Homomorphisms from \(L^ 1(G)\) for \(G\in[FIA]^ -\cup[\text{Moore}]\)
type: Item
instance of: scholarly article
title: Homomorphisms from \(L^ 1(G)\) for \(G\in[FIA]^ -\cup[\text{Moore}]\)
zbMATH Open document ID: 0805.43001
author: Volker Runde
published in: Journal of Functional Analysis
publication date: 1994-08-24T00:00:00Z
review text: One of the main problems in automatic continuity is the question of whether for a given Banach algebra \(A\) every homomorphism from \(A\) into an arbitrary Banach algebra \(B\) is continuous. Much is known in case \(A\) is a \(C^*\)-algebra. Although many \(C^*\)-algebras admit discontinuous homomorphisms, due to work of Sinclair and others there is a fair amount of information about the structure of homomorphisms. In the paper under review such problems are investigated for \(L^ 1\)- algebras \(L^ 1(G)\) of a locally compact group \(G\). Of course, one cannot expect substantial progress for general \(G\). The author obtains a number of interesting results for two classes of locally compact groups. The first one is [Moore], the class of groups all of whose irreducible unitary representations are finite dimensional, while the second, \([FIA]^ -\), consists of all groups \(G\) such that the group of inner automorphisms of \(G\) is relatively compact in the full automorphism group of \(G\), endowed with Birkhoff's topology. Both of these classes contain all abelian and all compact groups, and their structure and representation theory is well understood. If \(\Theta : A \to B\) is a homomorphism between Banach algebras, then the so-called continuity ideal \(I(\Theta)\) of \(\Theta\) is defined to be the set of all \(a \in A\) such that the mappings \(x \to \Theta(ax)\) and \(x \to \Theta(xa)\) from \(A\) into \(B\) are continuous. Using results from spectral synthesis it is shown that for \(G \in [\text{Moore}]\), \(\overline{I(\Theta)}\) has finite codimension in \(L^ 1(G)\). As a consequence, \(\Theta\) is continuous if and only if \(I(\Theta)\) is closed in \(L^ 1(G)\). Possibly, \(\Theta\) need not be continuous on \(I(\Theta)\). However, there always exists an ideal \(I\) in \(L^ 1(G)\) such that \(\overline{I}\) has finite codimension and the restriction \(\Theta| I\) is continuous. Turning to \(G \in [FIA]^ -\) and assuming that \(\overline{I(\Theta)}\) has fini…
reviewed by: Eberhard Kaniuth
Mathematics Subject Classification ID: 43A20
Mathematics Subject Classification ID: 46H40
Mathematics Subject Classification ID: 43A22
Mathematics Subject Classification ID: 22D15
Mathematics Subject Classification ID: 22D45
zbMATH DE Number: 599793
zbMATH Keywords: automatic continuity
zbMATH Keywords: discontinuous homomorphisms
zbMATH Keywords: \(L^ 1\)-algebras
zbMATH Keywords: locally compact group
zbMATH Keywords: irreducible unitary representations
zbMATH Keywords: group of inner automorphisms
MaRDI profile type: Publication
full work available at URL: jfan.1994.1060
OpenAlex ID: W2065985084
DOI: 10.1006/JFAN.1994.1060
description: scientific article; zbMATH DE number 599793