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

2017 ◽  
Vol 39 (1) ◽  
pp. 105-131 ◽  
Author(s):  
NATHAN BROWNLOWE ◽  
MITCHELL HAWKINS ◽  
AIDAN SIMS
Keyword(s):  
Extreme Points ◽  
Degenerate Case ◽  
Inverse Temperature ◽  
Transitive Action ◽  
Topological Graphs ◽  
Noncommutative Torus ◽  
Kms States ◽  
Kms State ◽  
Temperature Zero ◽  
Natural Dynamics

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.

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

2021 ◽  
pp. 1-17
Author(s):  
Soumalya Joardar ◽  
Arnab Mandal
Keyword(s):  
Automorphism Group ◽  
Automorphism Groups ◽  
Inverse Temperature ◽  
Circulant Graphs ◽  
Connected Graphs ◽  
Kms States ◽  
C Algebras ◽  
Kms State ◽  
Quantum Symmetry ◽  
Strongly Connected

Abstract 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.

scholarly journals Boundary quotients of the right Toeplitz algebra of the affine semigroup over the natural numbers

10.53733/90 ◽  
2021 ◽  
Vol 52 ◽  
pp. 109-143
Author(s):  
Astrid An Huef ◽  
Marcelo Laca ◽  
Iain Raeburn
Keyword(s):  
Regular Representation ◽  
Crossed Product ◽  
Ideal Structure ◽  
Toeplitz Algebra ◽  
Natural Numbers ◽  
Kms States ◽  
C Algebra ◽  
Affine Semigroup ◽  
Natural Dynamics ◽  
The Right

We study the Toeplitz $C^*$-algebra generated by the right-regular representation of the semigroup ${\mathbb N \rtimes \mathbb N^\times}$, which we call the right Toeplitz algebra. We analyse its structure by studying three distinguished quotients. We show that the multiplicative boundary quotient is isomorphic to a crossed product of the Toeplitz algebra of the additive rationals by an action of the multiplicative rationals, and study its ideal structure. The Crisp--Laca boundary quotient is isomorphic to the $C^*$-algebra of the group ${\mathbb Q_+^\times}\!\! \ltimes \mathbb Q$. There is a natural dynamics on the right Toeplitz algebra and all its KMS states factor through the additive boundary quotient. We describe the KMS simplex for inverse temperatures greater than one.

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

1998 ◽  
Vol 18 (4) ◽  
pp. 937-962 ◽  
Author(s):  
YOSHIKAZU KATAYAMA ◽  
KENGO MATSUMOTO ◽  
YASUO WATATANI
Keyword(s):  
Real Number ◽  
Topological Entropy ◽  
Symbolic Dynamics ◽  
Partial Isometry ◽  
Inverse Temperature ◽  
Real Numbers ◽  
Gauge Action ◽  
C Algebra ◽  
C Algebras ◽  
Kms State

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 $1$.

scholarly journals Equilibrium states on higher-rank Toeplitz non-commutative solenoids

2019 ◽  
Vol 40 (11) ◽  
pp. 2881-2912 ◽  
Author(s):  
ZAHRA AFSAR ◽  
ASTRID AN HUEF ◽  
IAIN RAEBURN ◽  
AIDAN SIMS
Keyword(s):  
Continuous Functions ◽  
Equilibrium States ◽  
Probability Measures ◽  
Inverse Limits ◽  
Inverse Temperature ◽  
Higher Dimensional ◽  
Direct Limits ◽  
Higher Rank ◽  
Natural Dynamics

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.

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

2003 ◽  
Vol 15 (05) ◽  
pp. 447-489 ◽  
Author(s):  
J. Dereziński ◽  
V. Jakšić ◽  
C.-A. Pillet
Keyword(s):  
Perturbation Theory ◽  
Large Class ◽  
Kms States ◽  
Kms State ◽  
Bounded Perturbations

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.

KMS states on the -algebras of reducible graphs

2014 ◽  
Vol 35 (8) ◽  
pp. 2535-2558 ◽  
Author(s):  
ASTRID AN HUEF ◽  
MARCELO LACA ◽  
IAIN RAEBURN ◽  
AIDAN SIMS
Keyword(s):  
Critical Value ◽  
Inverse Temperature ◽  
Toeplitz Algebra ◽  
Dimensional Simplex ◽  
Kms States ◽  
Gauge Action ◽  
Finite Graphs ◽  
Underlying Graph ◽  
Finite Dimensional ◽  
Graph Algebra

We consider the dynamics on the $C^{\ast }$-algebras of finite graphs obtained by lifting the gauge action to an action of the real line. Enomoto, Fujii and Watatani [KMS states for gauge action on ${\mathcal{O}}_{A}$. Math. Japon.29 (1984), 607–619] proved that if the vertex matrix of the graph is irreducible, then the dynamics on the graph algebra admits a single Kubo–Martin–Schwinger (KMS) state. We have previously studied the dynamics on the Toeplitz algebra, and explicitly described a finite-dimensional simplex of KMS states for inverse temperatures above a critical value. Here we study the KMS states for graphs with reducible vertex matrix, and for inverse temperatures at and below the critical value. We prove a general result which describes all the KMS states at a fixed inverse temperature, and then apply this theorem to a variety of examples. We find that there can be many patterns of phase transition, depending on the behaviour of paths in the underlying graph.

scholarly journals KMS States on Nica-Toeplitz $$\hbox {C}^*$$-algebras

2020 ◽  
Vol 378 (3) ◽  
pp. 1875-1929
Author(s):  
Zahra Afsar ◽  
Nadia S. Larsen ◽  
Sergey Neshveyev
Keyword(s):  
Complete Classification ◽  
Inverse Temperature ◽  
Toeplitz Algebra ◽  
Gauge Invariant ◽  
Kms States ◽  
C Algebras ◽  
System A ◽  
System Of Inequalities ◽  
Tracial States

Abstract Given a quasi-lattice ordered group (G, P) and a compactly aligned product system X of essential $$\hbox {C}^*$$ C ∗ -correspondences over the monoid P, we show that there is a bijection between the gauge-invariant $$\hbox {KMS}_\beta $$ KMS β -states on the Nica-Toeplitz algebra $$\mathcal {NT}(X)$$ NT ( X ) of X with respect to a gauge-type dynamics, on one side, and the tracial states on the coefficient algebra A satisfying a system (in general infinite) of inequalities, on the other. This strengthens and generalizes a number of results in the literature in several directions: we do not make any extra assumptions on P and X, and our result can, in principle, be used to study KMS-states at any finite inverse temperature $$\beta $$ β . Under fairly general additional assumptions we show that there is a critical inverse temperature $$\beta _c$$ β c such that for $$\beta >\beta _c$$ β > β c all $$\hbox {KMS}_\beta $$ KMS β -states are of Gibbs type, hence gauge-invariant, in which case we have a complete classification of $$\hbox {KMS}_\beta $$ KMS β -states in terms of tracial states on A, while at $$\beta =\beta _c$$ β = β c we have a phase transition manifesting itself in the appearance of $$\hbox {KMS}_\beta $$ KMS β -states that are not of Gibbs type. In the case of right-angled Artin monoids we show also that our system of inequalities for traces on A can be reduced to a much smaller system, a finite one when the monoid is finitely generated. Most of our results generalize to arbitrary quasi-free dynamics on $$\mathcal {NT}(X)$$ NT ( X ) .

scholarly journals KMS states on C*-algebras associated to local homeomorphisms

2014 ◽  
Vol 25 (08) ◽  
pp. 1450066 ◽  
Author(s):  
Zahra Afsar ◽  
Astrid an Huef ◽  
Iain Raeburn
Keyword(s):  
Finite Graph ◽  
Critical Value ◽  
Toeplitz Algebra ◽  
Graph Algebras ◽  
Kms States ◽  
C Algebra ◽  
C Algebras ◽  
Backward Shift ◽  
The One ◽  
Natural Dynamics

For every Hilbert bimodule over a C*-algebra, there are natural gauge actions of the circle on the associated Toeplitz algebra and Cuntz–Pimsner algebra, and hence natural dynamics obtained by lifting these gauge actions to actions of the real line. We study the KMS states of these dynamics for a family of bimodules associated to local homeomorphisms on compact spaces. For inverse temperatures larger than a certain critical value, we find a large simplex of KMS states on the Toeplitz algebra, and we show that all KMS states on the Cuntz–Pimsner algebra have inverse temperature at most this critical value. We illustrate our results by considering the backward shift on the one-sided path space of a finite graph, where we can use recent results about KMS states on graph algebras to see what happens below the critical value. Our results about KMS states on the Cuntz–Pimsner algebra of the shift show that recent constraints on the range of inverse temperatures obtained by Thomsen are sharp.

scholarly journals KMS states on the C*-algebras of Fell bundles over groupoids

2019 ◽  
pp. 1-25
Author(s):  
ZAHRA AFSAR ◽  
AIDAN SIMS
Keyword(s):  
Equilibrium States ◽  
Cross Sectional ◽  
Kms States ◽  
C Algebras ◽  
Trace Space ◽  
Strongly Connected ◽  
The Cross ◽  
Natural Dynamics ◽  
Special Case

Abstract We consider fibrewise singly generated Fell bundles over étale groupoids. Given a continuous real-valued 1-cocycle on the groupoid, there is a natural dynamics on the cross-sectional algebra of the Fell bundle. We study the Kubo–Martin–Schwinger equilibrium states for this dynamics. Following work of Neshveyev on equilibrium states on groupoid C*-algebras, we describe the equilibrium states of the cross-sectional algebra in terms of measurable fields of states on the C*-algebras of the restrictions of the Fell bundle to the isotropy subgroups of the groupoid. As a special case, we obtain a description of the trace space of the cross-sectional algebra. We apply our result to generalise Neshveyev’s main theorem to twisted groupoid C*-algebras, and then apply this to twisted C*-algebras of strongly connected finite k-graphs.

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