Low‐dimensional properties of uniform Roe algebras

Li, Kang; Willett, Rufus

doi:10.1112/jlms.12100

Cited by 43 publications

(15 citation statements)

References 37 publications

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“…Notice that this strategy does not hold for general non-minimal almost finite groupoids since they usually do not have stable rank one (see e.g. [17,23,4] for examples and further results in this direction).…”

Section: Consider Nowmentioning

confidence: 99%

Strict comparison for $$C^*$$-algebras arising from almost finite groupoids

Ara

Bönicke

Bosa

et al. 2020

Banach J. Math. Anal.

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In this paper we show that for an almost finite minimal ample groupoid G, its reduced C * -algebra C * r (G) has real rank zero and strict comparison even though C * r (G) may not be nuclear in general. Moreover, if we further assume G being also second countable and non-elementary, then its Cuntz semigroup Cu(C * r (G)) is almost divisible and Cu(C * r (G)) and Cu(C * r (G) ⊗ Z) are canonically order-isomorphic, where Z denotes the Jiang-Su algebra.Almost finiteness for an ample groupoid was introduced by Matui in [20]. He studied their topological full groups as well as the applications of almost finiteness to the homology of étale groupoids (see [21] for a survey of results in this direction). In [12], David Kerr specialised to almost finite group actions and treated them as a topological analogue of probability measure preserving hyperfinite equivalence relations, with the ultimate goal of transferring ideas from the classification of equivalence relations and von Neumann algebras to the world of (amenable) topological dynamics and C * -algebras.Recently, the classification program for C * -algebras has culminated in the outstanding theorem that all separable, simple, unital, nuclear, Z-stable C * -algebras satisfying the universal coefficient theorem (UCT) are classified by their Elliott-invariant (see [30, Corollary D] and [8, Corollary D]). Recall that a C * -algebra is Z-stable if A ⊗ Z ∼ = A, where Z denotes the so-called Jiang-Su algebra. By the Toms-Winter conjecture Z-stability is conjecturally equivalent to strict comparison (or equivalently, almost unperforation of the Cuntz semigroup) for separable, simple, nuclear, non-elementary C * -algebras. It is known that Z-stability implies strict comparison in general and the converse is indeed the last remaining open step in the Toms-Winter conjecture (see [33] for an overview and [7] for the state of the art for the conjecture).Going back to topological dynamics, David Kerr's approach in [12] has seen dramatic success. He was able to show that a crossed product C(X) Γ associated to a free and minimal action of an (amenable) infinite group Γ is Z-stable provided that the action is almost finite (see [12, Theorem 12.4]). Combining this with the recent result in [13, Theorem 8.1], which states that every free action of a countably infinite (amenable) group with subexponential

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Section: Consider Nowmentioning

confidence: 99%

Strict comparison for $$C^*$$-algebras arising from almost finite groupoids

Ara

Bönicke

Bosa

et al. 2020

Banach J. Math. Anal.

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“…Recently, there has been substantial research about the interplay between coarsegeometric properties of a metric space X with bounded geometry and analytic properties of its uniform Roe algebra C * u (X) (e.g. [1,2,8,17,18,30,36,40]). A prototypical result in this direction comes from [13,21,30]: a metric space X has Property A if and only if C * u (X) is a nuclear C * -algebra.…”

Section: Introductionmentioning

confidence: 99%

Quasi-local algebras and asymptotic expanders

Nowak

Spakula

et al. 2021

Groups Geom. Dyn.

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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 metric space of bounded geometry. In the process, we introduce a weakening of the notion of expanders, called asymptotic expanders. We show that being a sequence of asymptotic expanders is a coarse property under certain connectedness condition, 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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“…Prototypical versions of this C * -algebra were introduced by Roe [31] for index-theoretic purposes. The theory was consolidated in the 1990s by Roe, Yu and others, and uniform Roe algebras have since found applications in index theory (for example, [36,16]), C * -algebra theory (for example, [34,26]), single operator theory (for example, [30,38]), topological dynamics (for example, [21,6]), and mathematical physics (for example, [10,23]).…”

Section: Introductionmentioning

confidence: 99%

Uniform Roe algebras of uniformly locally finite metric spaces are rigid

Baudier¹,

Braga²,

Farah³

et al. 2021

Preprint

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We show that if X and Y are uniformly locally finite metric spaces whose uniform Roe algebras, C * u (X) and C * u (Y ), are isomorphic as C * -algebras, then X and Y are coarsely equivalent metric spaces. Moreover, we show that coarse equivalence between X and Y is equivalent to Morita equivalence between C * u (X) and C * u (Y ). As an application, we obtain that if Γ and Λ are finitely generated groups, then the crossed products ℓ∞(Γ) ⋊r Γ and ℓ∞(Λ) ⋊r Λ are isomorphic if and only if Γ and Λ are bi-Lipshitz equivalent.

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Low‐dimensional properties of uniform Roe algebras

Cited by 43 publications

References 37 publications

Strict comparison for $$C^*$$-algebras arising from almost finite groupoids

Strict comparison for $$C^*$$-algebras arising from almost finite groupoids

Quasi-local algebras and asymptotic expanders

Uniform Roe algebras of uniformly locally finite metric spaces are rigid

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