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 Reconstructing directed graphs from generalized gauge actions on their Toeplitz algebras

2019 ◽  
Vol 150 (5) ◽  
pp. 2632-2641
Author(s):  
Nathan Brownlowe ◽  
Marcelo Laca ◽  
Dave Robertson ◽  
Aidan Sims
Keyword(s):  
Directed Graph ◽  
Directed Graphs ◽  
Toeplitz Algebra ◽  
Independent Copy ◽  
Gauge Action ◽  
Abelian Subalgebra ◽  
Finite Dimensional

AbstractWe show how to reconstruct a finite directed graph E from its Toeplitz algebra, its gauge action, and the canonical finite-dimensional abelian subalgebra generated by the vertex projections. We also show that if E has no sinks, then we can recover E from its Toeplitz algebra and the generalized gauge action that has, for each vertex, an independent copy of the circle acting on the generators corresponding to edges emanating from that vertex. We show by example that it is not possible to recover E from its Toeplitz algebra and gauge action alone.

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 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.

scholarly journals When is the algebra of regular sets for a finitely additive borel measure a α-algebra?

1982 ◽  
Vol 33 (3) ◽  
pp. 374-385 ◽  
Author(s):  
Thomas E. Armstrong
Keyword(s):  
Borel Measure ◽  
Convex Set ◽  
Additive Measure ◽  
Dimensional Simplex ◽  
Finitely Additive Measure ◽  
Regular Algebra ◽  
Finite Dimensional ◽  
Regular Sets ◽  
Dirac Measures

AbstractIt is shown that hte algebra of regular sets for a finitely additive Borel measure μ on a compact Hausdroff space is a σ-algebra only if it includes the Baire algebra and μ is countably additive onthe σ-algebra of regular sets. Any infinite compact Hausdroff space admits a finitely additive Borel measure whose algebra of regular sets is not a σ-algebra. Although a finitely additive measure with a σ-algebra of regular sets is countably additive on the Baire σ-algebra there are examples of finitely additive extensions of countably additive Baire measures whose regular algebra is not a σ-algebra. We examine the particular case of extensions of Dirac measures. In this context it is shown that all extensions of a {0, 1}-valued countably additive measure from a σ-algebra to a larger σ-algebra are countably additive if and only if the convex set of these extensions is a finite dimensional simplex.

Dynamical Properties of Quadratic Homeomorphisms of a Finite-Dimensional Simplex

2020 ◽  
Vol 245 (3) ◽  
pp. 398-402
Author(s):  
R. N. Ganikhodzhaev ◽  
M. A. Tadzhieva ◽  
D. B. Eshmamatova
Keyword(s):  
Dimensional Simplex ◽  
Finite Dimensional ◽  
Dynamical Properties

scholarly journals KMS STATES ON NICA–TOEPLITZ ALGEBRAS OF PRODUCT SYSTEMS

2012 ◽  
Vol 23 (12) ◽  
pp. 1250123 ◽  
Author(s):  
JEONG HEE HONG ◽  
NADIA S. LARSEN ◽  
WOJCIECH SZYMAŃSKI
Keyword(s):  
Finite Type ◽  
Lattice Ordered Group ◽  
Toeplitz Algebra ◽  
Product System ◽  
Natural Numbers ◽  
Ordered Group ◽  
Kms States ◽  
The Core ◽  
Product Systems ◽  
Affine Semigroup

We investigate KMS states of Fowler's Nica–Toeplitz algebra [Formula: see text] associated to a compactly aligned product system X over a semigroup P of Hilbert bimodules. This analysis relies on restrictions of these states to the core algebra which satisfy appropriate scaling conditions. The concept of product system of finite type is introduced. If (G, P) is a lattice ordered group and X is a product system of finite type over P satisfying certain coherence properties, we construct KMSβ states of [Formula: see text] associated to a scalar dynamics from traces on the coefficient algebra of the product system. Our results were motivated by, and generalize some of the results of Laca and Raeburn obtained for the Toeplitz algebra of the affine semigroup over the natural numbers.

scholarly journals The nonlinear limit control of EDSQOs on finite dimensional simplex

Automatika ◽  
2019 ◽  
Vol 60 (4) ◽  
pp. 404-412 ◽  
Author(s):  
Rawad Abdulghafor ◽  
Shahrum Shah Abdullah ◽  
Sherzod Turaev ◽  
Raini Hassan
Keyword(s):  
Dimensional Simplex ◽  
Finite Dimensional

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 Isometric Dilations of Non-Commuting Finite Rank n-Tuples

2001 ◽  
Vol 53 (3) ◽  
pp. 506-545 ◽  
Author(s):  
Kenneth R. Davidson ◽  
David W. Kribs ◽  
Miron E. Shpigel
Keyword(s):  
Finite Rank ◽  
Dimensional Space ◽  
Finite Dimensional Space ◽  
Toeplitz Algebra ◽  
Finite Dimensional ◽  
Finite Set

AbstractA contractive n-tuple A = (A1,…,An) has a minimal joint isometric dilation S = (S1,…,Sn) where the Si’s are isometries with pairwise orthogonal ranges. This determines a representation of the Cuntz-Toeplitz algebra. When A acts on a finite dimensional space, the wot-closed nonself-adjoint algebra generated by S is completely described in terms of the properties of A. This provides complete unitary invariants for the corresponding representations. In addition, we show that the algebra is always hyper-reflexive. In the last section, we describe similarity invariants. In particular, an n-tuple B of d × d matrices is similar to an irreducible n-tuple A if and only if a certain finite set of polynomials vanish on B.

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