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Relative Commutant Pictures of Roe Algebras

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Abstract

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 \({\infty}\)), and (ii) it consists exactly of quasi-local operators, that is, ones which have finite \({\epsilon}\)-propogation (in the sense of Roe) for every \({\epsilon > 0}\). These descriptions hold both for the usual Roe algebra and for the uniform Roe algebra.

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Acknowledgements

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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Authors and Affiliations

  1. Mathematical Sciences, University of Southampton, Highfield, Southampton, SO17 1BJ, UK

    Ján Špakula

  2. Department of Mathematics and Statistics, University of Ottawa, Ottawa, ON, K1N 6N5, Canada

    Aaron Tikuisis

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  1. Ján Špakula
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  2. Aaron Tikuisis
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Correspondence to Aaron Tikuisis.

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Communicated by Y. Kawahigashi

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Špakula, J., Tikuisis, A. Relative Commutant Pictures of Roe Algebras. Commun. Math. Phys. 365, 1019–1048 (2019). https://doi.org/10.1007/s00220-019-03313-x

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