A characterization of minimal homogeneous Banach spaces

Mathematics, 2022

This paper is supposed to form a keystone towards a new and alternative approach to Fourier analysis over LCA (locally compact Abelian) groups G. In an earlier paper the author has already shown that one can introduce convolution and the Fourier–Stieltjes transform on (M(G),∥·∥M), the space of bounded measures (viewed as a space of linear functionals) in an elementary fashion over Rd. Bounded uniform partitions of unity (BUPUs) are easily constructed in the Euclidean setting (by dilation). Moving on to general LCA groups, it becomes an interesting challenge to find ways to construct arbitrary fine BUPUs, ideally without the use of structure theory, the existence of a Haar measure and even Lebesgue integration. This article provides such a construction and demonstrates how it can be used in order to show that any so-called homogeneous Banach space (B,∥·∥B) on G, such as Lp(G),∥·∥p, for 1≤p

Bulletin of the Australian Mathematical Society, 2007

Let G be a locally compact group and be the Banach space of all essentially bounded measurable functions on G vansihing an infinity. Here, we study some families of right completely continuous elements in the Banach algebra equipped with an Arens type product. As the main result, we show that has a certain right completely continuous element if and only if G is compact.

Journal of Mathematical Analysis and Applications, 1977

2015

Our main result is that the simple Lie group G = Sp(n, 1) acts properly isometrically on Lp(G) if p> 4n+2. To prove this, we introduce property BPV0, for V be a Banach space: a locally compact group G has property BPV0 if every affine isometric action of G on V, such that the linear part is a C0-representation of G, either has a fixed point or is metrically proper. We prove that solvable groups, connected Lie groups, and linear algebraic groups over a local field of characteristic zero, have property BPV0. As a consequence for unitary representations, we characterize those groups in the latter classes for which the first cohomology with respect to the left regular representation on L2(G) is non-zero; and we characterize uniform lattices in those groups for which the first L2-Betti number is non-zero.

Proceedings of the American Mathematical Society, 1976

Given a locally compact (and possibly non-Abelian) group G, we denote by B(G) the set of linear combinations of continuous positivedefinite functions on G and by AP(G) the set of continuous almost periodic functions on G. In this paper the sets B(G) and B(G) n AP(G) are characterized in terms of convolutions with measures. Specifically, let U consist of those measures ¡i £ M(G) for which \\ir( /j,)|| < 1, whenever it is a continuous unitary representation of G. It is proved that a function / e L°°(G) belongs to (i.e. is equal locally almost everywhere to a function in) B(G) if and only if the convolutions ¡i */, n ranging over U, form a relatively weakly compact set in LX(G). The same holds if we confine our attention to either the finitely supported or the absolutely continuous measures in U. Moreover, it is shown that any of these three sets of convolutions is relatively norm compact if and only if/belongs to B(G) n AP(G). / G C(G) is in F(G) if and only if the set {> */|/x G Mdd(G), ||/I|I < 1} is

Journal für die reine und angewandte Mathematik (Crelles Journal), 2000

We construct finitely generated groups with arbitrary prescribed Hilbert space compression α ∈ [0, 1]. This answers a question of E. Guentner and G. Niblo. For a large class of Banach spaces E (including all uniformly convex Banach spaces), the E-compression of these groups coincides with their Hilbert space compression. Moreover, the groups that we construct have asymptotic dimension at most 2, hence they are exact. In particular, the first examples of groups that are uniformly embeddable into a Hilbert space (moreover, of finite asymptotic dimension and exact) with Hilbert space compression 0 are given. These groups are also the first examples of groups with uniformly convex Banach space compression 0.

Acta Mathematica Hungarica, 2010

We study weakly compact left and right multipliers on the Banach algebra L ∞ 0 (G) * of a locally compact group G. We prove that G is compact if and only if L ∞ 0 (G) * has either a non-zero weakly compact left multiplier or a certain weakly compact right multiplier on L ∞ 0 (G) *. We also give a description of weakly compact multipliers on L ∞ 0 (G) * in terms of weakly completely continuous elements of L ∞ 0 (G) *. Finally we show that G is finite if and only if there exists a multiplicative linear functional n on L ∞ 0 (G) such that n is a weakly completely continuous element of L ∞ 0 (G) * .

2021

Let ω be a weight function on a locally compact group G and let M∗(G,ω) be the subspace of M(G,ω)∗ consisting of all functionals that vanish at infinity. In this paper, we first investigate the Arens regularity of M∗(G,ω) ∗ and show that M∗(G,ω) ∗ is Arens regular if and only if G is finite or Ω is zero cluster. This result is an answer to the question posed and it improves some well-known results. We also give necessary and sufficient criteria for the weight function spaces Wap(G, 1/ω) and Ap(G, 1/ω) to be equal to Cb(G, 1/ω). We prove that for non-compact group G, the Banach algebra M∗(G,ω) ∗ is Arens regular if and only if Wap(G, 1/ω) = Cb(G, 1/ω). We then investigate amenability of M∗(G,ω) ∗ and prove that M∗(G,ω) ∗ is amenable and Arens regular if and only if G is finite.

Georgian Mathematical Journal

We prove that, under certain restrictions, every locally compact group equipped with a nonzero, σ-finite, regular left Haar measure can be decomposed into two small sets, one of which is small in the sense of measure and the other is small in the sense of category, and all such decompositions originate from a generalised notion of a Lebesgue point. Incidentally, such class of topological groups for which this happens turns out to be metrisable. We also observe an interesting connection between Luzin sets in such spaces and decompositions of the above type.

Proceedings of the American Mathematical Society, 1976