Relative commutant pictures of Roe algebras

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.

35 pages. Minor changes. To appear in Comm. Math. Phys

References in corpus (3)

• The topology of positive scalar curvature
• The nuclear dimension of C*-algebras
• Index theory of uniform pseudodifferential operators

Cited by in corpus (8)

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• Quasi-locality for étale groupoids
• Extreme cases of limit operator theory on metric spaces
• On the structure of asymptotic expanders
• The strongly quasi-local coarse Novikov conjecture and Banach spaces with Property (H)