West semigroups as compactifications of locally compact abelian groups

Abstract

In this paper, we will identify certain subsemigroups of the unit ball of 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 , and the weakly almost periodic compactification, , 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 and , it is proved that and contain uncountably many idempotents and the set of idempotents is not closed.