Characterising weakly almost periodic functionals on the measure algebra

Tom 204 / 2011

Matthew Daws Studia Mathematica 204 (2011), 213-234 MSC: Primary 43A10, 46L89, 46G10; Secondary 43A20, 43A60, 81R50. DOI: 10.4064/sm204-3-2


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 $M(G)$.


  • Matthew DawsSchool of Mathematics
    University of Leeds
    Leeds, LS2 9JT, United Kingdom

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