Measured asymptotic expanders and rigidity for Roe algebras
Measured asymptotic expanders and rigidity for Roe algebras
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 which coarsely embed into some Lp-space for p∈ [1,∞). 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 Lp-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*(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.
Li, Kang
62945651-4b08-4fa3-a1fa-0eadaddbf6c5
Spakula, Jan
c43164e4-36a7-4372-9ce2-9bfbba775d77
Zhang, Jiawen
aa149f14-dd1d-42b0-b863-623d1fedd1f5
15 September 2022
Li, Kang
62945651-4b08-4fa3-a1fa-0eadaddbf6c5
Spakula, Jan
c43164e4-36a7-4372-9ce2-9bfbba775d77
Zhang, Jiawen
aa149f14-dd1d-42b0-b863-623d1fedd1f5
Li, Kang, Spakula, Jan and Zhang, Jiawen
(2022)
Measured asymptotic expanders and rigidity for Roe algebras.
International Mathematics Research Notices.
(doi:10.1093/imrn/rnac242).
Abstract
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 which coarsely embed into some Lp-space for p∈ [1,∞). 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 Lp-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*(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.
Text
rigidity-meas-asym-exp
- Accepted Manuscript
More information
Accepted/In Press date: 18 August 2022
Published date: 15 September 2022
Additional Information:
For the purpose of open access, the author has applied a Creative
Commons Attribution (CC BY) licence to any Author Accepted Manuscript version arising.
Identifiers
Local EPrints ID: 469292
URI: http://eprints.soton.ac.uk/id/eprint/469292
ISSN: 1687-0247
PURE UUID: 6d58ec1a-b995-4723-ae95-ecf2c67ba1ee
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Date deposited: 13 Sep 2022 16:35
Last modified: 17 Mar 2024 03:33
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Author:
Kang Li
Author:
Jiawen Zhang
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