Maximal and reduced Roe algebras of coarsely embeddable spaces

Ján Špakula, Rufus Willett
2013 Journal für die Reine und Angewandte Mathematik  
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 in 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
more » ... 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-Trout [12], [11] and Yu [28] . MSC: primary 46L80. * The first author was supported by the Deutsche Forschungsgemeinschaft (SFB 478 and SFB 878).
doi:10.1515/crelle.2012.019 fatcat:4gdckfnxgfal5lsfu4aslnv324