Characterising weakly almost periodic functionals on the measure algebra
Volume 204 / 2011
Abstract
Let $G$ be a locally compact group, and consider the weakly almost periodic functionals on $M(G)$, the measure algebra of $G$, denoted by ${\rm WAP}(M(G))$. This is a C$^*$-subalgebra of the commutative C$^*$-algebra $M(G)^*$, and so has character space, say $K_{\rm WAP}$. In this paper, we investigate properties of $K_{\rm WAP}$. We present a short proof that $K_{\rm WAP}$ can naturally be turned into a semigroup whose product is separately continuous; at the Banach algebra level, this product is simply the natural one induced by the Arens products. This is in complete agreement with the classical situation when $G$ is discrete. A study of how $K_{\rm WAP}$ is related to $G$ is made, and it is shown that $K_{\rm WAP}$ is related to the weakly almost periodic compactification of the discretisation of $G$. Similar results are shown for the space of almost periodic functionals on $M(G)$.
