Let G be a compact quantum group. We show that given a G-equivariant $\textrm {C}^*$-correspondence E, the Pimsner algebra $\mathcal {O}_E$ can be naturally made into a G-$\textrm {C}^*$-algebra. We also provide sufficient conditions under which it is guaranteed that a G-action on the Pimsner algebra $\mathcal {O}_E$ arises in this way, in a suitable precise sense. When G is of Kac type, a KMS state on the Pimsner algebra, arising from a quasi-free dynamics, is G-equivariant if and only if the tracial state obtained from restricting it to the coefficient algebra is G-equivariant, under a natural condition. We apply these results to the situation when the $\textrm {C}^*$-correspondence is obtained from a finite, directed graph and draw various conclusions on the quantum automorphism groups of such graphs, both in the sense of Banica and Bichon.

Let $C_c^{*}(\mathbb{N}^{2})$ be the universal $C^{*}$-algebra generated by a semigroup of isometries $\{v_{(m,n)}\,:\, m,n \in \mathbb{N}\}$ whose range projections commute. We analyse the structure of KMS states on $C_{c}^{*}(\mathbb{N}^2)$ for the time evolution determined by a homomorphism $c\,:\,\mathbb{Z}^{2} \to \mathbb{R}$. In contrast to the reduced version $C_{red}^{*}(\mathbb{N}^{2})$, we show that the set of KMS states on $C_{c}^{*}(\mathbb{N}^{2})$ has a rich structure. In particular, we exhibit uncountably many extremal KMS states of type I, II and III.

Given a self-similar set K defined from an iterated function system $\Gamma =(\gamma _{1},\ldots ,\gamma _{d})$ and a set of functions $H=\{h_{i}:K\to \mathbb {R}\}_{i=1}^{d}$ satisfying suitable conditions, we define a generalized gauge action on Kajiwara–Watatani algebras $\mathcal {O}_{\Gamma }$ and their Toeplitz extensions $\mathcal {T}_{\Gamma }$ . We then characterize the KMS states for this action. For each $\beta \in (0,\infty )$ , there is a Ruelle operator $\mathcal {L}_{H,\beta }$ , and the existence of KMS states at inverse temperature $\beta $ is related to this operator. The critical inverse temperature $\beta _{c}$ is such that $\mathcal {L}_{H,\beta _{c}}$ has spectral radius 1. If $\beta <\beta _{c}$ , there are no KMS states on $\mathcal {O}_{\Gamma }$ and $\mathcal {T}_{\Gamma }$ ; if $\beta =\beta _{c}$ , there is a unique KMS state on $\mathcal {O}_{\Gamma }$ and $\mathcal {T}_{\Gamma }$ which is given by the eigenmeasure of $\mathcal {L}_{H,\beta _{c}}$ ; and if $\beta>\beta _{c}$ , including $\beta =\infty $ , the extreme points of the set of KMS states on $\mathcal {T}_{\Gamma }$ are parametrized by the elements of K and on $\mathcal {O}_{\Gamma }$ by the set of branched points.

We consider groupoids constructed from a finite number of commuting local homeomorphisms acting on a compact metric space and study generalized Ruelle operators and $ C^{\ast } $ -algebras associated to these groupoids. We provide a new characterization of $ 1 $ -cocycles on these groupoids taking values in a locally compact abelian group, given in terms of $ k $ -tuples of continuous functions on the unit space satisfying certain canonical identities. Using this, we develop an extended Ruelle–Perron–Frobenius theory for dynamical systems of several commuting operators ( $ k $ -Ruelle triples and commuting Ruelle operators). Results on KMS states on $ C^{\ast } $ -algebras constructed from these groupoids are derived. When the groupoids being studied come from higher-rank graphs, our results recover existence and uniqueness results for KMS states associated to the graphs.

We study the invariance of KMS states on graph $C^{\ast }$-algebras coming from strongly connected and circulant graphs under the classical and quantum symmetry of the graphs. We show that the unique KMS state for strongly connected graphs is invariant under the quantum automorphism group of the graph. For circulant graphs, it is shown that the action of classical and quantum automorphism groups preserves only one of the KMS states occurring at the critical inverse temperature. We also give an example of a graph $C^{\ast }$-algebra having more than one KMS state such that all of them are invariant under the action of classical automorphism group of the graph, but there is a unique KMS state which is invariant under the action of quantum automorphism group of the graph.

Let $\phi :X\to X$ be a homeomorphism of a compact metric space X. For any continuous function $F:X\to \mathbb {R}$ there is a one-parameter group $\alpha ^{F}$ of automorphisms (or a flow) on the crossed product $C^*$ -algebra $C(X)\rtimes _{\phi }\mathbb {Z}$ defined such that $\alpha ^{F}_{t}(fU)=fUe^{-itF}$ when $f \in C(X)$ and U is the canonical unitary in the construction of the crossed product. In this paper we study the Kubo--Martin--Schwinger (KMS) states for these flows by developing an intimate relation to the ergodic theory of non-singular transformations and show that the structure of KMS states can be very rich and complicated. Our results are complete concerning the set of possible inverse temperatures; in particular, we show that when $C(X) \rtimes _{\phi } \mathbb Z$ is simple this set is either $\{0\}$ or the whole line $\mathbb R$ .

We consider a family of higher-dimensional non-commutative tori, which are twisted analogues of the algebras of continuous functions on ordinary tori and their Toeplitz extensions. Just as solenoids are inverse limits of tori, our Toeplitz non-commutative solenoids are direct limits of the Toeplitz extensions of non-commutative tori. We consider natural dynamics on these Toeplitz algebras, and we compute the equilibrium states for these dynamics. We find a large simplex of equilibrium states at each positive inverse temperature, parametrized by the probability measures on an (ordinary) solenoid.