On rigidity of Roe algebras
By Ján Špakula, Rufus Willett. 2011
72 citations recorded by the source corpus.
Roe algebras are C � -algebras built using large-scale (or ‘coarse’) aspects of a metric space (X,d). In the special case that X = is a finitely generated group and d is a word metric, the simplest Roe algebra associated to ( ,d) is isomorphic to the crossed product C � -algebra l 1 () ⋊r . Roe algebras are ‘coarse invariants’, in the sense that if X and Y are coarsely equivalent metric spaces, then their Roe algebras are isomorphic. Motivated in part by the coarse Baum-Connes conjecture, we ask if there is a converse to the above statement: that is, if X and Y are metric spaces with isomorphic Roe algebras, must X and Y be coarsely equivalent? We show that for very large classes of spaces the answer to this question, and some related questions, is ‘yes’. This can be thought of as a ‘C � rigidity result’: it shows that the Roe algebra construction preserves a large amount of information about the space, and is thus surprisingly ‘rigid’. As an example of our results, in the group case we have that if and � are finitely generated elementary amenable, hyperbolic, or linear, groups
Subfields: Algebra and Number Theory, Mathematical Physics, Statistical and Nonlinear Physics
View Paper: https://www.semanticscholar.org/paper/18d56787ff398f131b53b8bad1e3a08ad898ed45
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In Algebra and Number Theory
In Mathematical Physics
In Statistical and Nonlinear Physics
Citations in the corpus
- Cartan subalgebras in C*-algebras. Existence and uniqueness (2017, background): “In Section 6, we present such examples by combining rigidity results by Spakula and Willett [57] and by Whyte [65] with recent results of the first named author: Under the assumption that the finitely generated groups Γ and Λ are exact, the isomorphism of their stable”