Direct sums of spaces of functions on semigroups
Let \(F\) be an algebra of continuous bounded functions on a semigroup \(S\) which has a separately continuous multiplication. Denote by \(S^ F\) the carrier space of \(F\), so that \(S^ F\) contains an image \(\varepsilon(S)\) of \(S\); \(S^ F\) is compact and has a natural semigroup multiplication for which right translations are continuous if \(F\) satisfies the condition known as \(m\)-admissibility. If \(G\) is an \(m\)-admissible subalgebra of \(F\), there is a continuous surjective homomorphism \(\theta : S^ F \to S^ G\). The main theorem of the paper is that, in this situation, \(F\) is a direct sum \(G \oplus H\) where \(H\) is an ideal in \(F\) invariant under left translation by elements of \(S\) if and only if there is a continuous homomorphism \(\widetilde{\theta} : S^ G \to S^ F\) such that \(\widetilde{\theta}(S^ G)\) is a left ideal in \(S^ F\) and \(\theta \circ \widetilde{\theta}\) is the identity on \(S^ G\); also \(H\) is \(m\)-admissible if and only if \(\widetilde{\theta} (S^ G)\) is an ideal. Several general consequences are described, including de Leeuw and Glicksberg's classical result about the decomposition of weakly almost periodic functions. The author also applies his theorem to give a generalization of the Furstenberg-Namioka structure theorem for compact right topological groups. This applies to compact right topological semigroups which have dense topological centres and unique minimal right ideals. The general idea is that such a semigroup can be expressed as a projective limit of a well ordered family of quotient semigroups, in which between each semigroup and its successor a joint continuity property holds, but for the (complicated) details the paper should be consulted.
scientific article; zbMATH DE number 611120