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 KWAP. In this paper, we investigate properties of KWAP. We present a short proof that KWAP 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 KWAP is related to G is made, and it is shown that KWAP 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).