Intertwining operators over \(L^ 1(G)\) for \(G\in [PG]\cap [SIN]\)
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.
scientific article; zbMATH DE number 922490