Absolutely continuous functions on idempotent semigroups in the locally convex setting

Abstract

Let $E$ be a locally convex space and let $T$ be a semigroup of semicharacters on an idempotent semigroup. It is shown that there exists an isomorphism between the space of $E$-valued functions on $T$ and the space of all $E$-valued finitely additive measures on a certain algebra of sets. The space of all $E$-valued functions on $T$ which are absolutely continuous with respect to a positive definite function $F$ is identified with the space of all $E$-valued measures which are absolutely continuous with respect to the measure ${m_F}$ corresponding to $F$. Finally a representation is given for the operators on the set of all $E$-valued finitely additive measures on an algebra of sets which are absolutely continuous with respect to a positive measure.