Characterising weakly almost periodic functionals on the measure algebra

Matthew Daws¹

¹ School of Mathematics University of Leeds Leeds, LS2 9JT, United Kingdom

Studia Mathematica, Tome 204 (2011) no. 3, pp. 213-234

Résumé

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 $M(G)$.

Détail

DOI : 10.4064/sm204-3-2

Keywords: locally compact group consider weakly almost periodic functionals measure algebra denoted wap * subalgebra commutative * algebra * has character space say wap paper investigate properties wap present short proof wap naturally turned semigroup whose product separately continuous banach algebra level product simply natural induced arens products complete agreement classical situation discrete study wap related made shown wap related weakly almost periodic compactification discretisation similar results shown space almost periodic functionals