Amenability and uniform Roe algebras

Pere Ara, Kang Li, Fernando Lledó, Jianchao Wu

Research output: Contribution to journalArticleResearchpeer-review

7 Citations (Scopus)

Abstract

© 2017 Elsevier Inc. Amenability for groups can be extended to metric spaces, algebras over commutative fields and C⁎-algebras by adapting the notion of Følner nets. In the present article we investigate the close ties among these extensions and show that these three pictures unify in the context of the uniform Roe algebra Cu⁎(X) over a metric space (X,d) with bounded geometry. In particular, we show that the following conditions are equivalent: (1) (X,d) is amenable; (2) the translation algebra generating Cu⁎(X) is algebraically amenable (3) Cu⁎(X) has a tracial state; (4) Cu⁎(X) is not properly infinite; (5) [1]0≠[0]0 in the K0-group K0(Cu⁎(X)); (6) Cu⁎(X) does not contain the Leavitt algebra as a unital ⁎-subalgebra; (7) Cu⁎(X) is a Følner C⁎-algebra in the sense that it admits a net of unital completely positive maps into matrices which is asymptotically multiplicative in the normalized trace norm. We also show that every possible tracial state of the uniform Roe algebra Cu⁎(X) is amenable.
Original languageEnglish
Pages (from-to)686-716
JournalJournal of Mathematical Analysis and Applications
Volume459
Issue number2
DOIs
Publication statusPublished - 15 Mar 2018

Keywords

  • Amenability
  • Coarse space
  • Følner conditions
  • Semi-pre-C -algebra ⁎
  • Traces
  • Uniform Roe algebras

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