2014/10/31 by Emmanuel Breuillard, Mehrdad Kalantar, Matthew Kennedy +1 · 3 citations
Mathematics · #Advanced Algebra and Geometry #Advanced Operator Algebra Research #Algebraic number #Characterization (materials science) #Class (philosophy) #Generalization #Geometric and Algebraic Topology #Group (periodic table) #Product (mathematics) #Property (philosophy) #Simplicity #TRACE (psycholinguistics) #math.GR #math.OA #msc:20F65 #msc:37A20 #msc:43A07 #msc:46L35
paper · pdf · doi:10.1007/s10240-017-0091-2
published as Publications mathématiques de l'IHÉS 126 (2017), no. 1, 35-71 · 40 pages; major restructuring; four new sections added, including a new characterization of C*-simplicity, a new proof of the Kalantar-Kennedy characterization of C*-simplicity and a discussion of the Connes-Sullivan property
openalex created_date 2016/06/24 · arxiv created 2016/10/26 · openalex publication_date 2017/06/28 · crossref created 2017/06/28 · crossref issued 2017/11/07 · crossref published 2017/11/07 · crossref published-online 2017/11/07 · arxiv updated 2017/12/14 · crossref deposited 2026/02/17 · crossref indexed 2026/08/04 · openalex updated_date 2026/08/05
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.