Relative Commutant Pictures of Roe Algebras

Jan Spakula, Aaron Peter Tikuisis* (Corresponding Author)

*Corresponding author for this work

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Let X be a proper metric space, which has finite asymptotic dimension in the sense of Gromov (or more generally, straight finite decomposition complexity of Dranishnikov and Zarichnyi). New descriptions are provided of the Roe algebra of X: (i) it consists exactly of operators which essentially commute with diagonal operators coming from Higson functions (that is, functions on X whose oscillation tends to 0 at ∞), and (ii) it consists exactly of quasi-local operators, that is, ones which have finite -propogation (in the sense of Roe) for every > 0. These descriptions hold both for the usual Roe algebra and for the uniform Roe algebra.
Original languageEnglish
Pages (from-to)1019-1048
Number of pages30
JournalCommunications in Mathematical Physics
Issue number3
Early online date30 Jan 2019
Publication statusPublished - Feb 2019

Bibliographical note

AT was supported by EPSRC EP/N00874X/1.
JS was supported by Marie Curie FP7-PEOPLE-2013-CIG Coarse Analysis (631945). We would like to thank Ulrich Bunke, Alexander Engel,
John Roe, Thomas Weighill, Stuart White, and Rufus Willett for comments and discussion relating to this piece.


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