Coarse Baum-Connes conjecture and rigidity for Roe algebras

Braga, Bruno M.; Chung, Yeong Chyuan; Li, Kang

doi:10.1016/j.jfa.2020.108728

Cited by 9 publications

(21 citation statements)

References 17 publications

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“…where Υ is the standard conditional expectation of B( 2 (Γ)) onto ∞ (Γ). (3) Then clearly a = g∈S m a g u g , as required.…”

Section: Examples Of Nonmetrizable Coarse Spacesmentioning

confidence: 90%

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General Uniform Roe algebra rigidity

Braga¹,

Farah²,

Vignati³

2022

Annales De l'Institut Fourier

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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.Résumé. -Nous généralisons tous les résultats connus sur la rigidité des algèbres de Roe uniformes au cas des espaces grossiers uniformément localement finis arbitraires. Nous montrons que l'isomorphisme entre les algèbres de Roe uniformes des espaces grossiers uniformément localement finis dont les algèbres de Roe uniformes ne contiennent que des projections fantômes compactes implique que les espaces de base sont grossièrement équivalents. De plus, si l'un des espaces a la propriété A, alors les espaces de base sont bijectivement grossièrement équivalents. Nous fournissons également une caractérisation de l'existence d'un plongement sur les sous-algèbres héréditaires en termes de sous-espaces. Comme application, nous répondons partiellement à une question de White et Willett sur les sous-algèbres de Cartan des algèbres de Roe uniformes.

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“…where Υ is the standard conditional expectation of B( 2 (Γ)) onto ∞ (Γ). (3) Then clearly a = g∈S m a g u g , as required.…”

Section: Examples Of Nonmetrizable Coarse Spacesmentioning

confidence: 90%

“…Therefore Lemma 4.3 implies that this graph has uniformly bounded degree, and we can find a colouring X = l−1 i=0 X i where each X i is monochromatic. Clearly, the pieces of this partition satisfy (3). A partition of Y satisfying (4) is obtained in the same exact way.…”

Section: The "Rigid" Embedding Scenariomentioning

confidence: 99%

General Uniform Roe algebra rigidity

Braga¹,

Farah²,

Vignati³

2022

Annales De l'Institut Fourier

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“…n∈N has "A-by-CE" structure, namely, the coarse disjoint union [21] of the sequence (N n ) n∈N with the induced metric from the word metrics of (G n ) n∈N has Yu's property A, and the coarse disjoint union of the sequence (Q n ) n∈N with the quotient metrics coarsely embeds into Hilbert space (denoted briefly, CE), then the coarse Baum-Connes conjecture holds for the coarse disjoint union of the sequence (G n ) n∈N . It follows that the coarse Baum-Connes conjecture, and hence the coarse Novikov conjecture, holds for the relative expanders in [2], the group extensions in [3] mentioned above (see [5,23] for an alternative proof for these group extensions), and certain box spaces of free groups in [8], which do not coarsely embed into Hilbert space and yet do not coarsely contain any weakly embedded expander.…”

Section: Introductionmentioning

confidence: 87%

The coarse Novikov conjecture for extensions of coarsely embeddable groups

Wang¹,

Zhang²

2021

Preprint

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Let (1 → Nn → Gn → Qn → 1) n∈N be a sequence of extensions of countable discrete groups.Endow (Gn) n∈N with metrics associated to proper length functions on (Gn) n∈N respectively such that the sequence of metric spaces (Gn) n∈N have uniform bounded geometry. We show that if (Nn) n∈N and (Qn) n∈N are coarsely embeddable into Hilbert space, then the coarse Novikov conjecture holds for the sequence (Gn) n∈N , which may not admit a coarse embedding into Hilbert space.

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“…the fundamental group of a closed manifold) equipped with a word length metric, and the Gromov's zero-in-the-spectrum conjecture and the positive scalar curvature conjecture when X is a Riemannian manifold. See [55] for a comprehensive survey for the coarse Baum-Connes conjecture, and [5,9,18,19,20,21,22,24,41,48,49,50,51,53] for some recent developments.…”

Section: Introductionmentioning

confidence: 99%

“…It satisfies the strong Baum-Connes conjecture [29] which is strictly stronger than the Baum-Connes conjecture with coefficients [7], and consequently, the coarse Baum-Connes conjecture. It has been proved in [5,Proposition 2.11] by B. M. Braga, Y. C. Chung and K. Li that the second group Z 2 ≀ G (H ×F n ) constructed in [4] also satisfies the Baum-Connes conjecture with coefficients by applying a permanence result of H. Oyono-Oyono [40], and hence the coarse Baum-Connes conjecture. Since both groups are A-by-CE split extensions, our result provides an alternative proof to these facts.…”

Section: Introductionmentioning

confidence: 99%

The coarse Baum-Connes conjecture for certain relative expanders

Deng¹,

Wang²,

Yu³

2021

Preprint

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Let (1 → Nn → Gn → Qn → 1) n∈N be a sequence of extensions of finitely generated groups with uniformly finite generating subsets. We show that if the sequence (Nn)n∈N with the induced metric from the word metrics of (Gn) n∈N has property A, and the sequence (Qn) n∈N with the quotient metrics coarsely embeds into Hilbert space, then the coarse Baum-Connes conjecture holds for the sequence (Gn) n∈N , which may not admit a coarse embedding into Hilbert space. It follows that the coarse Baum-Connes conjecture holds for the relative expanders and group extensions exhibited by G. Arzhantseva and R. Tessera, and special box spaces of free groups discovered by T. Delabie and A. Khukhro, which do not coarsely embed into Hilbert space, yet do not contain a weakly embedded expander. This in particular solves an open problem raised by G. Arzhantseva and R. Tessera [3]. Contents

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Coarse Baum-Connes conjecture and rigidity for Roe algebras

Cited by 9 publications

References 17 publications

General Uniform Roe algebra rigidity

General Uniform Roe algebra rigidity

The coarse Novikov conjecture for extensions of coarsely embeddable groups

The coarse Baum-Connes conjecture for certain relative expanders

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