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

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.Comment: 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 propert

C⁎-simplicity of free products with amalgamation and radical classes of groups

“…Since we have assumed that int G is C * -simple, it follows from [7,Theorem 1.4] that G is C * -simple. Next, let C G (int G) denote the centralizer of int G in G, and suppose that g ∈ C G (int G) \ H. In particular, this means that g commutes with all elements in K 0 and K 1 , so…”

“…The result of [7] implies that the class of groups with the unique trace property is the residual class "dual" to the radical class of amenable groups. We then show that the class of C * -simple groups is also a residual class, giving rise to a "predual" radical class of groups that contains all the amenable groups, and we will call a group "amenablish" if it belongs to this class.…”

“…A long-standing open problem was whether the two properties were equivalent, but fairly recently it has been shown that C * -simplicity is strictly stronger than the unique trace property. The "stronger" part is due to Breuillard, Kalantar, Kennedy, and Ozawa [7,22] and the "strictly" part is due to Le Boudec [5]. The former showed that the unique trace property is equivalent to having a trivial amenable radical, a property already known to be weaker than C * -simplicity, see [18].…”

“…We complete this section by some facts about boundary actions and refer to [7,22,27] for further details. An action of G on a compact space X is called a boundary action if it is minimal (i.e., the orbits are dense) and strongly proximal (i.e., the orbit-closure in P (X) of every probability measure on X contains a point mass).…”

“…The former showed that the unique trace property is equivalent to having a trivial amenable radical, a property already known to be weaker than C * -simplicity, see [18]. Both of the works [7,22] use extensively the theory of boundary actions developed by Furstenberg and Hamana. Even more recently, some other more operator-theoretical characterizations have been obtained by Haagerup [15] and Kennedy [23].…”