On uniqueness of invariant means

Bekka, M. Bachir

doi:10.1090/s0002-9939-98-04044-1

Cited by 29 publications

(31 citation statements)

References 24 publications

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“…The following is inspired by Bekka's work [3]. Let G be a semisimple real Lie group with finite centre and without compact factors, having property (T ) (cf.…”

Section: Proof Take Asmentioning

confidence: 99%

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Some Extremely Amenable Groups Related to Operator Algebras and Ergodic Theory

Giordano¹,

Pestov²

2006

JMJ

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A topological group G is called extremely amenable if every continuous action of G on a compact space has a fixed point. This concept is linked with geometry of high dimensions (concentration of measure). We show that a von Neumann algebra is approximately finite dimensional if and only if its unitary group with the strong topology is the product of an extremely amenable group with a compact group, which strengthens a result by de la Harpe. As a consequence, a C * -algebra A is nuclear if and only if the unitary group U (A) with the relative weak topology is strongly amenable in the sense of Glasner. We prove that the group of automorphisms of a Lebesgue space with a non-atomic measure is extremely amenable with the weak topology and establish a similar result for groups of non-singular transformations. As a consequence, we prove extreme amenability of the groups of isometries of L p (0, 1), 1 p < ∞, extending a classical result of Gromov and Milman (p = 2). We show that a measure class preserving equivalence relation R on a standard Borel space is amenable if and only if the full group [R], equipped with the uniform topology, is extremely amenable. Finally, we give natural examples of concentration to a non-trivial space in the sense of Gromov occurring in the automorphism groups of injective factors of type III.

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“…The following is inspired by Bekka's work [3]. Let G be a semisimple real Lie group with finite centre and without compact factors, having property (T ) (cf.…”

Section: Proof Take Asmentioning

confidence: 99%

“…Let G be a semisimple real Lie group with finite centre and without compact factors, having property (T ) (cf. [3]; e.g. SL 3 (R)).…”

Section: Proof Take Asmentioning

confidence: 99%

Some Extremely Amenable Groups Related to Operator Algebras and Ergodic Theory

Giordano¹,

Pestov²

2006

JMJ

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“…In a first version of this paper, I had sketched a proof, similar to [18, §VII.1], relying on Margulis' arithmeticity theorem and the Harish-Chandra-Borel-Behr-Harder reduction theorem for S-arithmetic lattices. François Maucourant explained to me that this is a direct consequence of property (T ) (actually even of spectral gap), and that it applies more generally to all Lie groups and all simple algebraic groups over local fields [2], [5], [4]. I thank him for allowing me to include this proof here.…”

Section: Facts On Latticesmentioning

confidence: 93%

Strong property (T) for higher rank lattices

Salle

2019

Acta Mathematica

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la salle in) semisimple groups rather than simple groups, and also to some non semisimple Lie groups (Remark 4.3).In the whole article, local field will mean commutative, non-discrete locally compact topological field. So a local field is a finite extension of R (in which case it is Archimedean), or of Q p or F p ((t)) for some prime number p (in which case it is non-Archimedean). Higher-rank simple group will mean either real connected simple Lie group of real rank 2, or connected almost F-simple algebraic group of F-split rank 2 over a local field F. Higher-rank group will stand for a finite product of higherrank simple groups. We warn the reader that for us, products of rank-1 groups such as SL 2 (R)×SL 2 (Q p ) are not of higher rank. We refer to [18, Chapter I] for the terminology. Note that real connected simple Lie group or real rank 2 is more general than connected almost simple algebraic group of split rank 2 over R. It includes for example some groups with infinite center, as the infinite covering group of Sp 2n (R).Recall that a lattice in a locally compact group G is a discrete subgroup Γ such that G/Γ carries a G-invariant Borel probability measure.Theorem 1.1. Every lattice in a higher-rank group has strong property (T ).Examples of lattices in higher-rank groups include SL n (Z), SL n (F p [t]), SL n (Z[1/p]), for n 3, and Sp 2n (Z), Sp 2n (Z) (the preimage of Sp 2n (Z) in the universal cover of Sp 2n (R)), Sp 2n (F p [X]), for n 2. None of these examples is a cocompact lattice, so for all these cases Theorem 1.1 is new.

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“…Denote by λ 0 G/H the restriction of λ G/H to L 2 0 (G/H, µ) (in case µ is infinite, It is a standard fact that L 2 (G/H) has a spectral gap when H is cocompact in G (see [Marg91, Chapter III, Corollary 1.10]). When G is a semisimple Lie group, Theorem 1 is an easy consequence of Lemma 3 in [Bekk98]. Our proof is by reduction to these two cases.…”

Section: Introductionmentioning

confidence: 90%