• DOI:10.1007/s10240-017-0091-2
• Corpus ID: 119599985

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

@article{Breuillard2014CsimplicityAT,
  title={C*-simplicity and the unique trace property for discrete groups},
  author={Emmanuel Breuillard and Mehrdad Kalantar and Matthew Kennedy and Narutaka Ozawa},
  journal={Publications math{\'e}matiques de l'IH{\'E}S},
  year={2014},
  volume={126},
  pages={35-71}
}

• E. Breuillard, Mehrdad Kalantar, +1 author N. Ozawa
• Published 9 October 2014
• Mathematics
• Publications mathématiques de l'IHÉS

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.