9

Consider the following digraph $\varGamma ^0$:

Let $h > 0$. By adding return paths to $v_0$ as described in Sect. 5.2 we obtain a strongly connected recurrent digraph $\varGamma$ with Gurevich entropy $h(\varGamma ) = h$. Then $\varGamma$ is meager with one end represented by the unique ray r emitted from $v_0$. Note that r is $\beta$-summable for all $\beta > h$ and that there is a unique $v_0$-normalized $e^{\beta }$-conformal measure m on $P(\varGamma )$ when $\beta > h$ by Theorem 10. The exit defined by r is not $\beta$-summable in the sense of [43], and since all rays in $\varGamma$ are tail-equivalent to r it follows therefore from Proposition 5.6 in [43] that $m ({\text {Ray}}(\varGamma )) = 0$.

Appendix

In this appendix we consider the tensor product of the examples from Theorem 16 with the examples constructed by Bratteli, Elliott and Herman in [5]. This will lead to

Theorem 21

Let F be a subset of positive real numbers which is closed as a subset of ${\mathbb {R}}$. There is a simple unital $C^*$-algebra with a continuous one-parameter group of automorphisms such that the Choquet simplex $S_{\beta }$ of $\beta$-KMS states is non-empty if and only if $\beta \in F$, and for $\beta ,\beta ' \in F$ the simplices $S_{\beta }$ and $S_{\beta '}$ are affinely homeomorphic only when $\beta = \beta '$.

The input from [5] is the following, cf. Theorem 3.2 in [5].

Theorem 22

(Bratteli, Elliott and Herman) Let L be a closed subset of ${\mathbb {R}}$. There is a simple unital $C^*$-algebra B with a continuous one-parameter group $\gamma$ of automorphisms of B such that there is a $\beta$-KMS state for $\gamma$ if and only if $\beta \in L$, and for each $\beta \in L$ the $\beta$-KMS state is unique.

The algebra B in Theorem 22 is a corner in the crossed product of an AF-algebra by a single automorphism, constructed via the classification of AF-algebras which had just been completed around the time [5] was written. The one-parameter group $\gamma$ is the restriction to the corner of the dual action on the crossed product.

In order to prove Theorem 21 we make first some elementary observations about KMS-states for tensor product actions. Let A and B be unital $C^*$-algebras, $\alpha$ a continuous one-parameter group of automorphisms of A and $\gamma$ a continuous one-parameter group of automorphisms of B. On the minimal (or spatial) tensor product $A \otimes B$ we consider the tensor product action $\alpha \otimes \gamma$ defined such that

$$\begin{aligned} \left( \alpha \otimes \gamma \right) _t(a \otimes b) = \alpha _t(a) \otimes \gamma _t(b) \end{aligned}$$

when $t \in {\mathbb {R}}, \ a \in A$ and $b \in B$. When $\eta$ is a $\beta$-KMS for $\alpha$ and $\rho$ a $\beta$-KMS state for $\gamma$, the tensor product state $\eta \otimes \rho$ on $A \otimes B$, defined such that

$$\begin{aligned} \left( \eta \otimes \rho \right) (a \otimes b) = \eta (a)\rho (b), \end{aligned}$$

when $a \in A$ and $b \in B$, is easily seen to be a $\beta$-KMS state for $\alpha \otimes \gamma$. Conversely, when $\omega$ is $\beta$-KMS state of $\alpha \otimes \gamma$ the restrictions of $\omega$ to $A \subseteq A \otimes B$ and $B \subseteq A \otimes B$ are $\beta$-KMS states of $\alpha$ and $\gamma$, respectively. From these two observations it follows that the set of inverse temperatures of $(A\otimes B, \alpha \otimes \gamma )$, by which we mean the set of real numbers $\beta$ for which there is a $\beta$-KMS state for $\alpha \otimes \gamma$, is the intersection of the set of inverse temperatures for $(A,\alpha )$ and the set of inverse temperatures for $(B,\gamma )$. In the following proof these general observations will be combined with special features of the gauge action on graph $C^*$-algebras and the actions considered in [5].

Proof of Theorem 21

Let r be positive irrational number and set

$$\begin{aligned} L = r F = \left\{ r s : \ s \in F \right\} . \end{aligned}$$

Let $(B,\gamma )$ be the $C^*$-algebra and one-parameter group arising from Theorem 22 for this choice of L and set

$$\begin{aligned} \gamma '_t = \gamma _{rt}. \end{aligned}$$

Choose $h > 0$ such that $h < s$ for all $s \in F$ and let $\varGamma$ be the strongly connected row-finite graph from Theorem 16 corresponding to this h. Let $\alpha$ be the restriction to $P_{v_0}C^*(\varGamma )P_{v_0}$ of the gauge action on $C^*(\varGamma )$. We will show that the $C^*$-algebra $(P_{v_0}C^*(\varGamma )P_{v_0}) \otimes B$ and the one-parameter group $\alpha \otimes \gamma '$ have the properties specified in Theorem 21. First observe that since $P_{v_0}C^*(\varGamma )P_{v_0}$ and B are simple it follows from a result of Takesaki [40], that $(P_{v_0}C^*(\varGamma )P_{v_0}) \otimes B$ is simple. Next note that a state on B is a $\beta$-KMS state for $\gamma '$ iff it is a $r\beta$-KMS state for $\gamma$, and that consequently F is the set of inverse temperatures for $\gamma '$. By Theorem 16 the set of inverse temperatures for $\alpha$ is $[h,\infty )$; a set which contains F. It follows that F is also the set of inverse temperatures for $\alpha \otimes \gamma '$. For $\beta \in F$ let $S_{\beta }$ be the simplex of $\beta$-KMS states for $\alpha$ and $\rho$ the unique $\beta$-KMS state for $\gamma '$. By Theorem 16$S_{\beta }$ is not affinely homeomorphic to $S_{\beta '}$ when $\beta \ne \beta '$. It suffices therefore now to prove the following $\square$

Observation 23

Let $\beta \in F$. The map $S_{\beta } \ni \omega _{\beta } \ \mapsto \ \omega _{\beta } \otimes \rho$ is an affine homeomorphism from $S_{\beta }$ onto the simplex of $\beta$-KMS states for $\alpha \otimes \gamma '$.

Of course, only the surjectivity of the map in Observation 23 requires proof. For this we set ${\mathcal {P}} = \{ \mu \in P_f(\varGamma ): \ s(\mu ) =v_0\}$; the set of finite paths in $\varGamma$ emitted from $v_0$. Then $\{S_{\mu }S_{\nu }^* : \ \mu ,\nu \in {\mathcal {P}}\}$ spans a dense $*$-subalgebra of $P_{v_0}C^*(\varGamma )P_{v_0}$. When $\mu , \nu \in {\mathcal {P}}$ we find that

$$\begin{aligned} \begin{aligned}&R^{-1}\int _0^R\alpha _t\left( S_{\mu }S_{\nu }^*\right) \ \mathrm {d} t \\&\quad = {\left\{ \begin{array}{ll} S_{\mu }S_{\nu }^* \ , \ &{} \ \text {when} \ |\mu | = |\nu | \\ \\ \frac{1}{i R(|\mu | - |\nu |)} \left( e^{iR (|\mu | - |\nu |)} -1\right) S_{\mu }S_{\nu }^* \ , \ &{} \ \text {when} \ |\mu | \ne |\nu | \ \end{array}\right. } \end{aligned} \end{aligned}$$

for all $R > 0$. It follows that the limit

$$\begin{aligned} Q(a) = \lim _{R \rightarrow \infty } R^{-1}\int _0^R\alpha _t\left( a\right) \ \mathrm {d} t \end{aligned}$$

exists for all $a \in P_{v_0}C^*(\varGamma )P_{v_0}$, defining a conditional expectation

$$\begin{aligned} Q : P_{v_0}C^*(\varGamma )P_{v_0} \rightarrow \left( P_{v_0}C^*(\varGamma )P_{v_0}\right) ^{\alpha } \end{aligned}$$

onto the fixed-point $C^*$-algebra $\left( P_{v_0}C^*(\varGamma )P_{v_0}\right) ^{\alpha }$ of $\alpha$ with the property that $Q(S_{\mu }S_{\nu }^*) = 0$ when $|\mu | \ne |\nu |$. Let $E_k, \ k \in {\mathbb {Z}}$, be the eigenspaces for $\gamma$, i.e.

$$\begin{aligned} E_k = \left\{ b \in B : \ \gamma _t(b) = e^{ ik t} b \ \ \forall t \in {\mathbb {R}} \right\} . \end{aligned}$$

Then, as was already pointed out just before the statement of Theorem 3.2 in [5], the $C^*$-algebra B is the closed linear span of the eigenspaces $E_k, \ k \in {\mathbb {Z}}$. Now let $\omega$ be a $\beta$-KMS state for $\alpha \otimes \gamma '$. Then $\omega \circ (\alpha \otimes \gamma ')_t = \omega$ for all t by Proposition 5.3.3 in [7]. When $b_k \in E_k$, $\mu ,\nu \in {\mathcal {P}}$ and $|\mu | \ne |\nu |$, it follows that $|\mu | - |\nu | + kr \ne 0$ since r is irrational and hence also that

$$\begin{aligned} \begin{aligned}&\omega \left( (S_{\mu }S_{\nu }^*) \otimes b_k\right) \\&\quad = \frac{1}{R} \int _0^R \omega \left( \alpha _t(S_{\mu }S_{\nu }^*) \otimes \gamma '_t(b_k)\right) \ \mathrm {d} t \\ \\&\quad = \frac{\omega \left( (S_{\mu }S_{\nu }^*) \otimes b_k\right) }{i (|\mu | - |\nu | + k r)R} \left( e^{i (|\mu | - |\nu | + k r)R} - 1\right) \ \end{aligned} \end{aligned}$$

for all $R > 0$. Letting $R \rightarrow \infty$ it follows that

$$\begin{aligned} \omega \left( (S_{\mu }S_{\nu }^*) \otimes b_k\right) = 0 = \omega \left( (Q\left( S_{\mu }S_{\nu }^*\right) ) \otimes b_k\right) . \end{aligned}$$

Thus

$$\begin{aligned} \omega \left( (S_{\mu }S_{\nu }^*) \otimes b_k\right) = \omega \left( Q\left( S_{\mu }S_{\nu }^*\right) \otimes b_k\right) \end{aligned}$$

for all $\mu ,\nu \in {\mathcal {P}}$ since the identity is trivially true when $|\mu | = |\nu |$. As $\mu ,\nu \in {\mathcal {P}}, \ k \in {\mathbb {Z}}$ and $b_k \in E_k$ were all arbitrary, it follows by linearity and continuity that $\omega$ factorises through $Q \otimes {\text {id}}_B$, i.e.

$$\begin{aligned} \omega = \omega \circ \left( Q\otimes {\text {id}}_B\right) . \end{aligned}$$

When $d \in (P_{v_0}C^*(\varGamma )P_{v_0})^{\alpha }$ is a positive element the functional $\omega _d : B \rightarrow {\mathbb {C}}$ defined such that

$$\begin{aligned} \omega _d(b) = \omega (d \otimes b) \end{aligned}$$

is a non-negative multiple of a $\beta$-KMS state for $\gamma '$ and it must therefore be a non-negative multiple of $\rho$, i.e.

$$\begin{aligned} \omega (d \otimes b) = s(d)\rho (b) \quad \forall b \in B, \end{aligned}$$

for some $s(d) \ge 0$. It follows that s extends to a state $\omega '$ on $\left( P_{v_0}C^*(\varGamma )P_{v_0}\right) ^{\alpha }$ such that $\omega (d\otimes b) = \omega '(d)\rho (b)$ for all $d \in \left( P_{v_0}C^*(\varGamma )P_{v_0}\right) ^{\alpha }$ and $b \in B$. Set $\omega _{\beta } = \omega ' \circ Q$ and note that $\omega = \omega _{\beta } \otimes \rho$. Since $\omega$ is a $\beta$-KMS state for $\alpha \otimes \gamma '$ it follows that $\omega _{\beta }$ is a $\beta$-KMS state for $\alpha$, completing the proof of Observation 23 and hence also of Theorem 21.