Banach spaces related to integrable group representations and their atomic decompositions, I (original) (raw)

Construction of generalized atomic decompositions in Banach spaces

International Journal of Advanced Mathematical Sciences, 2014

G-atomic decompositions for Banach spaces with respect to a model space of sequences have been introduced and studied as a generalization of atomic decompositions. Examples and counter example have been provided to show its existence. It has been proved that an associated Banach space for G-atomic decomposition always has a complemented subspace. The notion of a representation system is introduced and exhibits its relation with G-atomic decomposition. Also It has been observed that G-atomic decompositions are exactly compressions of Schauder decompositions for a larger Banach space. We give a characterization for finite G-atomic decomposition in terms of finite-dimensional expansion of identity.

On an atomic decomposition in Banach spaces

An atomic decomposition is considered in Banach space. A method for constructing an atomic decomposition of Banach space, starting with atomic decomposition of subspaces is presented. Some relations between them are established. The proposed method is used in the study of the frame properties of systems of eigenfunc-tions and associated functions of discontinuous differential operators .

Disintegration of Group Representations on Direct Integrals of Banach Spaces

Scientific Research Publishing , 2019

In this paper, let G be a Polish locally compact group acting on a Polish space X with a G-invariant probability measures j j µ ∑. Factorize the integral with respect to j j µ ∑ in terms of the integrals with respect to the ergodic measures on X, and showed that () () () 1 , , 0 j j L X µ + ≤ < ∞ ∑   are G-equivar-iantly isometric ally lattice isomorphic to an () 1 L +-direct integral of the spaces () () 1 , j L X λ + , where j λ ranges over the ergodic measures on X. This yields a disintegration of the canonical representation of G as isometric lattice auto orphisms of () () 1 , j j L X µ + ∑  as an () 1 L +-direct integral of order in-decomposable representations. If () , j j X µ ′ ′ ∑ are probability space, and, for some 0 ≤ < ∞  , G acts in a strongly continuous manner on () () 1 , j j L X µ + ′ ′ ∑  as isometric lattice auto orphisms that leave the constants fixed, then G acts on () () 1 , j j L X µ + ′ ′ ∑  in a similar fashion for all 0 ≤ < ∞ . Moreover, there exists an alternative model in which these representations originate from a continuous action of G on a compact Hausdorff space. If () , j j X µ ′ ′ ∑ are separable, the representation of G on () () 1 , j j L X µ + ′ ′ ∑  can then be disintegrated into order indecomposable representations. The notions of () 1 L +-direct integrals of Banach spaces and representation is developed for extend those in the literature.

An imprimitivity theorem for representations of locally compact groups on arbitrary Banach spaces

Pacific Journal of Mathematics, 1998

We prove a general version of Mackey's Imprimitivity Theorem for induced representations of locally compact groups. Let G be a locally compact group and let H be a closed subgroup. Following Rieffel we show, using Morita equivalence of Banach algebras, that systems of imprimitivity for induction from strongly continuous Banach H−modules to strongly continuous Banach G−modules can be described in terms of an action on the induced module of C 0 (G/H), the algebra of complex continuous functions on G/H vanishing at ∞, which is compatible with the G−homogeneous structure of G/H and the strong operator topology continuity of the module action of G.

Representations of compact groups on Banach algebras

1984

ABSTRACT. Let a compact group U act by automorphisms of a commutative regular and Wiener Banach algebra A. We study representations Ru of U on quotient spaces A/I (üj), where u> is an orbit of U in the Gelfand space X of A and L (uj) is the minimal closed ideal with hull w C X. The main result of the paper is: if A= AP (X) is a weighted Fourier algebra on a LCA group X= A with a subpolynomial weight p on A, and U acts by affine transformations on X, then for any orbit w CX the representation Ru has finite multiplicity.

Banach algebras on groups and semigroups

2018

This thesis concerns the theory of Banach algebras, particularly those coming from abstract harmonic analysis. The focus for much of the thesis is the theory of the ideals of these algebras. In the final chapter we use semigroup algebras to solve an open probelm in the theory of C*-algebras. Throughout the thesis we are interested in the interplay between abstract algebra and analysis. Chapters 2, 4, and 5 are closely based upon the articles [88], [89], and [56], respectively. In Chapter 2 we study (algebraic) finite-generation of closed left ideals in Banach algebras. Let G be a locally compact group. We prove that the augmentation ideal in L 1 pGq is finitely-generated as a left ideal if and only if G is finite. We then investigate weighted versions of this result, as well as a version for semigroup algebras. Weighted measure algebras are also considered. We are motivated by a recent conjecture of Dales and Żelazko, which states that a unital Banach algebra in which every maximal ...

Homogeneous Banach Spaces as Banach Convolution Modules over M(G)

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

Topological group actions by group automorphisms and Banach representations

2021

For every Banach space V , we have the topological group of Isol(V ) of linear isometries (in its strong operator topology) and its canonical dual action on the weak-star compact unit ball BV ∗ of the dual Banach space V ∗. We study Banach representability for actions of topological groups on groups by automorphisms (in particular, actions of groups on itself by conjugations). Every such action is Banach representable on some Banach space. The natural question is to examine when we can find representations on low complexity Banach spaces. In contrast to the standard left action of a locally compact second countable group G on itself, the conjugation action need not be reflexively representable even for SL2(R). We show that the linear action of GL(n,R) on R, for every n ≥ 2, is not representable on Asplund Banach spaces. On the other hand, this action is representable on a Rosenthal Banach space (not containing an isomorphic copy of l1). For every cyclic subgroup G of GL(2,R), the re...

Coorbit Description and Atomic Decomposition of Besov Spaces

Numerical Functional Analysis and Optimization, 2012

Function spaces are central topic in analysis. Often those spaces and related analysis involves symmetries in form of an action of a Lie group. Coorbit theory as introduced by Feichtinger and Gröchenig and then later extended in [3] gives a unified method to construct Banach spaces of functions based on representations of Lie groups. In this article we identify the homogeneous Besov spaces on stratified Lie groups introduced in [13] as coorbit spaces in the sense of [3] and use this to derive atomic decompositions for the Besov spaces.

Representations of dynamical systems on Banach spaces not containing l1l_{1}l1

Transactions of the American Mathematical Society, 2012

For a topological group G, we show that a compact metric G-space is tame if and only if it can be linearly represented on a separable Banach space which does not contain an isomorphic copy of l1 (we call such Banach spaces, Rosenthal spaces). With this goal in mind we study tame dynamical systems and their representations on Banach spaces.