Conditional expectations on \(CV_{p}(G)\). Applications
type: Item
instance of: scholarly article
title: Conditional expectations on \(CV_{p}(G)\). Applications
zbMATH Open document ID: 1127.43003
author: Antoine Derighetti
published in: Journal of Functional Analysis
publication date: 2007-06-26T00:00:00Z
review text: For a locally compact group \(G\), let \(A_p(G)\) be the Figà-Talamanca-Herz algebra and let \(CV_p(G)\) be the Banach algebra of \(p\)-convolution operators on \(L^p(G)\). For any closed subgroup \(H\) of \(G\), the author finds a condition on the pair of \((G,H)\), denoted by \(G\in [{\mathcal A}]_H\), which implies the existence of a conditional expectation \({\mathcal P}\) of \(CV_p(G)\) onto \(\{T\in CV_p(G):\text{supp}\,T\subset H\}\), i.e., a projection \({\mathcal P}\) from \(CV_p(G)\) onto \(\{T\in CV_p(G): \text{supp}\,T\subset H\}\) such that \({\mathcal P}(uT)=u{\mathcal P} (T)\) for \(u \in A_p(G)\) and \(T\in CV_p(G)\). In addition, this projection satisfies \({\mathcal P} (T)\in PM_p(G)\) if \(T\in PM_p(G)\) and \({\mathcal P} (\mu)=1_H \mu\) if \(\mu\) is a bounded measure. It is proved in this paper that if \(H\) is neutral in \(G\) or \(H\) is amenable, then \(G\in[{\mathcal A}]_H\). As a consequence the author obtains that for \(G\in[{\mathcal A}]_H\), \(H\) is locally \(p\)-Ditkin in \(G\). This generalizes several existing results. Another consequence is that for \(G\in[{\mathcal A}]_H\), the Banach algebra \(cv_p(H)\) is canonically isometrically isomorphic to \(\{T\in cv_p (G): \text{supp}\,T\subset H\}\), where \(cv_p(H)\) \([cv_p(G)]\) is the norm closure of the set of all convolution operators in \(CV_p(H)\) \([CV_p(G)]\) with compact support. This generalizes a theorem of Reiter for abelian locally compact groups \(G\).
reviewed by: Tianxuan Miao
Mathematics Subject Classification ID: 43A15
Mathematics Subject Classification ID: 43A45
Mathematics Subject Classification ID: 43A46
zbMATH DE Number: 5167469
zbMATH Keywords: locally compact groups
zbMATH Keywords: amenable groups
zbMATH Keywords: Fourier algebra,
zbMATH Keywords: Figà-Talamanca algebra
zbMATH Keywords: convolution operators
zbMATH Keywords: pseudomeasures
MaRDI profile type: Publication
full work available at URL: j.jfa.2007.03.003
OpenAlex ID: W2085096227
cites work: Non-abelian Pontriagin duality
cites work: Operator biflatness of the Fourier algebra and approximate indicators for subgroups
cites work: Q5587430
cites work: Uniform Closures of Fourier-Stieltjes Algebras
cites work: Invariant projections and convolution operators
cites work: Q3949499
cites work: Quelques observations concernant les ensembles de Ditkin d'un groupe localement compact. (Ditkin sets on locally compact groups)
cites work: Q4207212
cites work: On {H}erz's projection theorem
cites work: Ideals with bounded approximate identities in Fourier algebras.
cites work: Multipliers and ideals in second conjugate algebras related to locally compact groups
cites work: Q5591039
cites work: Measures on Locally Compact Groups Whose Fourier-Stieltjes Transforms Vanish at Infinity and Group $C^*$-Algebras.
cites work: A separation property of positive definite functions on locally compact groups and applications to Fourier algebras
cites work: On neutral subgroups of topological groups
cites work: Estimations \(L^ p\) des coefficients de représentation et opérateurs de convolution
cites work: On the connection between sets of operator synthesis and sets of spectral synthesis for locally compact groups
cites work: Q5829828
cites work: Contributions to harmonic analysis
cites work: Q4520727
cites work: Q4741226
DOI: 10.1016/J.JFA.2007.03.003
description: scientific article; zbMATH DE number 5167469