Some Extremely Amenable Groups Related to Operator Algebras and Ergodic Theory

Giordano, Thierry; Pestov, Vladimir

doi:10.1017/s1474748006000090

Cited by 71 publications

(67 citation statements)

References 50 publications

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“…(As pointed out to me by Philip Dowerk and Andreas Thom, and independently by Sven Raum, a von Neumann algebra M is finite if and only if the group U(M) s is SIN.) In the separable case, amenability of U(M) 2 means M = R is the hyperfinite factor of type II 1 , in which case the group U(R) 2 is in fact extremely amenable [22]. We will show that the group U(R) 2 does not have strong property (T ), however we were unable to verify whether it has property (T ).…”

Section: U C (H)mentioning

confidence: 76%

“…We mark the weak topology with the subscript w. The group Aut * (X, µ) w is Polish. If µ is diffuse, then Aut * (X, µ) w is extremely amenable [22]. From here it is easy to deduce, with a view of Example 3.11, that for every measure µ the group Aut * (X, µ) w is amenable.…”

Section: U C (H)mentioning

confidence: 80%

“…This topology is always complete, and is Polish if and only if M is separable. The group U(M) s is amenable if and only if M is injective [15,29], in which case U(M) s decomposes into a direct product of an extremely amenable group and a compact group [22]. In some cases, U(M) s has property (T ) (e.g.…”

Section: U C (H)mentioning

confidence: 99%

“…(3.1) and the corresponding topology is a complete SIN group. If µ is diffuse, the group is nonseparable, and it is non-amenable [22]. This topological group clearly has property (OB), in fact even the Bergman property [18].…”

Section: U C (H)mentioning

confidence: 99%

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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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Section: U C (H)mentioning

confidence: 76%

Section: U C (H)mentioning

confidence: 80%

Section: U C (H)mentioning

confidence: 99%

Section: U C (H)mentioning

confidence: 99%

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

Pestov

2018

Trans. Amer. Math. Soc.

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“…Many concrete examples of Lévy groups are known by the works of S. Glasner [8], H. Furstenberg and B. Weiss (unpublished), T. Giordano and V. Pestov [6], [7], and Pestov [25], [26]. For examples, groups of measurable maps from the standard Lebesgue measure space to compact groups, unitary groups of some von Neumann algebras, groups of measure and measure-class preserving automorphisms of the standard Lebesgue measure space, full groups of amenable equivalence relations, and the isometry groups of the universal Urysohn metric spaces are Lévy groups (see the recent monograph [24] for precise statements).…”

Section: Introductionmentioning

confidence: 99%

Concentration of $1$-Lipschitz maps into an infinite dimensional $\ell ^p$-ball with the $\ell ^q$-distance function

Funano

2009

Proc. Amer. Math. Soc.

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Fraïssé Limits, Ramsey Theory, and topological dynamics of automorphism groups

Kechris¹,

Pestov

Todorčević

2005

GAFA, Geom. funct. anal.

339

736

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Introduction(A) We study in this paper some connections between the Fraïssé theory of amalgamation classes and ultrahomogeneous structures, Ramsey theory, and topological dynamics of automorphism groups of countable structures.A prime concern of topological dynamics is the study of continuous actions of (Hausdorff) topological groups G on (Hausdorff) compact spaces X. These are usually referred to as (compact) G-flows. Of particular interest is the study of minimal Gflows, those for which every orbit is dense. Every G-flow contains a minimal subflow. A general result of topological dynamics asserts that every topological group G has a universal minimal flow M(G), a minimal G-flow which can be homomorphically mapped onto any other minimal G-flow. Moreover, this is uniquely determined, by this property, up to isomorphism. (As usual a homomorphism π : X → Y between G-flows is a continuous G-map and an isomorphism is a bijective homomorphism.) For separable, metrizable groups G, which are the ones that we are interested in here, the universal minimal flow of G is an inverse limit of manageable, i.e., metrizable G-flows, but itself may be very complicated, for example non-metrizable. In fact, for the "simplest" infinite G, i.e., the countable discrete ones, M(G) is a very complicated compact G-invariant subset of the space βG of ultrafilters on G and is always non-metrizable.Rather remarkably, it turned out that there are non-trivial topological groups G for which M(G) is actually trivial, i.e., a singleton. This is equivalent to saying that G has a very strong fixed point property, namely every G-flow has a fixed point (i.e., a point x such that g • x = x, ∀g ∈ G). (For separable, metrizable groups this is also equivalent to the fixed point property restricted to metrizable G-flows.) Such groups are said to have the fixed point on compacta property or be extremely amenable. The latter name comes from one of the standard characterizations of second countable locally compact amenable groups. A second countable locally compact group G is amenable iff every metrizable G-flow has an invariant (Borel probability) measure. However, no non-trivial locally compact group can be extremely amenable, because, by a theorem of Veech [83], every such group admits a free G-flow (i.e., a flow for which g • x = x ⇒ g = 1 G ). Nontriviality of the universal minimal flow for locally compact groups also follows from the earlier results of . This probably explains the rather late emergence of extreme amenability. Note that the corresponding property

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Some Extremely Amenable Groups Related to Operator Algebras and Ergodic Theory

Cited by 71 publications

References 50 publications

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

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

Concentration of $1$-Lipschitz maps into an infinite dimensional $\ell ^p$-ball with the $\ell ^q$-distance function

Fraïssé Limits, Ramsey Theory, and topological dynamics of automorphism groups

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