Uniform Roe algebras of uniformly locally finite metric spaces are rigid

Baudier, Florent P.; Braga, Bruno de Mendonça; Farah, Ilijas; Khukhro, Ana; Vignati, Alessandro; Willett, Rufus

doi:10.1007/s00222-022-01140-x

Cited by 4 publications

(3 citation statements)

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“…(1) X is coarsely equivalent to Y; Remark 6.14. Very recently, it has been shown in [6] that (1) to (4) in Theorem 6.13 are all equivalent for general metric spaces with bounded geometry. At this moment, we believe that Theorem 6.13 is still the best-known result on the equivalence of (5) with the other items.…”

Section: 3mentioning

confidence: 99%

“…1 About half a year after this paper was first announced, an unconditional positive answer to the rigidity problem for uniform Roe algebras is given in [6]. However, the method in [6] does not immediately apply to Roe algebras. To the best of the authors' knowledge, the rigidity problem for Roe algebras is still open in general.…”

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confidence: 99%

“…Theorem B is the fundamental theorem of this paper, and its proof is rather technical. To briefly summarise, we start with a structure theorem inspired by [24,Theorem 3.5] that measured asymptotic expanders admit a uniform exhaustion by measured expander graphs with bounded measure ratios (see Corollary 4.21) 6 . Next, we show that certain Laplacians associated to measured expanders with bounded measure ratios have spectral gap (see Proposition 5.4).…”

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confidence: 99%

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Measured Asymptotic Expanders and Rigidity for Roe Algebras

Spakula

Zhang

2022

International Mathematics Research Notices

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In this paper, we give a new geometric condition in terms of measured asymptotic expanders to ensure rigidity of Roe algebras. Consequently, we obtain the rigidity for all bounded geometry spaces that coarsely embed into some $L^p$-space for $p\in [1,\infty )$. Moreover, we also verify rigidity for the box spaces constructed by Arzhantseva–Tessera and Delabie–Khukhro even though they do not coarsely embed into any $L^p$-space. The key step in our proof of rigidity is showing that a block-rank-one (ghost) projection on a sparse space $X$ belongs to the Roe algebra $C^{\ast }(X)$ if and only if $X$ consists of (ghostly) measured asymptotic expanders. As a by-product, we also deduce that ghostly measured asymptotic expanders are new sources of counterexamples to the coarse Baum–Connes conjecture.

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Section: 3mentioning

confidence: 99%

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