Maximal and reduced Roe algebras of coarsely embeddable spaces

Abstract

In [7], Gong, Wang and Yu introduced a maximal, or universal, version of the Roe C *-algebra associated to a metric space. We study the relationship between this maximal Roe algebra and the usual version, in both the uniform and non-uniform cases. The main result is that if a (uniformly discrete, bounded geometry) metric space X coarsely embeds into a Hilbert space, then the canonical map between the maximal and usual (uniform) Roe algebras induces an isomorphism on K -theory. We also give a simple proof that if X has property A, then the maximal and usual (uniform) Roe algebras are the same. These two results are natural coarse-geometric analogues of certain well-known implications of a-T-menability and amenability for group C *-algebras. The techniques used are E -theoretic, building on work of Higson, Kasparov and Trout [11], [12] and Yu [28].