Boundedness in uniform spaces, topological groups, and homogeneous spaces

Atkin, Chris

doi:10.1007/bf01903672

Cited by 13 publications

(36 citation statements)

References 4 publications

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“…It is bounded, so has property (OB) (cf. Remark 6.9 in [1], and Proposition 6.5 and Remark 6.8 in [2]). In particular, the Polish group U C (H) has property (F H), while we will show that it does not have property (T ).…”

Section: U C (H)mentioning

confidence: 82%

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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: 82%

“…If dim H = ∞, the group U(H) u is non-separable, nor is it amenable [14]. The group U(H) u is bounded ( [2], (3.5)), so has property (OB). It remains unknown if U(ℓ 2 ) u has property (T ) [5].…”

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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“…But x α / ∈ V K for all α ∈ I implies that m(f ) = lim α f (x α ) = 1, a contradiction. Hence, for any symmetric (open) neighbourhood V ⊆ G of e there is α V ∈ I such that x α V ∈ V K. It now follows easily from the compactness of K that, if V runs through a descending symmetric (open) neighbourhood base of e in G, some subnet of (x α V ) converges to some y ∈ K, whence m = y ∈ G. 2 We are now ready to present another main result of this paper. Proof.…”

Section: (Ii) If G Is ω-Bounded and Unbounded Then Z T (Luc(g) * ) ⊆mentioning

confidence: 87%

“…This is [25, Proposition 3.2.6]; see also [28]. 2 We now come to the factorization theorems which constitute the main tool of the paper, and are of interest in their own right. They show that the factorization result [25, Satz 3.2.7] (cf.…”

Section: Lemma 32 Let G Be a Topological Group Consider A Family Omentioning

confidence: 98%

On the topological centre of the algebra LUC(G)∗ for general topological groups

Ferri

Neufang

2007

Journal of Functional Analysis

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We consider the Banach algebra LUC(G) * for a not necessarily locally compact topological group G. Our goal is to characterize the topological centre Z t (LUC(G) * ) of LUC(G) * . For locally compact groups G, it is well known that Z t (LUC(G) * ) equals the measure algebra M(G). We shall prove that for every second countable (not precompact) group G, we have Z t (LUC(G) * ) = M( G), where G denotes the completion of G with respect to its right uniform structure (if G is precompact, then Z t (LUC(G) * ) = LUC(G) * , of course). In fact, this will follow from our more general result stating that for any separable (or any precompact) group G, we have Z t (LUC(G) * ) = Leb(G), where Leb(G) denotes the algebra of uniform measures. The latter result also partially answers a conjecture made by I. Csiszár 35 years ago [I. Csiszár, On the weak * continuity of convolution in a convolution algebra over an arbitrary topological group, Studia Sci. Math. Hungar. 6 (1971) 27-40]. We shall give similar results for the topological centre Λ(G LUC ) of the LUC-compactification G LUC of G. In particular, we shall prove that for any second countable (not precompact) group G admitting a group completion, we have Λ(G LUC ) = G (if G is precompact, then Λ(G LUC ) = G LUC ). Finally, we shall show that every linear (left) LUC(G) * -module map on LUC(G) is automatically continuous whenever G is, e.g., separable and not precompact.

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“…Proposition 3.6). In Section 4, we introduce the highest weight representations (π λ , H λ ) of U (∞) as a direct limit of highest weight representations of the subgroups U (n) and we directly conclude 1 Using the terminology of [At91], this can be expressed as G being bounded in the right uniformity of the topological group G. If G is connected, one can always use F = {e} which leads some authors to adopt this as a requirement in their definition.…”

Section: Introductionmentioning

confidence: 99%

On the First Order Cohomology of Infinite-Dimensional Unitary Groups

Herbst¹,

Neeb²

2018

Ann. inst. Fourier

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The irreducible unitary highest weight representations (π λ , H λ ) of the group U (∞), which is the countable direct limit of the compact unitary groups U (n), are classified by the orbits of the weights λ ∈ Z N under the Weyl group S (N) of finite permutations. Here, we determine those weights λ for which the first cohomology space H 1 (U (∞), π λ , H λ ) vanishes. For finitely supported λ = 0, we find that the first cohomology space H 1 (U (∞), π λ , H λ ) never vanishes. For these λ, the highest weight representations extend to norm-continuous irreducible representations of the full unitary group U (H) (for H := ℓ 2 (N, C)) endowed with the strong operator topology and to norm-continuous representations of the unitary groups Up(H) (p ∈ [1, ∞]) consisting of those unitary operators g ∈ U (H) for which g−½ is of pth Schatten class. However, not every 1-cocycle on U (∞) automatically extends to one on these unitary groups, so we may not conclude that the first cohomology spaces of the extended representations are non-vanishing. On the contrary, for the groups U (H) and U∞(H), all first cohomology spaces vanish. This is different for the groups Up(H) with 1 ≤ p < ∞, where only the identical representation on H and its dual representation have vanishing first cohomology spaces.

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Boundedness in uniform spaces, topological groups, and homogeneous spaces

Cited by 13 publications

References 4 publications

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

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

On the topological centre of the algebra LUC(G)∗ for general topological groups

On the First Order Cohomology of Infinite-Dimensional Unitary Groups

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