On K-theoretic invariants of semigroup C*-algebras from actions of congruence monoids

Bruce, Chris; Li, Xin

doi:10.48550/arxiv.1906.00445

Cited by 3 publications

(13 citation statements)

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“…This allows us to characterize the field itself using C*-dynamical invariants. That these quantities are invariant has been shown via K-theory, without any assumptions about time evolutions, in [11]. Here we reiterate the point that KMS states very often lead to results quite similar to those obtained via K-theory, through methods that are quite different, and that some of the number-theoretic invariants are computable directly from the representations associated to KMS states.…”

Section: Introductionsupporting

confidence: 69%

“…) H 0 , we obtain an isometry S a b : H a → H ba satisfying (11). Now S b := a∈k,a⊆R S a b has the desired properties.…”

Section: Type Classification For Low-temperature Kms Statesmentioning

confidence: 95%

“…The associated group of quotients is 19,Ch.2.11] for the general background on congruence monoids and also [9, Section 2.2] and [11] for the details of this construction in relation to C*-algebras.…”

Section: C*-algebras From Actions Of Congruence Monoidsmentioning

confidence: 99%

“…)), which can be regarded as a generalized 'class group', plays a key role in the parametrization of low-temperature KMS states in [10, Theorem 3.2(iii)] and in the K-theory formula in [11,Theorem 4.1].…”

Section: C*-algebras From Actions Of Congruence Monoidsmentioning

confidence: 99%

“…These systems have been introduced in [9] and include the Toeplitz type systems from [15] as a particular case. They exhibit a low-temperature phase transition [10] parametrized by the tracial states on certain group C*-algebras, which, intriguingly, also play a central role in the K-theory computation in [11]; reinforcing the connection made in the discussion following [18,Theorem 6.6.1] between the K-theory computations from [16,17] for the case of the full ax + b-semigroup and the phase transition described in [15]. Here we focus on the extremal KMS β states for β > 2 and classify them according to type I or II, thus completing the type classification of KMS states started in [10, Theorem 3.2(ii)], see also [10,Remark 3.3(g)].…”

Section: Introductionmentioning

confidence: 99%

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Partition functions as C*-dynamical invariants and actions of congruence monoids

Bruce,

Laca,

Takeishi

2019

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We study the phase transition of KMS states for the C*-algebras of ax + b-semigroups of algebraic integers in which the multiplicative part is restricted to a congruence monoid, as in recent work of Bruce generalizing earlier work of Cuntz, Deninger, and Laca. Here we realize the extremal low-temperature KMS states as generalized Gibbs states by constructing concrete representations induced from extremal traces of certain group C*-algebras. We use these representations to compute the Murray-von Neumann type of extremal KMS states and we determine explicit partition functions for the type I factor states. The collection of partition functions that arise this way is an invariant under R-equivariant isomorphism of C*-dynamical systems, which produces further invariants through the analysis of the topological structure of the KMS state space. As an application we characterize several features of the underlying number field and congruence monoid in terms of these invariants. In most cases our systems have infinitely many type I factor KMS states and at least one type II factor KMS state at the same inverse temperature and there are infinitely many partition functions. This low-temperature phase transition exhibited by our system does not respect the energy levels associated to extremal states. In order to deal with this multiplicity, we establish, in the context of general C*-dynamical systems, a precise way to associate partition functions to extremal KMS states that are of type I, and we then show that for our systems these partition functions depend only on connected components in the KMS simplex. The discussion of partition functions of general C*-dynamical systems may be of interest by itself and is likely to have applications in other contexts, so we include it in a self-contained initial section that is partly expository and is independent of the numbertheoretic background and of the technical results about congruence monoids.

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Section: Introductionsupporting

confidence: 69%

“…) H 0 , we obtain an isometry S a b : H a → H ba satisfying (11). Now S b := a∈k,a⊆R S a b has the desired properties.…”

Section: Type Classification For Low-temperature Kms Statesmentioning

confidence: 95%

Section: C*-algebras From Actions Of Congruence Monoidsmentioning

confidence: 99%

Section: C*-algebras From Actions Of Congruence Monoidsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Partition functions as C*-dynamical invariants and actions of congruence monoids

Bruce,

Laca,

Takeishi

2019

Preprint

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Nuclearity of semigroup C*-algebras

Huef

Nucinkis

Sehnem

et al. 2019

Preprint

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We study the semigroup C * -algebra of a positive cone P of a weakly quasi-lattice ordered group. That is, P is a subsemigroup of a discrete group G with P ∩ P −1 = {e} and such that any two elements of P with a common upper bound in P also have a least upper bound. We find sufficient conditions for the semigroup C * -algebra of P to be nuclear. These conditions involve the idea of a generalised length function, called a "controlled map", into an amenable group.Here we give a new definition of a controlled map and discuss examples from different sources. We apply our main result to establish nuclearity for semigroup C * -algebras of a class of one-relator semigroups, motivated by a recent work of Li, Omland and Spielberg. This includes all the Baumslag-Solitar semigroups. We also analyse semidirect products of weakly quasi-lattice ordered groups and use our theorem in examples to prove nuclearity of the semigroup C * -algebra. Moreover, we prove that the graph product of weak quasi-lattices is again a weak quasi-lattice, and show that the corresponding semigroup C * -algebra is nuclear when the underlying groups are amenable.

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Constructing number field isomorphisms from -isomorphisms of certain crossed product C-algebras

Bruce¹,

Takeishi²

2022

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We prove that the class of crossed product C*-algebras associated with the action of the multiplicative group of a number field on its ring of finite adeles is rigid in the following explicit sense: Given any *-isomorphism between two such C*-algebras, we construct an isomorphism between the underlying number fields. As an application, we prove an analogue of the Neukirch-Uchida theorem using topological full groups, which gives a new class of discrete groups associated with number fields whose abstract isomorphism class completely characterises the number field.

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On K-theoretic invariants of semigroup C*-algebras from actions of congruence monoids

Cited by 3 publications

References 32 publications

Partition functions as C*-dynamical invariants and actions of congruence monoids

Partition functions as C*-dynamical invariants and actions of congruence monoids

Nuclearity of semigroup C*-algebras

Constructing number field isomorphisms from -isomorphisms of certain crossed product C-algebras

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On K-theoretic invariants of semigroup C*-algebras from actions of congruence monoids

Cited by 3 publications

References 32 publications

Partition functions as C*-dynamical invariants and actions of congruence monoids

Partition functions as C*-dynamical invariants and actions of congruence monoids

Nuclearity of semigroup C*-algebras

Constructing number field isomorphisms from *-isomorphisms of certain crossed product C*-algebras

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Product

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Constructing number field isomorphisms from -isomorphisms of certain crossed product C-algebras