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.
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DOI: 10.1007/s10240-017-0091-2
Emmanuel Breuillard 1; Mehrdad Kalantar 2; Matthew Kennedy 3; Narutaka Ozawa 4
@article{PMIHES_2017__126__35_0,
author = {Emmanuel Breuillard and Mehrdad Kalantar and Matthew Kennedy and Narutaka Ozawa},
title = {C*-simplicity and the unique trace property for discrete groups},
journal = {Publications Math\'ematiques de l'IH\'ES},
pages = {35--71},
year = {2017},
publisher = {Springer Berlin Heidelberg},
address = {Berlin/Heidelberg},
volume = {126},
doi = {10.1007/s10240-017-0091-2},
mrnumber = {3735864},
zbl = {1391.46071},
language = {en},
url = {https://pmihes.centre-mersenne.org/articles/10.1007/s10240-017-0091-2/}
}
TY - JOUR AU - Emmanuel Breuillard AU - Mehrdad Kalantar AU - Matthew Kennedy AU - Narutaka Ozawa TI - C*-simplicity and the unique trace property for discrete groups JO - Publications Mathématiques de l'IHÉS PY - 2017 SP - 35 EP - 71 VL - 126 PB - Springer Berlin Heidelberg PP - Berlin/Heidelberg UR - https://pmihes.centre-mersenne.org/articles/10.1007/s10240-017-0091-2/ DO - 10.1007/s10240-017-0091-2 LA - en ID - PMIHES_2017__126__35_0 ER -
%0 Journal Article %A Emmanuel Breuillard %A Mehrdad Kalantar %A Matthew Kennedy %A Narutaka Ozawa %T C*-simplicity and the unique trace property for discrete groups %J Publications Mathématiques de l'IHÉS %D 2017 %P 35-71 %V 126 %I Springer Berlin Heidelberg %C Berlin/Heidelberg %U https://pmihes.centre-mersenne.org/articles/10.1007/s10240-017-0091-2/ %R 10.1007/s10240-017-0091-2 %G en %F PMIHES_2017__126__35_0
Emmanuel Breuillard; Mehrdad Kalantar; Matthew Kennedy; Narutaka Ozawa. C*-simplicity and the unique trace property for discrete groups. Publications Mathématiques de l'IHÉS, Volume 126 (2017), pp. 35-71. doi: 10.1007/s10240-017-0091-2
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