An exotic group as limit of finite special linear groups

Carderi, Alessandro; Thom, Andreas

doi:10.5802/aif.3160

Cited by 10 publications

(12 citation statements)

References 25 publications

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“…The first such example is given by Herer-Christensen [HC75]. See also the work of Megrelishvili [Me08] and Carderi-Thom [CT18] for more exotic groups with surprising properties. For abelian exotic groups, see also Banaszczyk's book [Ba].…”

Section: Introduction and Main Resultsmentioning

confidence: 98%

Polish groups of unitaries

Ando,

Matsuzawa

2019

Preprint

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We study the question of which Polish groups can be realized as subgroups of the unitary group of a separable infinite-dimensional Hilbert space. We also show that for a separable unital C * -algebra A, the identity component U0(A) of its unitary group has property (OB) of Rosendal (hence it also has property (FH)) if and only if the algebra has finite exponential length (e.g. if it has real rank zero), while in many cases the unitary group U(A) does not have property (T). On the other hand, the p-unitary group Up(M, τ ) where M is a properly infinite semifinite von Neumann algbera with separable predual, does not have property (FH) for any 1 ≤ p < ∞. This in particular solves a problem left unanswered in the work of Pestov [Pe18].

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Section: Introduction and Main Resultsmentioning

confidence: 98%

Polish groups of unitaries

Ando,

Matsuzawa

2019

Preprint

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“…Pour transférer le résultat au groupe de lacets basés H 1 0 (S 1 , K ), nous montrons que tout sousgroupe normal et co-compact d'un groupe moyennable en biais est moyennable en biais. Notons que en général, la moyennabilité en biais n'est pas transmise au sous-groupes topologiques fermés, comme le témoigne l'exemple dans [4].…”

Section: Esquisse De La Preuveunclassified

An amenability-like property of finite energy path and loop groups

Pestov¹

2021

Comptes Rendus. Mathématique

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We show that the groups of finite energy loops and paths (that is, those of Sobolev class H 1 ) with values in a compact connected Lie group, as well as their central extensions, satisfy an amenability-like property: they admit a left-invariant mean on the space of bounded functions uniformly continuous with regard to a left-invariant metric. Every strongly continuous unitary representation π of such a group (which we call skew-amenable) has a conjugation-invariant state on B (H π ).Résumé. Nous montrons que les groupes de lacets et de chemins à énergie finie (c.à.d. de classe H 1 de Sobolev) à valeurs dans un groupe de Lie compact et connexe, ainsi que leurs extensions centrales, satisfont une version de la moyennabilité: ils admettent une moyenne invariante à gauche sur l'espace de fonctions bornées uniformément continues par rapport a une métrique invariante à gauche. Chaque représentation unitaire continue, π, d'un tel groupe (que nous disons d'être "moyennable en biais") possède un état sur B (H π ) invariant sous conjugaison.

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“…It is well known that containment of a discrete subgroup being isomorphic to does not prevent a general (i.e., not necessarily locally compact) topological group from being amenable: among the most prominent examples of amenable topological groups admitting discrete free subgroups are the full symmetric group of any infinite set with the topology of point-wise convergence, the unitary group of any infinite-dimensional Hilbert space equipped with the strong operator topology (which is even extremely amenable by a famous result of Gromov and Milman [GM83]), as well as the automorphism group with the topology of point-wise convergence (which Pestov proved both to be extremely amenable and to contain a discrete copy of [Pes98]). Recently, the first example of an extremely amenable Polish group admitting a complete bi-invariant metric and containing an (even maximally) discrete free subgroup was given by Carderi and the second author [CT18].…”

Section: Perturbed Translations and The Topological Von Neumann Problemmentioning

confidence: 99%

On Følner sets in topological groups

Schneider¹,

Thom²

2018

Compositio Math.

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We extend Følner's amenability criterion to the realm of general topological groups. Building on this, we show that a topological group G is amenable if and only if its left translation action can be approximated in a uniform manner by amenable actions on the set G. As applications we obtain a topological version of Whyte's geometric solution to the von Neumann problem and give an affirmative answer to a question posed by Rosendal.

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An exotic group as limit of finite special linear groups

Cited by 10 publications

References 25 publications

Polish groups of unitaries

Polish groups of unitaries

An amenability-like property of finite energy path and loop groups

On Følner sets in topological groups

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