Abstract
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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First author supported by ERC Grant Number 617129.
Third author supported by NSERC Grant Number 418585.
Fourth author supported by JSPS KAKENHI Grant Number 26400114.
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Breuillard, E., Kalantar, M., Kennedy, M. et al. C*-simplicity and the unique trace property for discrete groups. Publ.math.IHES 126, 35–71 (2017). https://doi.org/10.1007/s10240-017-0091-2
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DOI: https://doi.org/10.1007/s10240-017-0091-2