Projections onto translation-invariant subspaces of \(L_ 1(G)\)
type: Item
instance of: scholarly article
title: Projections onto translation-invariant subspaces of \(L_ 1(G)\)
zbMATH Open document ID: 0581.43003
DOI: 10.1016/0022-1236(84)90074-0
author: Dale Alspach
author: Alec L. Matheson
author: J. Rosenblatt
published in: Journal of Functional Analysis
publication date: 1984-01-01T00:00:00Z
review text: The purpose of this paper is to extend the characterization of the complemented translation-\(invariant\) subspaces of \(L^ 1({\mathbb{R}})\) obtained by the first two authors to other LCA groups. Many partial results are proved. In particular an inductive procedure is given which produces projections in all cases where a projection is known to exist. Also the complemented translation-\(invariant\) subspaces of \(L^ 1(G)\) with discrete hull are completely characterized. There are some errors in the paper. Lemma 1.5 is false as stated and Lemmas 1.5 and 1.6 are given incorrect proofs. The hypothesis \(A\in \Omega_ c(\Gamma)\) should be added to Lemma 1.5. A correct proof of Lemma 1.5 can be given by repeated application of a lemma of Rosenthal. [See \textit{H. P. Rosenthal}, Ark. Mat. 7, 185-191 (1967; Zbl 0172.183).] Our new proof of Lemma 1.6 is quite lengthy. These corrections are contained in a new paper, Separating sets by the Fourier-Stieltjes transform, which is currently available from the authors in preprint form.
Mathematics Subject Classification ID: 43A20
Mathematics Subject Classification ID: 22B10
Mathematics Subject Classification ID: 43A25
Mathematics Subject Classification ID: 46B20
zbMATH DE Number: 3929652
zbMATH Keywords: locally compact abelian group
zbMATH Keywords: complemented translation-\(invariant\) subspaces
zbMATH Keywords: \(L^ 1({\mathbb{R}})\)
zbMATH Keywords: projections
MaRDI profile type: Publication
full work available at URL: 0022-1236(84)90074-0
OpenAlex ID: W2089184712
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description: scientific article; zbMATH DE number 3929652