Relative commutant pictures of Roe algebras

TL;DR

This paper provides new characterizations of Roe algebras for proper metric spaces with finite asymptotic dimension, showing they consist of operators that essentially commute with Higson functions and are quasi-local.

Contribution

It introduces novel descriptions of Roe algebras using commutation with Higson functions and quasi-locality, applicable to both usual and uniform Roe algebras.

Findings

Roe algebra equals operators commuting with Higson functions

Roe algebra equals quasi-local operators with finite epsilon propagation

Results hold for spaces with finite asymptotic dimension or decomposition complexity

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 infinity) 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.