### 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 ∞), 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 language | English |
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Pages (from-to) | 1019-1048 |

Number of pages | 30 |

Journal | Communications in Mathematical Physics |

Volume | 365 |

Issue number | 3 |

Early online date | 30 Jan 2019 |

DOIs | |

Publication status | Published - Feb 2019 |

### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics

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## Cite this

Spakula, J., & Tikuisis, A. P. (2019). Relative Commutant Pictures of Roe Algebras.

*Communications in Mathematical Physics*,*365*(3), 1019-1048. https://doi.org/10.1007/s00220-019-03313-x