West semigroups as compactifications of locally compact abelian groups (Q305761)

Let \(G\) be a non-compact locally compact abelian group. The dual group \(\widehat{G}\) of \(G\) contains a Cantor set \(K\) (this means that \(K\) is homeomorphic to the Cantor subset \(C\) of the interval \([0, 1]\)) which is a Kronecker set, or one which is of type \(K_q\). Let \(\mu\) be a nonzero continuous positive probability measure in the space \(M(\widehat{G})\) of bounded regular Borel measures on \(\widehat{G}\) such that \(\mathrm{supp}(\mu)= K\). The group \(G =\widehat{{\widehat{G}}}\) can be embedded into \(\mathrm{L}^\infty(\widehat{G}, \mu)\), equipped with its \(\text{weak}^*\) topology. The \(\text{weak}^*\) closure of \(G\) in \(\mathrm{L}^\infty(\widehat{G}, \mu)\), denoted by \(S_w(\widehat{G}, \mu)\), is called a West semigroup on \(G\). \(S_w(\widehat{G}, \mu)\) is a compact semitopological semigroup containing a dense homomorphic image of \(G\), and so is a semigroup compactification of \(G\). Let \(m\) be the Lebesgue measure on the interval \([0, 1]\) and let \((\mathrm{L}^\infty)_1\) be the norm closed unit ball of \(\mathrm{L}^\infty([0, 1], m)\), equipped with the relative \(\text{weak}^*\) topology and pointwise multiplication. In the paper under review, the author proves that, if \(\widehat{G}\) is an \(I\)-group (that is, every neighborhood of the identity of \(\widehat{G}\) contains an element of infinite order) then \(S_w(\widehat{G}, \mu)\) is isomorphic to \((\mathrm{L}^\infty)_1\) (Proposition 3.1). A similar result is also established in the case of non-\(I\)-groups (Proposition 3.5). As a consequence, every non-compact locally compact abelian group \(G\) has a West semigroup, in which the set of idempotent elements is not closed (Corollary 3.7).