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.

中文翻译：

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C *简单性和离散组的唯一跟踪属性

如果离散基团的简化C *代数简单，则称其为C *简单；如果离散化基团的简化C *代数具有唯一的族态，则称其具有唯一的迹性质。第二名和第三名作者最近获得了C *简单性的动态特征。在本文中，我们介绍了用于处理群乘积和交叉乘积C *代数的新方法，这些方法使我们能够进一步研究C *单纯性，并且用独特的方法解决了长期存在的表征群体特征的开放问题。跟踪属性。我们给出了上述C *-简单性表征的新的且独立的证明。就准正则表示的弱约束而言，这产生了C *简单性的新特征。我们引入了一个便利的代数条件，它暗示了C *-简单性，并表明，这个条件可以由众多的群体来满足，这些群体包括几乎所有以前已知的示例以及许多新的示例。我们还解决了Skandalis和de la Harpe关于减少交叉产品的简单性的问题。最后，我们为离散群引入了一个与C *简单性密切相关的新属性，并用它来证明Zimmer定理的广泛推广，该定理最初是由Connes和Sullivan猜想的，涉及可取动作。