Intertwining operators over \(L^ 1(G)\) for \(G\in [PG]\cap [SIN]\)
type: Item
instance of: scholarly article
title: Intertwining operators over \(L^ 1(G)\) for \(G\in [PG]\cap [SIN]\)
zbMATH Open document ID: 0853.43006
DOI: 10.1007/PL00004255
author: Volker Runde
published in: Mathematische Zeitschrift
publication date: 1997-01-06T00:00:00Z
full work available at URL: https://eudml.org/doc/174862
review text: Let \(A\) be a Banach algebra, let \(E\) be a Banach \(A\)-module, and let \(F\) be a weak Banach \(A\)-module. A linear operator \(\theta : E \to F\) is called intertwining if the mappings \(E \ni x \mapsto \theta (a.x) - a. \theta (x)\) and \(E \ni x \mapsto \theta (x.a) - \theta (x).a\) are continuous for each \(a \in A\). The class of intertwining operators includes all \(A\)-module homomorphisms, algebra homomorphisms, and derivations. Suppose in the above situation that \(A = L^1 (G)\) for an [SIN]-group \(G\) with polynomial growth. In case \(G\) is compactly generated, we show that \({\mathcal I} (\theta)^-\) has finite codimension in \(L^1 (G)\), where \({\mathcal I} (\theta)\) is the continuity ideal of \(\theta\). An important rôle in the proof of this assertion is played by a result due to \textit{T. Pytlik} [Studia Math. 73, 169-176 (1982; Zbl 0504.43005)]. In case \(F\) is a Banach \(L^1 (G)\)-module as well, the additional demand that \(G\) be compactly generated can be dropped. As a consequence, we obtain the automatic continuity of all derivations from \(L^1 (G)\) into a Banach \(L^1 (G)\)-module when \(G\) is an [SIN]-group with polynomial growth. Finally, we give an application of our results to the cohomology comparison problem for Banach algebras.
reviewed by: Volker Runde
Mathematics Subject Classification ID: 43A20
Mathematics Subject Classification ID: 46H40
Mathematics Subject Classification ID: 22D15
Mathematics Subject Classification ID: 18G99
Mathematics Subject Classification ID: 43A45
Mathematics Subject Classification ID: 46M99
Mathematics Subject Classification ID: 47B47
Mathematics Subject Classification ID: 47B48
zbMATH DE Number: 922490
zbMATH Keywords: Banach \(L^ 1 (G)\)-module
zbMATH Keywords: Banach algebra
zbMATH Keywords: intertwining operators
zbMATH Keywords: polynomial growth
zbMATH Keywords: continuity ideal
zbMATH Keywords: [SIN]-group
zbMATH Keywords: cohomology comparison problem
MaRDI profile type: Publication
OpenAlex ID: W2090484615
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description: scientific article; zbMATH DE number 922490