Invariance of KMS states on graph C*-algebras under classical and quantum symmetry

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.

A von Neumann algebraic approach to self-similar group actions

We study some relations between self-similar group actions and operator algebras. We see that [Formula: see text] or [Formula: see text], where [Formula: see text] denotes the Bernoulli measure and [Formula: see text] the set of [Formula: see text]-generic points. In the case [Formula: see text], we get a unique KMS state for the canonical gauge action on the Cuntz–Pimsner algebra constructed from a self-similar group action by Nekrashevych. Moreover, if [Formula: see text], there exists a unique tracial state on the gauge invariant subalgebra of the Cuntz–Pimsner algebra. We also consider the GNS representation of the unique KMS state and compute the type of the associated von Neumann algebra.

The Toeplitz noncommutative solenoid and its Kubo–Martin–Schwinger states

We use Katsura’s topological graphs to define Toeplitz extensions of Latrémolière and Packer’s noncommutative-solenoid $C^{\ast }$-algebras. We identify a natural dynamics on each Toeplitz noncommutative solenoid and study the associated Kubo–Martin–Schwinger (KMS) states. Our main result shows that the space of extreme points of the KMS simplex of the Toeplitz noncommutative torus at a strictly positive inverse temperature is homeomorphic to a solenoid; indeed, there is an action of the solenoid group on the Toeplitz noncommutative solenoid that induces a free and transitive action on the extreme boundary of the KMS simplex. With the exception of the degenerate case of trivial rotations, at inverse temperature zero there is a unique KMS state, and only this one factors through Latrémolière and Packer’s noncommutative solenoid.

A class of simple C*-algebras arising from certain non-sofic subshifts

We present a class of subshifts ZN,N=1,2,…, whose associated C*-algebras 𝒪ZN are simple, purely infinite and not stably isomorphic to any Cuntz–Krieger algebra nor to the Cuntz algebra 𝒪∞. The class of the subshifts is the first example whose associated C*-algebras are not stably isomorphic to any Cuntz–Krieger algebra nor to the Cuntz algebra 𝒪∞. The subshifts ZN are coded systems whose languages are context free. We compute the topological entropy for the subshifts and show that a KMS-state (a state satisfying the Kubo–Martin–Schwinger condition) for gauge action on the associated C*-algebra 𝒪ZN exists if and only if the logarithm of the inverse temperature is the topological entropy for the subshift ZN, and the corresponding KMS-state is unique.

CORE SYMMETRIES OF A FLOW

For a flow α on a C*-algebra one defines a symmetry as the group of automorphisms γ such that γαγ-1 is a cocycle perturbation of α. We propose to define a core of this symmetry, which acts trivially on the set of equivalence classes of KMS state representations, but may act non-trivially on the set of equivalence classes of covariant irreducible representations. In particular this core acts transitively on the set of those which induce faithful representations of the crossed product by α.

Perturbation Theory of W*-Dynamics, Liouvilleans and KMS-States

Given a W*-algebra [Formula: see text] with a W*-dynamics τ, we prove the existence of the perturbed W*-dynamics for a large class of unbounded perturbations. We compute its Liouvillean. If τ has a β-KMS state, and the perturbation satisfies some mild assumptions related to the Golden–Thompson inequality, we prove the existence of a β-KMS state for the perturbed W*-dynamics. These results extend the well known constructions due to Araki valid for bounded perturbations.

Simple ${\bi C}^*$-algebras arising from ${\beta}$-expansion of real numbers

Given a real number $\beta > 1$, we construct a simple purely infinite $C^*$-algebra ${\cal O}_{\beta}$ as a $C^*$-algebra arising from the $\beta$-subshift in the symbolic dynamics. The $C^*$-algebras $\{{\cal O}_{\beta} \}_{1<\beta \in {\Bbb R}}$ interpolate between the Cuntz algebras $\{{\cal O}_n\}_{1 < n \in {\Bbb N}}$. The K-groups for the $C^*$-algebras ${\cal O}_{\beta}$, $1 < \beta \in {\Bbb R}$, are computed so that they are completely classified up to isomorphism. We prove that the KMS-state for the gauge action on ${\cal O}_{\beta}$ is unique at the inverse temperature $\log \beta$, which is the topological entropy for the $\beta$-shift. Moreover, ${\cal O}_{\beta}$ is realized to be a universal $C^*$-algebra generated by $n-1=[\beta]$ isometries and one partial isometry with mutually orthogonal ranges and a certain relation coming from the sequence of $\beta$-expansion of $1$.

Entropy Density of One-Dimensional Quantum Lattice Systems

We prove that the entropy density of a KMS state of one-dimensional quantum lattice systems is equal to the thermodynamical limit of the entropy of local Gibbs states.