TY - JOUR

T1 - Amenability and uniform Roe algebras

AU - Ara, Pere

AU - Li, Kang

AU - Lledó, Fernando

AU - Wu, Jianchao

N1 - Funding Information:
The second-named author is partially supported by the DFG ( SFB 878 ) and he wishes to thank James Gabe and Mikael Rørdam for helpful discussions on the subject of properly infinite -algebras. The third-named author thanks Wilhelm Winter for his kind invitation to the Mathematics Department of the University of Münster in April 2014 and March–June 2016. Financial support was provided by the DFG through SFB 878 , as well as, by a DAAD-grant during these visits. Part of the research was conducted during visits and workshops at Universitat Autònoma de Barcelona, University of Copenhagen, University of Münster and Institut Mittag-Leffler. The authors owe many thanks and great appreciation to these institutes and hosts for their hospitality.
Publisher Copyright:
© 2017 Elsevier Inc.

PY - 2018/3/15

Y1 - 2018/3/15

N2 - 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.

AB - 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.

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U2 - 10.1016/j.jmaa.2017.10.063

DO - 10.1016/j.jmaa.2017.10.063

M3 - Article

AN - SCOPUS:85033379288

SN - 0022-247X

VL - 459

SP - 686

EP - 716

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

IS - 2

ER -