Amenable unitary representations of locally compact groups

Bekka, Mohammed E. B.

doi:10.1007/bf01231192

Cited by 101 publications

(147 citation statements)

References 16 publications

Supporting

Mentioning

142

Contrasting

Unclassified

Order By: Relevance

“…Bekka [3]. Bekka proved analogues of all of the classical invariance properties of amenable groups, including the Følner conditions, and interpreted several amenability theories in terms of amenable representations.…”

Section: Introductionmentioning

confidence: 99%

Amenable representations and coefficient subspaces of Fourier-Stieltjes algebras

Stokke¹

2006

MATH. SCAND.

View full text Add to dashboard Cite

Amenable unitary representations of a locally compact group, G, are studied in terms of associated coefficient subspaces of the Fourier-Stieltjes algebra B(G), and in terms of the existence of invariant and multiplicative states on associated von Neumann and C * -algebras. We introduce Fourier algebras and reduced Fourier-Stieltjes algebras associated to arbitrary representations, and study amenable representations in relation to these algebras.

show abstract

Section: Introductionmentioning

confidence: 99%

Amenable representations and coefficient subspaces of Fourier-Stieltjes algebras

Stokke¹

2006

MATH. SCAND.

View full text Add to dashboard Cite

show abstract

“…A topological group G is called a SIN group (from Small Invariant Neighbourhoods) if neighbourhoods of the identity invariant under conjugation form a neighbourhood basis, or, equivalently, if the left and the right uniform structures on G coincide. Using Bekka's concept of an amenable unitary representation [4] and a result from [22], we prove the following. Theorem 1.2.…”

Section: Theorem 11 a Locally Compact Group G With Property (T ) Ismentioning

confidence: 99%

“…Every strongly continuous unitary representation of an amenable locally compact group is amenable [4]. This is no longer true for non-locally compact topological groups.…”

Section: Amenable Representationsmentioning

confidence: 99%

“…A unitary representation π of a group G in a Hilbert space H is amenable (in the sense of Bekka [4]) if there exists a state φ on the von Neumann algebra B(H) invariant under the action of G by conjugations: for all T ∈ B(H) and all g ∈ G, one has…”

Section: Amenable Representationsmentioning

confidence: 99%

“…If we assume that π is amenable, with an invariant state φ, then the rule ψ(f ) = φ(m f ) determines an invariant mean on F 2 , which is a contradiction. (More generally, the left regular representation of a locally compact group G is amenable if and only if G is amenable [4]. )…”

Section: Amenable Representationsmentioning

confidence: 99%

See 2 more Smart Citations

Amenability versus property (𝑇) for non-locally compact topological groups

Pestov

2018

Trans. Amer. Math. Soc.

View full text Add to dashboard Cite

Abstract. For locally compact groups amenability and Kazhdan's property (T) are mutually exclusive in the sense that a group having both properties is compact. This is no longer true for more general Polish groups. However, a weaker result still holds for SIN groups (topological groups admitting a basis of conjugationinvariant neighbourhoods of identity): if such a group admits sufficiently many unitary representations, then it is precompact as soon as it is amenable and has the strong property (T ) (i.e. admits a finite Kazhdan set). If an amenable topological group with property (T ) admits a faithful uniformly continuous representation, then it is maximally almost periodic. In particular, an extremely amenable SIN group never has strong property (T ), and an extremely amenable subgroup of unitary operators in the uniform topology is never a Kazhdan group. This leads to first examples distinguishing between property (T ) and property (F H) in the class of Polish groups. Disproving a 2003 conjecture by Bekka, we construct a complete, separable, minimally almost periodic topological group with property (T ), having no finite Kazhdan set. Finally, as a curiosity, we observe that the class of topological groups with property (T ) is closed under arbitrary infinite products with the usual product topology.

show abstract