Maximal and reduced Roe algebras of coarsely embeddable spaces

By Ján Špakula, Rufus Willett. 2011

51 citations recorded by the source corpus.

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].

Subfields: Algebra and Number Theory, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/0c0cccb0db5b14f6b63833836de5aeeb95733e46

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In Algebra and Number Theory

In Mathematical Physics

In Statistical and Nonlinear Physics

Citations in the corpus

Coarse Non-Amenability and Coarse Embeddings (2011, background): “If X has Property A, this map is an isomorphism [2, 15]; the converse is an open question, a particular case of an open question about groupoid C*-algebras.”
The maximal coarse Baum-Connes conjecture for spaces which admit a fibred coarse embedding into Hilbert space (2012, background): “9 in [37] for essentially the same arguments which can be used to show that the norms in these two C-algebras agree), as desired in the first item.”