Contractive projections on Banach algebras
type: Item
instance of: scholarly article
title: Contractive projections on Banach algebras
zbMATH Open document ID: 1149.46040
author: A. T.-M. Lau
author: R. J. Loy
published in: Journal of Functional Analysis
publication date: 2008-05-19T00:00:00Z
review text: If \(A\) is a Banach algebra, then a projection \(P\) on \(A\) is called a conditional expectation (or quasi-expectation) if it satisfies \[ P(b_1 a b_2 )= b_1 (Pa) b_2 \qquad (b_1, b_2 \in P(A), \quad a \in A). \] A classical theorem of J. Tomiyama asserts that, if \(A\) is a \(C^\ast\)-algebra and \(P\) is a contractive projection onto a \(C^\ast\)-subalgebra of \(A\), then \(P\) is automatically a conditional expectation. Consequently, the authors define a Banach algebra \(A\) to have the Tomiyama property if every contractive projection on \(A\) whose range is a subalgebra of \(A\) is a conditional expectation. Most of the paper is devoted to investigate, for a plethora of Banach algebras occurring naturally in abstract harmonic analysis, when those algebras have the Tomiyama property (or a variant of it). For instance, the authors prove that, for any locally compact group \(G\), the group algebra \(L^1(G)\) has the Tomiyama property for \(L\)-projections on \(^\ast\)-subalgebras, i.e., if \(P\) is projection on \(L^1(G)\) with \(\| f \| = \| P f \| + \| f - Pf \| \) for \(f \in L^1(G)\) with a \(^\ast\)-subalgebra as its range, then \(P\) is a conditional expectation.
reviewed by: Volker Runde
Mathematics Subject Classification ID: 46H05
Mathematics Subject Classification ID: 43A20
Mathematics Subject Classification ID: 47B48
Mathematics Subject Classification ID: 47L50
Mathematics Subject Classification ID: 22C05
Mathematics Subject Classification ID: 22D15
zbMATH DE Number: 5276379
zbMATH Keywords: contractive projections
zbMATH Keywords: conditional expectations
zbMATH Keywords: Tomiyama property
MaRDI profile type: Publication
full work available at URL: j.jfa.2008.02.008
OpenAlex ID: W2045318660
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DOI: 10.1016/J.JFA.2008.02.008
description: scientific article; zbMATH DE number 5276379