Approximate identities for ideals of Segal algebras on a compact group
type: Item
instance of: scholarly article
title: Approximate identities for ideals of Segal algebras on a compact group
zbMATH Open document ID: 1004.43004
author: Yong Zhang
published in: Journal of Functional Analysis
publication date: 2002-09-08T00:00:00Z
review text: A left approximate identity \((e_\alpha)\) of a normed algebra \(A\) satisfies condition \((U)\) if for every compact set \(K\subset A\) \(\|e_\alpha- a\|\to 0\) uniformly for \(a\in K\). It is proved that for a compact group \(G\) every closed two-sided ideal of \(L^1(G)\) has an approximate identity that lies in the center of \(L^1(G)\) and satisfies condition \((U)\). Moreover it is proved that every closed left (right) ideal \(E\) of \(L^1(G)\) has a right (left) approximate identity satisfying condition \((U)\) if and only if it is an approximately complemented subspace in \(L^1(G)\). The last property means that there is a net of continuous operators \(P_\alpha: X\to E\) such that \(\lim P_\alpha x=x\) uniformly on compact subsets of \(E\). The same results are proved for a Segal algebra \(S^1(G)\) of a compact group \(G\).
reviewed by: Leszek Skrzypczak
Mathematics Subject Classification ID: 43A20
zbMATH DE Number: 1797585
zbMATH Keywords: approximate identity
zbMATH Keywords: Segal algebra
MaRDI profile type: Publication
full work available at URL: jfan.2001.3872
OpenAlex ID: W1995452472
cites work: Banach algebras which are ideals in a Banach algebra
cites work: Closed Ideals in Subalgebras of Banach Algebras. I
cites work: Q5568755
cites work: Approximate identities and factorization in Banach modules
cites work: Segal Algebras and Left Normed Ideals
cites work: Q4059391
cites work: Q4843641
cites work: Q5615626
cites work: Segal Algebras on Non-Abelian Groups
cites work: Projections and Approximate Identities for Ideals in Group Algebras
cites work: \(L^ 1\)-algebras and Segal algebras
cites work: Q4520727
cites work: Approximate Units in Finite Codimensional Ideals of Group Algebras
cites work: Nilpotent ideals in a class of Banach algebras
cites work: Unbounded approximate identities in algebras of compact operators on Banach spaces
DOI: 10.1006/JFAN.2001.3872
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description: scientific article; zbMATH DE number 1797585