Symmetries of the KMS Simplex

Christensen, Jørgen Grønnegård

doi:10.1007/s00220-018-3250-5

Cited by 10 publications

(23 citation statements)

References 11 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In the terminology of [Chr17] or [Nes13], Proposition 3.5 above implies that µ y,β is the unique measure on Λ ∞ which is e −βc y,β -quasi-invariant. Therefore, Lemma 4.1 and Theorem 5.2 of [Chr17] imply that the extremal KMS β states for α y,β are in bijection with a certain subgroup B of T k . Theorem 4.8 below establishes that B is equal to PerΛ ⊆ Z k , the periodicity group of Λ.…”

Section: R + -Functors and Measures On The Infinite Path Spacementioning

confidence: 99%

“…To describe the strategy of our proof, let us recall that in the groupoid perspective, as explained by Renault already in [Ren80], time evolutions (dynamics) on the C * -algebra of a groupoid G are implemented by continuous cocycles on G, and the task of understanding the KMS states for these dynamics on C * (G) requires, at a minimum, identifying the measures on the unit space of G which are quasi-invariant with respect to the cocycle. There are now refinements of Renault's result, see for example [Nes13,Tho14,Chr17]. In particular, Christensen's recent preprint [Chr17] combines quasi-invariant measures with a certain group of symmetries to describe KMS states on groupoid C * -algebras.…”

Section: Introductionmentioning

confidence: 99%

“…There are now refinements of Renault's result, see for example [Nes13,Tho14,Chr17]. In particular, Christensen's recent preprint [Chr17] combines quasi-invariant measures with a certain group of symmetries to describe KMS states on groupoid C * -algebras. This perspective is particularly well suited to our case of interest, namely, the KMS states associated to generalized gauge actions on k-graph C * -algebras.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Generalized gauge actions on $\kappa$-graph $C*-Algebras: KMS states and Hausdorff structure

Farsi¹,

Gillaspy²,

Larsen³

et al. 2021

Indiana Univ. Math. J.

View full text Add to dashboard Cite

In this paper, we consider non-standard dynamics on the C * -algebra associated to a higher-rank graph Λ. These dynamics were first introduced by McNamara in his thesis and arise from a functor y : Λ → R+. We show that the KMS states associated to these dynamics are parametrized by the periodicity group of the higher-rank graph and a family of Borel probability measures on the infinite path space; an analogous parametrization was earlier obtained by an Huef, Laca, Raeburn, and Sims in the case of the standard dynamics. The aforementioned Borel probability measures also arise as Hausdorff measures on the infinite path space of the higher-rank graph, and the associated Hausdorff dimension is intimately linked to the inverse temperatures at which KMS states exist. Our construction of the metrics underlying the Hausdorff structure uses the functors y; the stationary k-Bratteli diagram associated to Λ; and a new concept of exponentially self-similar weights on Bratteli diagrams.

show abstract

Section: R + -Functors and Measures On The Infinite Path Spacementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Generalized gauge actions on $\kappa$-graph $C*-Algebras: KMS states and Hausdorff structure

Farsi¹,

Gillaspy²,

Larsen³

et al. 2021

Indiana Univ. Math. J.

View full text Add to dashboard Cite

show abstract

“…Typically, these Toeplitz extensions exhibit more interesting KMS structure. This recent work has covered Toeplitz algebras of directed graphs and their higher-rank analogues [7,8,13,15,16] (after earlier work in [11]), Toeplitz algebras arising in number theory [9], the Nica-Toeplitz extensions of Cuntz-Pimsner algebras [1,4,[17][18][19] and Toeplitz algebras associated to self-similar actions [22,23]. In [6], Brownlowe, Hawkins and Sims described Toeplitz extensions of the non-commutative solenoids from [24], and they considered a natural dynamics on this extension.…”

Section: Z Afsar Et Almentioning

confidence: 99%

Equilibrium states on higher-rank Toeplitz non-commutative solenoids

Afsar¹,

Huef²,

Raeburn³

et al. 2019

Ergod. Th. Dynam. Sys.

View full text Add to dashboard Cite

We consider a family of higher-dimensional noncommutative 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 noncommutative solenoids are direct limits of the Toeplitz extensions of noncommutative tori. We consider natural dynamics on these Toeplitz algebras, and compute the equilibrium states for these dynamics. We find a large simplex of equilibrium states at each positive inverse temperature, parametrised by the probability measures on an (ordinary) solenoid.

show abstract

“…which gives the required approximation. 6 For every compactly aligned product system X there is a universal Nica covariant Toeplitz representation i X : X → N T (X), see [11]. The C * -algebra N T (X) is called the Nica-Toeplitz algebra of X; it is denoted by T cov (X) in op.…”

Section: Introductionmentioning

confidence: 99%

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

Afsar

Larsen

Neshveyev

2020

Commun. Math. Phys.

View full text Add to dashboard Cite

Given a quasi-lattice ordered group (G, P ) and a compactly aligned product system X of essential C * -correspondences over the monoid P , we show that there is a bijection between the gauge-invariant KMS β -states on the Nica-Toeplitz algebra N T (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 β. Under fairly general additional assumptions we show that there is a critical inverse temperature βc such that for β > βc all KMS β -states are of Gibbs type, hence gauge-invariant, in which case we have a complete classification of KMS β -states in terms of tracial states on A, while at β = βc we have a phase transition manifesting itself in the appearance of 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 N T (X).Acknowledgement. This research was initiated when Z.A. visited the Department of Mathematics at the University of Oslo. She thanks her colleagues for their hospitality.

show abstract

Symmetries of the KMS Simplex

Cited by 10 publications

References 11 publications

Generalized gauge actions on $\kappa$-graph $C*-Algebras: KMS states and Hausdorff structure

Generalized gauge actions on $\kappa$-graph $C*-Algebras: KMS states and Hausdorff structure

Equilibrium states on higher-rank Toeplitz non-commutative solenoids

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

Contact Info

Product

Resources

About