The Dual of Semigroup Algebras with Certain Locally Convex Topologies

Abstract

In this paper, we deal with the dual of the semigroup algebras \(M_a(S)\) for an extensive class of locally compact semigroups S under certain locally convex topologies. We first introduce and study a locally convex topology on \(M_a(S)\) under which the Banach space \(L^\infty_0(S,M_a(S))\) can be identified with its strong dual. We then show that, except for the case where S is finite, there are infinitely many such locally convex topologies \(\tau\) on \(M_a(S)\). Finally, we characterize the spectrum of \((M_a(S),\tau)\) in terms of semicharacters on S.