The Arens regularity of certain Banach algebras related to compactly cancellative foundation semigroups

Abstract

We study in this paper the space $L^\infty_0({\cal S},M_a({\cal S}))$ of a locally compact semigroup ${\cal S}$. That space consists of all $\mu$-measurable ($\mu\in M_a({\cal S})$) functions vanishing at infinity, where $M_a({\cal S})$ denotes the algebra of all measures with continuous translations. We introduce an Arens multiplication on the dual $L^\infty_0({\cal S},M_a({\cal S}))^*$ of $L^\infty_0({\cal S},M_a({\cal S}))$ under which $M_a({\cal S})$ is an ideal. We then give some characterizations for Arens regularity of $M_a({\cal S})$ and $L^\infty_0({\cal S},M_a({\cal S}))^*$. As the main result, we show that $M_a({\cal S})$ or $L^\infty_0({\cal S},M_a({\cal S}))^*$ is Arens regular if and only if ${\cal S}$ is finite.