Topological Brandt semigroups

In the present paper, the properties of a locally compact Hausdorff topological Brandt semigroup, and the relation between its semigroup algebras and ℓ1-Munn algebras over group algebras are investigated. It is proved that for each locally compact Hausdorff topological group G, and each index set I, there exists a locally compact Hausdorff topological Brandt semigroup S=B(G,I) such that the Banach algebras \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal {LM}_{I}(M(G))$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{LM}_{I}(L^{1}(G))$\end{document} are isometrically isomorphic to M(S)/ℓ1({0}) and Ma(S)/ℓ1({0}), respectively.