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

Ivanov, Nikolay A.; Omland, Tron

doi:10.1016/j.jfa.2016.12.011

Cited by 7 publications

(18 citation statements)

References 17 publications

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“…In particular, they established a bijective correspondence between maximal ideals of the reduced crossed product and maximal invariant ideals of the underlying C*-algebra. Finally, recent work of Raum explores the C*-simplicity of non-discrete groups [55,56], and very recent work of Ivanov and Omland [34] provides further examples of non-C*-simple groups, among other things.…”

Section: Introductionmentioning

confidence: 99%

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

et al. 2017

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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

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Section: Introductionmentioning

confidence: 99%

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

et al. 2017

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“…Graphs of groups with only one edge are the most studied examples, and their fundamental groups are of two types, either an amalgamated free product or an HNN extension. The former case was investigated in [22] and the latter case is studied in detail in the remaining sections of this paper, where we define two "quasi-kernels" of an HNN extension in order to determine C *simplicity. Theorem 4.10 gives combinatorial properties analog to Proposition 3.8, and Proposition 4.12 shows that in certain cases, C * -simplicity is equivalent to the group being icc.…”

Section: Annales De L'institut Fouriermentioning

confidence: 99%

“…In this section, we first recall the theory of boundary actions and extreme boundary actions, and their relation to C * -simplicity and the unique trace property. We employ the terminology of [22,Section 5] to formalize the results.…”

Section: Preliminaries On Boundary Actions and C * -Simplicitymentioning

confidence: 99%

C * -simplicity of HNN extensions and groups acting on trees

Bryder¹,

Ivanov²,

Omland³

2021

Annales De l'Institut Fourier

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“…In particular, in free probability theory, studying semicircularity of free random variables is one of the main branches (e.g., [1,9,[11][12][13][14]16] and [17]). …”

Section: Motivationmentioning

confidence: 99%

“…. , t N 9) where tr N is the restricted trace (6.8) on C ⊕N , and E W is the conditional expectation…”

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confidence: 99%