## Bulletin of the Belgian Mathematical Society - Simon Stevin

### 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.

#### Article information

Source
Bull. Belg. Math. Soc. Simon Stevin, Volume 16, Number 2 (2009), 205-221.

Dates
First available in Project Euclid: 3 June 2009

https://projecteuclid.org/euclid.bbms/1244038134

Digital Object Identifier
doi:10.36045/bbms/1244038134

Mathematical Reviews number (MathSciNet)
MR2541036

Zentralblatt MATH identifier
1171.43002

#### Citation

Maghsoudi, S.; Nasr-Isfahani, R. The Arens regularity of certain Banach algebras related to compactly cancellative foundation semigroups. Bull. Belg. Math. Soc. Simon Stevin 16 (2009), no. 2, 205--221. doi:10.36045/bbms/1244038134. https://projecteuclid.org/euclid.bbms/1244038134