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Measured asymptotic expanders and rigidity for Roe algebras

Measured asymptotic expanders and rigidity for Roe algebras
Measured asymptotic expanders and rigidity for Roe algebras
Our main result about rigidity of Roe algebras is the following: if X and Y are metric spaces with bounded geometry such that their Roe algebras are ∗-isomorphic, then X and Y are coarsely equivalent provided that either X or Y contains no sparse subspaces consisting of ghostly measured asymptotic expanders. Note that this geometric condition generalises the existing technical assumptions used for rigidity of Roe algebras.
Consequently, we show that the rigidity holds for all bounded geometry spaces which coarsely embed into some Lp-space for p∈[1,∞). Moreover, we also verify the rigidity for the box spaces constructed by Arzhantseva-Tessera and Delabie-Khukhro even though they do \emph{not} coarsely embed into any Lp-space.
The key step towards our proof for the rigidity is to show that a block-rank-one (ghost) projection on a sparse space X belongs to the Roe algebra C∗(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.
Spakula, Jan
c43164e4-36a7-4372-9ce2-9bfbba775d77
Zhang, Jiawen
aa149f14-dd1d-42b0-b863-623d1fedd1f5
Li, Kang
62945651-4b08-4fa3-a1fa-0eadaddbf6c5
Spakula, Jan
c43164e4-36a7-4372-9ce2-9bfbba775d77
Zhang, Jiawen
aa149f14-dd1d-42b0-b863-623d1fedd1f5
Li, Kang
62945651-4b08-4fa3-a1fa-0eadaddbf6c5

Spakula, Jan, Zhang, Jiawen and Li, Kang (2020) Measured asymptotic expanders and rigidity for Roe algebras. arXiv. (In Press)

Record type: Article

Abstract

Our main result about rigidity of Roe algebras is the following: if X and Y are metric spaces with bounded geometry such that their Roe algebras are ∗-isomorphic, then X and Y are coarsely equivalent provided that either X or Y contains no sparse subspaces consisting of ghostly measured asymptotic expanders. Note that this geometric condition generalises the existing technical assumptions used for rigidity of Roe algebras.
Consequently, we show that the rigidity holds for all bounded geometry spaces which coarsely embed into some Lp-space for p∈[1,∞). Moreover, we also verify the rigidity for the box spaces constructed by Arzhantseva-Tessera and Delabie-Khukhro even though they do \emph{not} coarsely embed into any Lp-space.
The key step towards our proof for the rigidity is to show that a block-rank-one (ghost) projection on a sparse space X belongs to the Roe algebra C∗(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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More information

In preparation date: 21 October 2020
Accepted/In Press date: 21 October 2020

Identifiers

Local EPrints ID: 445428
URI: http://eprints.soton.ac.uk/id/eprint/445428
PURE UUID: b067b39b-67c6-44fd-83c0-29c8a6691aed
ORCID for Jan Spakula: ORCID iD orcid.org/0000-0001-5775-9905

Catalogue record

Date deposited: 08 Dec 2020 17:32
Last modified: 18 Feb 2021 17:22

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Contributors

Author: Jan Spakula ORCID iD
Author: Jiawen Zhang
Author: Kang Li

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