The dual of semigroup algebras with certain locally convex topologies (Q871621)

Let \(S\) be a locally compact semigroup, \(M(S)\) the Banach space of all bounded complex regular Borel measures on \(S\), \(M_a(S)\) the space of all \(\mu \in M(S)\) for which the maps \(x \to \delta_x * | \mu| \) and \(x \to | \mu| *\delta_x\) from \(S\) into \(M(S)\) are weakly continuous, \(\delta_x\) denotes the Dirac measure at \(x\). For an extensive class of locally compact semigroups \(S\), a locally convex topology on \(M_a(S)\) is introduced so that the Banach space \(L^\infty_0 (S,M_a(S))\) can be identified with its strong dual. It is shown that, except for the case when \(S\) is finite, there are infinitely many such locally convex topologies \(\tau\) on \(M_a(S)\). The spectrum of \((M_a(S), \tau )\) is characterized in terms of semicharacters on \(S\).