KMS states on the<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" display="inline" overflow="scroll"><mml:msup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mo>∗</mml:mo></mml:mrow></mml:msup></mml:math>-algebras of finite graphs

Huef, Astrid an; Laca, Marcelo; Raeburn, Iain; Sims, Aidan

doi:10.1016/j.jmaa.2013.03.055

Cited by 40 publications

(151 citation statements)

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“…The following lemma will enable us to use Theorem 3.2 to compute theσ-KMS β states on C * λ (R R m,Γ )/I P m K . Its proof relies on an idea that is well-known to experts and goes back at least to Lemma 10.3]; this idea is explained in a more general setting, which is suitable for our purposes, in [aHLRS,Lemma 2.2].…”

Section: The Boundary Quotientmentioning

confidence: 99%

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Phase Transitions on C*-Algebras from Actions of Congruence Monoids on Rings of Algebraic Integers

Bruce

2020

International Mathematics Research Notices

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We compute the KMS (equilibrium) states for the canonical time evolution on C*-algebras from actions of congruence monoids on rings of algebraic integers. We show that for each β ∈ [1, 2], there is a unique KMS β state, and we prove that it is a factor state of type III1. There is a phase transition at β = 2: For each β ∈ (2, ∞], the set of extremal KMS β states decomposes as a disjoint union over a quotient of a ray class group in which the fibers are extremal traces on certain group C*-algebras associated with the ideal classes. Moreover, in most cases, there is a further phase transition at β = ∞ in the sense that there are ground states that are not KMS∞ states. Our computation of KMS and ground states generalizes the results of Cuntz, Deninger, and Laca for the full ax + b-semigroup over a ring of integers, and our type classification generalizes a result of Laca and Neshveyev in the case of the rational numbers and a result of Neshveyev in the case of arbitrary number fields.

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Section: The Boundary Quotientmentioning

confidence: 99%

“…The "only if" direction is obvious. For the other direction, we can apply [aHLRS,Lemma 2.2] to φ with, in the notation from [aHLRS, Lemma 2.2],…”

Section: The Boundary Quotientmentioning

confidence: 99%

Phase Transitions on C*-Algebras from Actions of Congruence Monoids on Rings of Algebraic Integers

Bruce

2020

International Mathematics Research Notices

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“…The KMS-state φ ω on C * (G) in Theorem 5.5 is the unique KMS-state by [23]. Numerous authors present constructions of this state and for more general graphs, eg [1,2,13,42].…”

Section: Graph C * -Algebrasmentioning

confidence: 99%

Constructing KMS states from infinite-dimensional spectral triples

Goffeng

Rennie

Usachev

2019

Journal of Geometry and Physics

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We construct KMS-states from Li 1 -summable semifinite spectral triples and show that in several important examples the construction coincides with well-known direct constructions of KMS-states for naturally defined flows. Under further summability assumptions the constructed KMS-state can be computed in terms of Dixmier traces. For closed manifolds, we recover the ordinary Lebesgue integral. For Cuntz-Pimsner algebras with their gauge flow, the construction produces KMS-states from traces on the coefficient algebra and recovers the Laca-Neshveyev correspondence. For a discrete group acting on its Stone-Cech boundary, we recover the Patterson-Sullivan measures on the Stone-Čech boundary for a flow defined from the Radon-Nikodym cocycle. *

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“…Moreover, there is a unique KMS ln(n) -state on T n which factors through ϕ, so taking the GNS representation of T n implements the canonical surjection T n → O n . (However, note that for each β > ln(n), there is a KMS β -state on T n by [aHLRS13] which does not factor through O n . )…”

Section: Definition 25 Define the Conditional Expectation E : T (Y)mentioning

confidence: 99%

“…If Λ is finite and the edge matrix is irreducible, by [EFW84], O Λ has a unique KMS state, and the only admissible β-value is ln(λ), where λ is the Frobenius-Perron eigenvalue of Λ. See [aHLRS13] for the KMS states on T Λ in this case. For infinite locally finite graphs, see also [Tho13].…”

Section: Cuntz-krieger Graph Algebras and Toeplitz Extensionsmentioning

confidence: 99%

C-algebras from planar algebras I: Canonical C-algebras associated to a planar algebra

Hartglass

Penneys

2016

Trans. Amer. Math. Soc.

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From a planar algebra, we give a functorial construction to produce numerous associated C ∗ ^* -algebras. Our main construction is a Hilbert C ∗ ^* -bimodule with a canonical real subspace which produces Pimsner-Toeplitz, Cuntz-Pimsner, and generalized free semicircular C ∗ ^* -algebras. By compressing this system, we obtain various canonical C ∗ ^* -algebras, including Doplicher-Roberts algebras, Guionnet-Jones-Shlyakhtenko algebras, universal (Toeplitz-) Cuntz-Krieger algebras, and the newly introduced free graph algebras. This is the first article in a series studying canonical C ∗ ^* -algebras associated to a planar algebra.

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KMS states on theC∗-algebras of finite graphs

Cited by 40 publications

References 19 publications

Phase Transitions on C*-Algebras from Actions of Congruence Monoids on Rings of Algebraic Integers

Phase Transitions on C*-Algebras from Actions of Congruence Monoids on Rings of Algebraic Integers

Constructing KMS states from infinite-dimensional spectral triples

C-algebras from planar algebras I: Canonical C-algebras associated to a planar algebra

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KMS states on theC∗-algebras of finite graphs

Cited by 40 publications

References 19 publications

Phase Transitions on C*-Algebras from Actions of Congruence Monoids on Rings of Algebraic Integers

Phase Transitions on C*-Algebras from Actions of Congruence Monoids on Rings of Algebraic Integers

Constructing KMS states from infinite-dimensional spectral triples

C*-algebras from planar algebras I: Canonical C*-algebras associated to a planar algebra

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Product

Resources

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C-algebras from planar algebras I: Canonical C-algebras associated to a planar algebra