Projections onto translation-invariant subspaces of \(L_ 1(G)\)
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.
complemented translation-\(invariant\) subspaces
scientific article; zbMATH DE number 3929652
scientific article