Compact right multipliers on a Banach algebra related to locally compact semigroups
A locally compact semigroup \(S\) is called compactly cancellative if \(C^{-1}D\) and \(CD^{-1}\) are compact for all compact subsets \(C\) and \(D\) of \(S\). In [Pac. J. Math. 14, 659--664 (1964; Zbl 0135.35803)], \textit{S. Sakai} has proved that, if \(G\) is a locally compact but not compact group, then zero is the only compact right multiplier on the group algebra \(L^{1}(G)\). The main result of this article is a characterization of right compact multipliers on the Banach algebra \(L_{0}^{\infty}(S, M_{a})^{*}\) equipped with an Arens product. This is done by putting together four equivalent definitions of compact right multipliers on \(S\), when \(S\) is a compactly cancellative semigroup with identity. An interesting example is provided at the end to show that there are nontrivial locally compact semigroups for which compact multipliers exist.
scientific article; zbMATH DE number 6018263
scientific article