Homomorphisms of convolution algebras
type: Item
instance of: scholarly article
title: Homomorphisms of convolution algebras
zbMATH Open document ID: 1236.43003
author: Ross Stokke
published in: Journal of Functional Analysis
publication date: 2011-11-24T00:00:00Z
review text: \textit{P. J.~Cohen} [``On homomorphisms of group algebras'', Am. J. Math. 82, 213--226 (1960; Zbl 0099.25601)] characterised all homomorphisms \(\phi: L^1(G)\to M(H)\) between the group and measure algebras of locally compact abelian groups \(G\) and \(H\). There has been a considerable effort to generalise this result to all locally compact groups. One may approach the question from two directions, dual to each other. First, we may simply leave out the condition that \(G\) and \(H\) are abelian. Secondly, we may consider homomorphisms \(\phi: A(G) \to B(H)\), where \(A(G)\) is the Fourier algebra of \(G\) and \(B(H)\) the Fourier-Stieltjes algebra of \(H\) (and again \(G\) and \(H\) are not necessarily abelian). As Cohen employed the Fourier-Stieltjes transform in his solution, perhaps the latter approach is more amenable. In this setting, \textit{M.~Ilie} and \textit{N.~Spronk} [``Completely bounded homomorphisms of the Fourier algebras'', J. Funct. Anal. 225, No.~2, 480--499 (2005; Zbl 1077.43004)] characterised all completely bounded homomorphisms \(\phi: A(G) \to B(H)\) for amenable \(G\) by corresponding these homomorphisms to continuous piece-wise affine maps \(\alpha: H\supseteq Y\to G\); this covers Cohen's theorem as well as earlier extensions by \textit{B.~Host} [``Le théorème des idempotents dans B(G)'', Bull. Soc. Math. Fr. 114, 215--223 (1986; Zbl 0606.43002)] and \textit{M.~Ilie} [``On Fourier algebra homomorphisms'', J. Funct. Anal. 213, No.~1, 88--110 (2004; Zbl 1052.43006)]. Recently, \textit{H. L.~Pham} [``Contractive homomorphisms of the Fourier algebras'', Bull. Lond. Math. Soc. 42, No.~5, 937--947 (2010; Zbl 1248.43003)] showed that every contractive homomorphism \(\phi: A(G) \to B(H)\) has a factorisation \(\phi = \ell_s\circ \iota\circ j_\theta \circ \ell_t\), where \(\ell_t\) is the left translation on \(A(G)\) by \(t\in G\), \(j_\theta: A(G)\to B(H_0)\) is the homomorphism \(u\mapsto u\circ \theta\) defined by a continuous homomorphism or …
reviewed by: Pekka Salmi
Mathematics Subject Classification ID: 43A20
Mathematics Subject Classification ID: 43A10
zbMATH DE Number: 5978501
zbMATH Keywords: locally compact group
zbMATH Keywords: group algebra
zbMATH Keywords: measure algebra
zbMATH Keywords: homomorphism
zbMATH Keywords: introverted subspace
MaRDI profile type: Publication
full work available at URL: j.jfa.2011.08.014
OpenAlex ID: W2016115637
cites work: Q4141642
cites work: On Homomorphisms of Group Algebras
cites work: Banach algebras and automatic continuity
cites work: The second duals of Beurling algebras
cites work: Second duals of measure algebras
cites work: Q3946357
cites work: Q4023638
cites work: L'algèbre de Fourier d'un groupe localement compact
cites work: INVOLUTIONS ON THE SECOND DUALS OF GROUP ALGEBRAS AND A MULTIPLIER PROBLEM
cites work: Fourier analysis on coset spaces
cites work: Relations between Locally Compact Abelian Groups with Isomorphic Group Algebras
cites work: On the Extension of Uniformly Continuous Functions(1)
cites work: Isometric Isomorphisms Between the Second Conjugate Algebras of Group Algebras
cites work: Isometric Isomorphisms Between Banach Algebras Related to Locally Compact Groups
cites work: Isomorphisms and multipliers on second dual algebras of Banach algebras
cites work: The Second Dual Algebra of the Measure Algebra of a Compact Group
cites work: Norm decreasing homomorphisms of group algebras
cites work: On Central Topological Groups
cites work: Q5723435
cites work: Le théorème des idempotents dans $B(G)$
cites work: On Fourier algebra homomorphisms
cites work: Completely bounded homomorphisms of the Fourier algebras
cites work: Weak*- continuous homomorphisms of Fourier–Stieltjes algebras
cites work: Extension of Fourier algebra homomorphisms to duals of algebras of uniformly continuous functionals
cites work: Isometric Isomorphisms of Measure Algebras
cites work: Homomorphisms of group algebras with norm less than \(\sqrt2\)
cites work: Norm decreasing homomorphisms between group algebras
cites work: Continuity of Arens multiplication on the dual space of bounded uniformly continuous functions on locally compact groups and topological semigroups
cites work: The \(C^ \ast\)-algebra generated by operators with compact support on a locally compact group
cites work: Weak amenability of Banach algebras on locally compact groups
cites work: Isomorphisms of Locally Compact Groups and Banach Algebras
cites work: Compactifications of semitopological semigroups
cites work: Contractive homomorphisms of the Fourier algebras
cites work: Q3293614
cites work: Amenability for dual Banach algebras
cites work: Dual Banach algebras: Connes-amenability, normal, virtual diagonals, and injectivity of the predual bimodule
cites work: Matrix coefficients of unitary representations and associated compactifications
cites work: Isometric isomorphisms of measure algebras
cites work: \(W^*\)-algebras and nonabelian harmonic analysis
cites work: Left centralizers and isomorphisms of group algebras
cites work: Small isomorphisms between group algebras
DOI: 10.1016/J.JFA.2011.08.014
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description: scientific article; zbMATH DE number 5978501