A mixed topology over the space of measures with continuous translations on a locally compact semigroup
Throughout this review \(S\) will always be a locally compact semigroup. Let \(M(S)\) denote the Banach space of all bounded complex-valued regular Borel measures on \(S\) with the total variation norm. Write \(n(S)\) for the topology generated by the total variation norm on \(M(S)\). The space of all \(\mu \in M(S)\) for which the functions \(x \mapsto | \mu | (x^{-1}K)\) and \(x \mapsto | \mu | (Kx^{-1})\) are continuous on \(S\), for all compact subsets \(K\), is denoted by \(M_a(S)\). Let \(\mathcal{C}\) be the set of increasing sequences of compact subsets of \(S\) and let \(\mathcal{R}\) be the set of increasing sequences \((r_n)\) of real numbers in \((0, \infty)\) with \(r_n \rightarrow \infty\). For any \((C_n) \in \mathcal{C}\) and \((r_n) \in \mathcal{R}\), set \[ U( (C_n), (r_n)) = \{ \mu \in M_a(S) : | \mu |(C_n) \leq r_n \text{ for all } n \geq 1 \}. \] The family \(\mathcal{U}\) of all sets \(U((C_n), (r_n))\) for all \((C_n) \in \mathcal{C}\) and \((r_n) \in \mathcal{R}\) is a neighborhood base for zero of a locally convex topology on \(M_a(S)\). This topology is denoted by \(\beta^1(S)\). Observe \(\beta^1(S) \leq n(S)\). Denote by \(\kappa(S)\) the locally convex topology on \(M_a(S)\) generated by the seminorms \(P_C(\mu) = | \mu | (C)\), where \(C\) runs through all compact subsets of \(S\). The author shows that the topology \(\beta^1(S)\) on \(M_a(S)\) is the mixed topology with respect to \(\kappa(S)\) and \(n(S)\). Using this result, a new proof is given to show that \((M_a(S), \beta^1(S))\) is a bornological, barrelled or a metrizable space precisely when \(S\) is compact, cf. [the author and \textit{R. Nasr-Isfahani}, Acta Math. Sin., Engl. Ser. 27, No. 5, 933--942 (2011; Zbl 1222.43003)]. A locally compact semigroup \(S\) is called foundation if the closure of the set \(\bigcup\{ \operatorname{supp}(\mu) : \mu \in M_a(S) \}\) coincides with \(S\). The semigroup \(S\) is called compactly cancellative if \(C^{-1}D\) and \(CD^{-1}\) are compact subsets of \(S\) for all compact subsets \(C\) and \(D\) of \(S\). The author proves that \((M_a(S), \beta^1(S))\) is a complete space if \(S\) is a compactly cancellative foundation semigroup with identity. The last main result of the paper is that, if \(S\) is a compactly cancellative foundation semigroup with identity, then \((M_a(S), \beta^1(S))\) with the convolution product is a complete semi-topological algebra.
scientific article; zbMATH DE number 6147877
scientific article