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Characterising weakly almost periodic functionals on the measure algebra

Let $G$ be a locally compact group, and consider the weakly-almost periodic functionals on $M(G)$, the measure algebra of $G$, denoted by $\wap(M(G))$. This is a C$^*$-subalgebra of the commutative C$^*$-algebra $M(G)^*$, and so has character space, say $K_\wap$. In this paper, we investigate properties of $K_\wap$. We present a short proof that $K_\wap$ 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 $K_\wap$ is related to $G$ is made, and it is shown that $K_\wap$ 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 $M(G)$.