The Arens regularity of certain Banach algebras related to compactly cancellative foundation semigroups
S. Maghsoudi and R. Nasr-Isfahani
Source: Bull. Belg. Math. Soc. Simon Stevin Volume 16, Number 2 (2009), 205-221.
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.
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Permanent link to this document: http://projecteuclid.org/euclid.bbms/1244038134
Zentralblatt MATH identifier:
05578794
Bulletin of the Belgian Mathematical Society - Simon Stevin