Amenable Actions and Weak Containment of Certain Representations of Discrete Groups

Kuhn, M. Gabriella

doi:10.2307/2160750

Cited by 13 publications

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“…Recall that if a non-singular action G (X, m) is amenable, then the Koopman unitary representation π of G on L 2 (X, m) is weakly regular [4,38]. (For the purposes of this paper, this is essentially all that one has to know about the amenability of a non-singular action.)…”

Section: Connes-sullivan Conjecturementioning

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: Connes-sullivan Conjecturementioning

confidence: 99%

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

et al. 2017

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“…This can be done by constructing a sequence of functions (f n ) on Γ × Ω satisfying condition (h) of [2: Théorème 4.9].. For the case where Γ is the free group on two generators this is done explicitly in the appendix of [16]. See also [15]. More generally the arguments of [1] show how to do it for any hyperbolic group Γ.…”

Section: Introductionmentioning

confidence: 99%

C*-Algebras Arising from Group Actions on the Boundary of a Triangle Building

Robertson

Steger

1996

Proceedings of the London Mathematical Society

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A subgroup of an amenable group is amenable. The C * -algebra version of this fact is false. This was first proved by M.-D. Choi [6] who proved that the non-nuclear C * -algebrais a subalgebra of the nuclear Cuntz algebra O 2 . A. Connes provided another example , based on a crossed product construction. More recently J. Spielberg [23] showed that these examples were essentially the same. In fact he proved that certain of the C *algebras studied by J. Cuntz and W. Krieger [10] can be constructed naturally as crossed product algebras. For example if the group Γ acts simply transitively on a homogeneous tree of finite degree with boundary Ω then C(Ω) ⋊ Γ is a Cuntz-Krieger algebra. Such trees may be regarded as affine buildings of type A 1 . The present paper is devoted to the study of the analogous situation where a group Γ acts simply transitively on the vertices of an affine building of type A 2 with boundary Ω [7]. The corresponding crossed product algebra C(Ω) ⋊ Γ is then generated by two Cuntz-Krieger algebras . (See §3.) Moreover we show that C(Ω) ⋊ Γ is simple and nuclear. This is a consequence of the facts that the action of Γ on Ω is minimal, topologically free, and amenable. ( See §4.)

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“…Before we proceed to the proof, note that since by [1] the action of Γ on ∂Γ is amenable, and by [18], ergodic amenable actions lead to quasi-regular representations which are weakly contained in the regular representation, Proposition 2.4 applies to the matrix coefficients of boundary representations.…”

Section: Property Rdmentioning

confidence: 99%

Asymptotic Schur orthogonality in hyperbolic groups with application to monotony

Boyer¹,

Garncarek²

2019

Trans. Amer. Math. Soc.

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We prove a generalization of Schur orthogonality relations for certain classes of representations of Gromov hyperbolic groups. We apply the obtained results to show that representations of non-abelian free groups associated to the Patterson-Sullivan measures corresponding to a wide class of invariant metrics on the group are monotonous in the sense introduced by Kuhn and Steger. This in particular includes representations associated to harmonic measures of a wide class of random walks.

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Amenable Actions and Weak Containment of Certain Representations of Discrete Groups

Cited by 13 publications

References 0 publications

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

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

C*-Algebras Arising from Group Actions on the Boundary of a Triangle Building

Asymptotic Schur orthogonality in hyperbolic groups with application to monotony

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