Endomorphisms and modular theory of 2-graph C*-algebras

Yang, Dilian

doi:10.1512/iumj.2010.59.3973

Cited by 23 publications

(54 citation statements)

References 30 publications

(52 reference statements)

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“…Here we illustrate our results by applying them to a 2-graph with a single vertex. Such graphs were first studied by Kribs and Power [16], and their C * -algebras have been extensively studied by Davidson and Yang [5,33,34]. Yang in particular has made a convincing case that these C * -algebras should be viewed as higher-rank anaologues of the Cuntz algebras, and share many of their properties.…”

Section: -Graphs With a Single Vertexmentioning

confidence: 99%

Spatial realisations of KMS states on theC⁎-algebras of higher-rank graphs

Huef

Kang

Raeburn

2015

Journal of Mathematical Analysis and Applications

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Several authors have recently been studying the equilibrium or KMS states on the Toeplitz algebras of finite higher-rank graphs. For graphs of rank one (that is, for ordinary directed graphs), there is a natural dynamics obtained by lifting the gauge action of the circle to an action of the real line. The algebras of higher-rank graphs carry a gauge action of a higher-dimensional torus, and there are many potential dynamics arising from different embeddings of the real line in the torus. Previous results show that there is nonetheless a "preferred dynamics" for which the system exhibits a particularly satisfactory phase transition, and that the unique KMS state at the critical inverse temperature can then be implemented by integrating vector states against a measure on the infinite path space of the graph. Here we obtain a similar description of the KMS state at the critical inverse temperature for other dynamics. Our spatial implementation is given by integrating against a measure on a space of paths which are infinite in some directions but finite in others. Our results are sharpest for the algebras of rank-two graphs.

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Section: -Graphs With a Single Vertexmentioning

confidence: 99%

Spatial realisations of KMS states on theC⁎-algebras of higher-rank graphs

Huef

Kang

Raeburn

2015

Journal of Mathematical Analysis and Applications

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“…For the second statement note that since by (1.5) the element i X (ξ) * i X (ζ) can be written as the sum of elements i X (η)i X (µ) * , with η ∈ X p −1 (p∨q) and µ ∈ X q −1 (p∨q) , and i i X (λ i )i X (λ i ) * i X (η) = i X (η), 31 we have x = i i X (λ i )i X (λ i ) * x. Therefore by applying φ and the KMS-condition we get…”

Section: Gauge-invariancementioning

confidence: 99%

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

Afsar

Larsen

Neshveyev

2020

Commun. Math. Phys.

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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.

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“…. , β c k ), following the recent conventions for k-graph algebras (see [45,46,22]), we call the common value β c := r −1 i β c i the critical inverse temperature. In particular, we are interested in r := (β c 1 , .…”

Section: By Proposition 43(b)mentioning

confidence: 99%

KMS states on C⁎-algebras associated to a family of ⁎-commuting local homeomorphisms

Afsar

Huef

Raeburn

2018

Journal of Mathematical Analysis and Applications

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We consider a family of * -commuting local homeomorphisms on a compact space, and build a compactly aligned product system of Hilbert bimodules. The Nica-Toeplitz algebra of this system carries a gauge action of a higher-dimensional torus, and there are many possible dynamics obtained by composing with different embeddings of the real line in this torus. We study the KMS states of these dynamics. For large inverse temperatures including ∞, we describe the simplex of KMS states on the Nica-Toeplitz algebra. We illustrate our main theorem by considering backward shifts on the infinite-path spaces of a class of k-graphs whose shift maps * -commute.

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Endomorphisms and modular theory of 2-graph C*-algebras

Cited by 23 publications

References 30 publications

Spatial realisations of KMS states on theC⁎-algebras of higher-rank graphs

Spatial realisations of KMS states on theC⁎-algebras of higher-rank graphs

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

KMS states on C⁎-algebras associated to a family of ⁎-commuting local homeomorphisms

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