Cartan subalgebras in uniform Roe algebras

White, Stuart; Willett, Rufus

doi:10.48550/arxiv.1808.04410

Cited by 7 publications

(14 citation statements)

References 33 publications

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“…Regarding 'weak rigidity', we generalize the findings of [5] and [4]. 1 These should not be confused with the concepts of rigidity and superrigidity, see [26,Remark 4.21].…”

Section: Introductionmentioning

confidence: 58%

“…The following simple corollary of Theorem 1.4 answers a question raised by White and Willett in [26,Remark 3.4] in the scenario of property A (see Corollary 6.3 below).…”

Section: Introductionmentioning

confidence: 65%

“…If A is unital then (3) is clearly redundant. In the context of Roe algebras, White and Willett introduced the following in [26,Definition 4.20]. Definition 6.2.…”

mentioning

confidence: 99%

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General uniform Roe algebra rigidity

Braga¹,

Farah²,

Vignati³

2020

Preprint

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We generalize all known results on rigidity of uniform Roe algebras to the setting of arbitrary uniformly locally finite coarse spaces. For instance, we show that isomorphism between uniform Roe algebras of uniformly locally finite coarse spaces whose uniform Roe algebras contain only compact ghost projections implies that the base spaces are coarsely equivalent. Moreover, if one of the spaces has property A, then the base spaces are bijectively coarsely equivalent. We also provide a characterization for the existence of an embedding onto hereditary subalgebra in terms of the underlying spaces. As an application, we partially answer a question of White and Willett about Cartan subalgebras of uniform Roe algebras. 5 We will eventually prove that this terminology was not necessary; see Remark 3.4.

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“…Regarding 'weak rigidity', we generalize the findings of [5] and [4]. 1 These should not be confused with the concepts of rigidity and superrigidity, see [26,Remark 4.21].…”

Section: Introductionmentioning

confidence: 58%

“…The following simple corollary of Theorem 1.4 answers a question raised by White and Willett in [26,Remark 3.4] in the scenario of property A (see Corollary 6.3 below).…”

Section: Introductionmentioning

confidence: 65%

See 1 more Smart Citation

General uniform Roe algebra rigidity

Braga¹,

Farah²,

Vignati³

2020

Preprint

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“…In particular, Theorem E shows that we can not use nuclearity to distinguish C * uq (X) from C * u (X). Moreover, we know that ℓ ∞ (X) ⊆ C * u (X) is always a Cartan subalgebra, and structural and uniqueness questions for Cartan subalgebras in uniform Roe algebras were intensively studied in [35].…”

Section: • X Has Property a If And Only If The Uniform Quasi-local Al...mentioning

confidence: 99%

Quasi-local Algebras and Asymptotic Expanders

Nowak

Spakula

et al. 2019

Preprint

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In this paper, we study the relation between the uniform Roe algebra and the uniform quasi-local algebra associated to a discrete metric space of bounded geometry. In the process, we introduce a weakening of the notion of expanders, called asymptotic expanders. We show that being asymptotic expanders is a coarse property, and it implies non-uniformly local amenability. Moreover, we also analyse some C * -algebraic properties of uniform quasi-local algebras. In particular, we show that a uniform quasi-local algebra is nuclear if and only if the underlying metric space has Property A.

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“…Moreover, for p ∈ [1, ∞), X = ℓ p and E the standard unit basis of ℓ p , the uniform Roe algebra of (d, E) is often denoted by B p u (N, d), or B p u (X). In recent years, rigidity questions for uniform Roe algebras have been extensively studied for ℓ p , with special emphasis for p = 2 (e.g., [4,6,10,30,32]). In layman's term, rigidity questions ask what kind of equivalence notions between two uniform Roe algebras are strong enough so that their existence implies that the base spaces of those algebras must be (bijectively) coarsely equivalent.…”

Section: Introductionmentioning

confidence: 99%

On Banach algebras of band-dominated operators and their order structure

Braga¹

2020

Preprint

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The goal of this paper is to study band-dominated operators on Banach spaces with Schauder basis with respect to uniformly locally finite metric spaces as well as the Banach algebras generated by them: the so called uniform Roe algebras. We investigate several kinds of isomorphisms between those Banach spaces (e.g., isomorphisms preserving norm, order, algebraic structure, etc) and prove several rigidity results on when a certain kind of isomorphism between the uniform Roe algebras implies that the base metric spaces are (bijectively) coarsely equivalent. Contents1. Introduction 1 2. Preliminaries 5 3. Properties of different bases in uniform Roe algebras 8 4. Regular uniform Roe algebra 12 5. Rigidity for order preserving equivalences 15 6. Property A and ghost operators 20 7. Rigidity for Banach algebra isomorphisms 25 8. Open questions 33 9. Appendix 34 References 36

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Cartan subalgebras in uniform Roe algebras

Cited by 7 publications

References 33 publications

General uniform Roe algebra rigidity

General uniform Roe algebra rigidity

Quasi-local Algebras and Asymptotic Expanders

On Banach algebras of band-dominated operators and their order structure

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