Module maps and invariant subsets of Banach modules of locally compact groups (Q2016111)

Here \(G\) denotes a locally compact group, \(C_0(G)\) and its Banach dual \(M(G)\), the complex regular Borel measures \(\mu\) on \(G\) with convolution product; also \(L^p(G), 0 \leq p < \infty\), while \(L^1(G)\) denotes the group convolution algebra, a closed ideal of \(M(G)\). The \(\mu_f, f \in L^1\), act on \(M(G)\)-modules via test functions in \(C_0(G)\). The article is based on results of \textit{A. T. M. Lau} [Trans. Am. Math. Soc. 232, 131--142 (1977; Zbl 0363.43002)] and \textit{A. Ghaffari} [Houston J. Math. 34, No. 4, 1225--1232 (2008; Zbl 1162.43004)]; they had been concerned with the action, for \(s \in G, f \in L^p, 0 \leq p <\infty\), in particular \(s \cdot f = \delta_s * f(t) = \delta s*f\) and \(s \cdot f(t) = \delta_s *f(t) =\Delta(s)^{1/p} f(s^{-1}ts)\). The author extends some of their results to include arbitrary left Banach \(G\)-modules, \(M(G)\) modules and \(L^1(G)\) modules; his results concern equivalence of statements relating the module mappings, usually needing changes of topologies. By \(T\) being a \textit{odular mapping} is meant \(T(s \cdot x) = s \cdot Tx\) for \(s \in G\) and \(x\) in some module. Given a bounded linear mapping \(T:X \to Y\) where \(X\) and \(Y\) are left \(G\)-modules the statements for \(T\) being a \(G\)-modular mapping, an \(M(G)\)-modular mapping or an \(L^1(G)\)-modular mapping are shown to be equivalent. Providing the duals \(X^*\) and \(Y^*\) with the \(\omega\)* topology, for \(T\) a continuous mapping \(Y^* \to X^*\) the statements that \(T\) is a right \(G\)-module mapping, a right \(M(G)\)-module mapping and a right \(L^1(G)\) mapping are equivalent. For a more involved result the author provides the dual Banach spaces with the uniform norm, and denotes by \(UC(X^*)\) the closed linear subspace of \(X^*\), i.e., those \(\phi \in X^*\) for which \(s \to \phi \cdot s\) is continuous. Restriction to \(UC(X^*)\) removes an ambiguity between the \(M(G)\)-module action and the dual \(M(G)\) action. Then, for a bounded linear operator \(UC(X^*) \to UC(Y^*)\) such that \(T(\phi \cdot s) = T\phi \cdot s\) for \(\phi \in X^*, s \in G\). This and the corresponding statements for \(M(G)\) and \(L^{1}(G)\) are equivalent. The probability measure algebra \(M(G)^+_1\) composed of probability measures and \(L^1(G)^+_1= M(G) \cap L^1(G)\), a closed convex subset \(C\) of a left Banach \(G\)-module \(X\) is said to be \(G\)-\textit{invariant} if \(s \cdot x \in C\) whenever \(s \in G\), \(x \in C\). The author shows that \(G\)-invariance can be equivalently expressed in terms of \(M(G)_1^+\) invariance or of \(L^1(G)^+_1\) invariance. A similar result holds for \(\omega^*\)-closed convex subsets of \(X^{*}\). The author follows Lau's [loc. cit.] Theorem 4.1 and Corollary 2; the corollary concerns the closed convex hulls of subsets of \(f \cdot x \) and \(\phi \cdot s\) for \(f \in L^1(G)^+_1\). The next section of the article concerns convex subsets of left Banach modules \(X\) and \(Y\). The author uses, as in Lau [loc. cit.], the vague or \(\tau\) topology on \(M(G)\), to get seminorms and a locally convex topology on \(M(G)\). Let \(B\) and \(C\) be closed \(G\)-invariant convex sets in \(X\) and \(Y\) respectively. For \(T: B \to C\) continuous and affine he finds conditions, in terms of \(M(G)^+_1\) and \(L^1(G)^+_1\), equivalent to \(T(s \cdot x) = s \cdot Tx\), \(s \in G\), \(x \in B\). Similar equivalences hold for \(\omega^*\)-closed subsets of \(X^*\). Reviewer's remark: The author's Theorem 4.4 is a little confusing; the result is an immediate consequence of Lau's Theorem 4.8, viz., that \(G\) is noncompact if and only if every weakly compact convex left or right invariant non-empty subset of \(L^1(G)\) consists of the origin only. Following Lau's Theorem 5.5, for \(C\) in a weakly compact closed left \(G\)-invariant subset of a left Banach \(G\)-module \(X\) and \(T:L^1(G)^+_1\) \(X\to C\) continuous affine then there exists \(x \in C\) such that \(T(u) = u \cdot x\) whenever \(u \in L^1(G)^+_1\). The last theorem of the article illustrates that his results do not necessarily apply if \(G\) is a non-abelian discrete group.