West semigroups as compactifications of locally compact abelian groups

Abstract

In this paper, we will identify certain subsemigroups of the unit ball of $L^{\infty }[0,1]$ as semitopological compactifications of locally compact abelian groups, using an idea of West (Proc R Ir Acad Sect A 67:27–37, 1968). Our result has been known for the additive group of integers since Bouziad et al. (Semigr Forum 62(1):98–102, 2001). We will construct a semitopological semigroup compactification for each locally compact abelian group G, depending on the algebraic properties of G. These compact semigroups can be realized as quotients of both the Eberlein compactification $G^e$, and the weakly almost periodic compactification, $G^w$, of G. The concrete structure of these compact quotients allows us to gain insight into known results by Brown (Bull Lond Math Soc 4:43–46, 1972) and Brown and Moran (Proc Lond Math Soc 22(3):203–216, 1971) and by Bordbar and Pym (Math Proc Camb Philos Soc 124(3):421–449, 1998), where for the groups $G=\mathbb {Z}$ and $G=\mathbb {Z}_q^{\infty }$, it is proved that $G^e$ and $G^w$ contain uncountably many idempotents and the set of idempotents is not closed.