Weakly almost periodic functionals on the measure algebra

TL;DR

This paper demonstrates that the set of weakly almost periodic functionals on the measure algebra of a commutative Hopf von Neumann algebra forms a C*-algebra, extending to measure algebras of locally compact groups.

Contribution

It establishes that weakly almost periodic functionals on measure algebras are C*-algebras, using a novel adaptation of corepresentation theory to reflexive Banach spaces.

Findings

Weakly almost periodic functionals form a C*-algebra.

Extension to measure algebras of locally compact groups.

Utilizes factorisation results for weakly compact module maps.

Abstract

It is shown that the collection of weakly almost periodic functionals on the convolution algebra of a commutative Hopf von Neumann algebra is a C$^*$-algebra. This implies that the weakly almost periodic functionals on $M(G)$, the measure algebra of a locally compact group $G$, is a C$^*$-subalgebra of $M(G)^* = C_0(G)^{**}$. The proof builds upon a factorisation result, due to Young and Kaiser, for weakly compact module maps. The main technique is to adapt some of the theory of corepresentations to the setting of general reflexive Banach spaces.