1-cohomologie des représentations unitaires des groupes de Lie semi-simples et résolubles. Produits tensoriels continus de représentations

Delorme, Patrick

doi:10.24033/bsmf.1853

Cited by 83 publications

(70 citation statements)

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“…The terminology is justified by the well-known theorem that every affine, isometric action of a property T group on a Euclidean space has bounded orbits [7], and by the recent observation [3] that every countable amenable discrete group admits a metrically proper, affine, isometric action on a Euclidean space. Other examples of a-T -menable groups are proper groups of isometries of real or complex hyperbolic space, finitely generated Coxeter groups, and groups which act properly on locally finite trees.…”

Section: Introductionmentioning

confidence: 99%

Operator 𝐾-theory for groups which act properly and isometrically on Hilbert space

Higson

Kasparov

1997

Electron. Res. Announc. Amer. Math. Soc.

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Section: Introductionmentioning

confidence: 99%

Operator 𝐾-theory for groups which act properly and isometrically on Hilbert space

Higson

Kasparov

1997

Electron. Res. Announc. Amer. Math. Soc.

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“…2 in [6]. While the implication (T ) ⇒ (F H) (due to Delorme [16]) holds for every topological group, the converse implication (F H) ⇒ (T ) (established by Guichardet [28] for sigma-compact locally compact groups) is in general invalid already for uncountable discrete groups.…”

Section: Property (F H)mentioning

confidence: 99%

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

Pestov

2018

Trans. Amer. Math. Soc.

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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.

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“…Real-valued conditionally negative definite functions can also be viewed as generators of semigroups of positive definite functions by Schoenberg's theorem. These equivalences then make possible certain connections between 1-cohomology, conditionally negative definite functions, and positive definite deformations, for example the Delorme-Guichardet theorem [Delorme 1977;Guichardet 1977], which states that a group has property (T) of Kazhdan [1967] if and only if the first cohomology vanishes for any unitary representation. It was Evans [1977] who introduced the notion of bounded conditionally completely positive/negative maps and related them to the study the infinitesimal generators of norm continuous semigroups of completely positive maps.…”

Section: Introductionmentioning

confidence: 99%

A 1-cohomology characterization of property (T) in von Neumann algebras

Peterson¹

2009

Pacific J. Math.

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We obtain a characterization of property (T) for von Neumann algebras in terms of 1-cohomology similar to the Delorme-Guichardet Theorem for groups. Introduction.The analogue of group representations in von Neumann algebras is the notion of correspondences which is due to Connes ([C2], [C3], [P1]), and has been a very useful in defining notions such as property (T) and amenability for von Neumann algebras. It is often useful to view group representations as positive definite functions which we obtain through a GNS construction. Correspondences of a von Neumann algebra N can also be viewed in two separate ways, as Hilbert N -N bimodules H, or as completely positive maps φ : N → N , and the equivalence of these two descriptions is also realized via a GNS construction. This allows one to characterize property (T) for von Neumann algebras in terms of completely positive maps.For a countable group G there is also a notion of conditionally negative definite functions ψ : G → C which satisfy ψ(g −1 ) = ψ(g) and the condition:Real valued conditionally negative definite functions can also be viewed as generators of semigroups of positive definite functions by Schoenberg's Theorem. These equivalences then make it possible for certain connections between 1-cohomology, conditionally negative definite functions, and positive definite deformations, for example the Delorme-Guichardet Theorem [De] [G] which states that a group has property (T) of 2000 Mathematics Subject Classification. 46L10, 46L57, 22D25.

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1-cohomologie des représentations unitaires des groupes de Lie semi-simples et résolubles. Produits tensoriels continus de représentations

Cited by 83 publications

References 2 publications

Operator 𝐾-theory for groups which act properly and isometrically on Hilbert space

Operator 𝐾-theory for groups which act properly and isometrically on Hilbert space

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

A 1-cohomology characterization of property (T) in von Neumann algebras

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