Amenability and uniform Roe algebras

Ara P, Lledo F, Wu J, Li K (2018)

Publication Type: Journal article

Publication year: 2018

Journal

Journal of Mathematical Analysis and Applications Elsevier

Book Volume: 459

Pages Range: 686-716

Journal Issue: 2

DOI: 10.1016/j.jmaa.2017.10.063

Abstract

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.

Authors with CRIS profile

Kang Li

Involved external institutions

Autonomous University of Barcelona (UAB) / Universitat Autònoma de Barcelona ES Spain (ES) Universidad Carlos III de Madrid (UC3M) ES Spain (ES) Pennsylvania State University (Penn State) US United States (USA) (US) Westfälische Wilhelms-Universität (WWU) Münster DE Germany (DE)

How to cite

APA:

Ara, P., Lledo, F., Wu, J., & Li, K. (2018). Amenability and uniform Roe algebras. Journal of Mathematical Analysis and Applications, 459(2), 686-716. https://dx.doi.org/10.1016/j.jmaa.2017.10.063

MLA:

Ara, Pere, et al. "Amenability and uniform Roe algebras." Journal of Mathematical Analysis and Applications 459.2 (2018): 686-716.

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