Projective operator spaces, almost periodicity and completely complemented ideals in the Fourier algebra
type: Item
instance of: scholarly article
title: Projective operator spaces, almost periodicity and completely complemented ideals in the Fourier algebra
zbMATH Open document ID: 1210.47098
DOI: 10.1216/RMJ-2011-41-1-155
published in: Rocky Mountain Journal of Mathematics
publication date: 2011-03-01T00:00:00Z
review text: The author considers projective operator spaces in the sense of [\textit{D.\,P.\thinspace Blecher}, ``The standard dual of an operator space'', Pac.\ J.\ Math.\ 153, No.\,1, 15--30 (1992; Zbl 0726.47030)]: an operator space space \(E\) is called \textit{projective} if, for every operator space \(X\), each closed subspace \(Y\) of \(X\), each complete contraction \(\Gamma : E \to X/Y\), and each \(\varepsilon > 0\), there is a lifting \(\tilde{\Gamma} : E \to X\) of \(\Gamma\) with \(\| \tilde{\Gamma} \|_{cb} \leq 1 + \varepsilon\). The paper focuses on projective operator spaces that arise naturally in abstract harmonic analysis, in particular on subspaces of the Fourier algebra \(A(G)\) or of the Fourier-Stieltjes algebra \(B(G)\) of a locally compact group \(G\). Among other things, the author shows that \(A(G)\) is projective if and only if \(B(G)\) is projective if and only if \(G\) is compact. Applications to (complete) invariant complementation of ideals of \(A(G)\) are also given.
reviewed by: Volker Runde
Mathematics Subject Classification ID: 47L25
Mathematics Subject Classification ID: 43A30
Mathematics Subject Classification ID: 46J20
Mathematics Subject Classification ID: 46L07
zbMATH DE Number: 5858883
zbMATH Keywords: projective operator spaces
zbMATH Keywords: Fourier algebra
zbMATH Keywords: Fourier-Stieltjes algebra
author: Brian Forrest
MaRDI profile type: Publication
cites work: A Characterization of the Complemented Translation Invariant Subspaces of L 1 (R2 )
cites work: Projections onto Translation-Invariant Subspaces of L 1 (R)
cites work: Projections onto translation-invariant subspaces of \(L_ 1(G)\)
cites work: Separating sets by Fourier-Stieltjes transforms
cites work: Q4141642
cites work: Biprojective algebras and operator spaces
cites work: Geometric properties of coefficient function spaces determined by unitary representations of a locally compact group
cites work: The standard dual of an operator space
cites work: The Dunford-Pettis Property in the Predual of a Von Neumann Algebra
cites work: A Note on Scattered C∗ -Algebras and the Radon-Nikodym Property
cites work: The Dunford-Pettis property in C*-algebras
cites work: Q4148368
cites work: L'algèbre de Fourier d'un groupe localement compact
cites work: Amenability and weak amenability of the Fourier algebra
cites work: Complemented Ideals in the Fourier Algebra and the Radon Nikodym Property
cites work: Ideals with bounded approximate identities in Fourier algebras.
cites work: Cohomology and the Operator Space Structure of the Fourier Algebra and its Second Dual
cites work: Une Caracterisation Vectorielle-Metrique Des Espaces L1
cites work: On linear topological properties of some \(C^*\)-algebras
cites work: Q5723435
cites work: Cohomology in Banach algebras
cites work: Non-Amenability of the Fourier Algebra of a Compact Group
cites work: Amenability in Group Algebras and Banach Algebras.
cites work: Q3687232
cites work: Q4890729
cites work: Some Geometric Properties on the Fourier and Fourier Stieltjes Algebras of Locally Compact Groups, Arens Regularity and Related Problems
cites work: On tensor products of Fourier algebras
cites work: Compactness of a Locally Compact Group G and Geometric Properties of Ap(G)
cites work: The Nonexistence of Projections from L 1 to H 1
cites work: Projections onto translation-invariant subspaces of 𝐿^{𝑝}(𝐺)
cites work: The Operator Amenability of A(G)
cites work: Projections on Invariant Subspaces
cites work: Operator weak amenability of the Fourier algebra
cites work: On the Beurling algebras \(A_{\alpha}^+(\mathbb{D})\)-derivations and extensions
cites work: The Operator Biprojectivity of the Fourier Algebra
cites work: Complemented ideals in the Fourier algebra of a locally compact group
description: scientific article; zbMATH DE number 5858883