C*-simplicity and the unique trace property for discrete groups

Breuillard, Emmanuel; Kalantar, Mehrdad; Kennedy, Matthew; Ozawa, Narutaka

doi:10.1007/s10240-017-0091-2

Breuillard, Emmanuel ¹ ; Kalantar, Mehrdad ² ; Kennedy, Matthew ³ ; Ozawa, Narutaka ⁴

¹ Mathematisches Institut, Universität Münster 48149 Münster Germany
² Department of Mathematics, University of Houston 77204-3008 Houston TX United States
³ Department of Pure Mathematics, University of Waterloo N2L 3G1 Waterloo ON Canada
⁴ Research Institute for Mathematical Sciences, Kyoto University 606-8502 Kyoto Japan

Publications Mathématiques de l'IHÉS, Tome 126 (2017), pp. 35-71.

Résumé

A discrete group is said to be C*-simple if its reduced C*-algebra is simple, and is said to have the unique trace property if its reduced C*-algebra has a unique tracial state. A dynamical characterization of C*-simplicity was recently obtained by the second and third named authors. In this paper, we introduce new methods for working with group and crossed product C*-algebras that allow us to take the study of C*-simplicity a step further, and in addition to settle the longstanding open problem of characterizing groups with the unique trace property. We give a new and self-contained proof of the aforementioned characterization of C*-simplicity. This yields a new characterization of C*-simplicity in terms of the weak containment of quasi-regular representations. We introduce a convenient algebraic condition that implies C*-simplicity, and show that this condition is satisfied by a vast class of groups, encompassing virtually all previously known examples as well as many new ones. We also settle a question of Skandalis and de la Harpe on the simplicity of reduced crossed products. Finally, we introduce a new property for discrete groups that is closely related to C*-simplicity, and use it to prove a broad generalization of a theorem of Zimmer, originally conjectured by Connes and Sullivan, about amenable actions.

MR Zbl

DOI : 10.1007/s10240-017-0091-2

Affiliations des auteurs :

Breuillard, Emmanuel ¹ ; Kalantar, Mehrdad ² ; Kennedy, Matthew ³ ; Ozawa, Narutaka ⁴

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     author = {Breuillard, Emmanuel and Kalantar, Mehrdad and Kennedy, Matthew and Ozawa, Narutaka},
     title = {C*-simplicity and the unique trace property for discrete groups},
     journal = {Publications Math\'ematiques de l'IH\'ES},
     pages = {35--71},
     publisher = {Springer Berlin Heidelberg},
     address = {Berlin/Heidelberg},
     volume = {126},
     year = {2017},
     doi = {10.1007/s10240-017-0091-2},
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     zbl = {1391.46071},
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     url = {http://archive.numdam.org/articles/10.1007/s10240-017-0091-2/}
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Breuillard, Emmanuel; Kalantar, Mehrdad; Kennedy, Matthew; Ozawa, Narutaka. C*-simplicity and the unique trace property for discrete groups. Publications Mathématiques de l'IHÉS, Tome 126 (2017), pp. 35-71. doi : 10.1007/s10240-017-0091-2. http://archive.numdam.org/articles/10.1007/s10240-017-0091-2/

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