Let G be a non-discrete locally compact group, A(G) the Fourier algebra of G, VN(G) the von Neumann algebra generated by the left regular representation of G which is identified with A(G)*, and WAP(Ĝ) the space of all weakly almost periodic functionals on A(G). We show that there exists a directed family ℋ of open subgroups of G such that: (1) for each H ∈ ℋ, A(H) is extremely non-Arens regular; (2) $VN\left(G\right)={\bigcup }_{H\in \mathscr{H}}VN\left(H\right)$ and $VN\left(G\right)/WAP\left(Ĝ\right)={\bigcup }_{H\in \mathscr{H}}[VN\left(H\right)/WAP\left(Ĥ\right)]$; (3) $A\left(G\right)={\bigcup }_{H\in \mathscr{H}}A\left(H\right)$ and it is a WAP-strong inductive union in the sense that the unions in (2) are strongly...

Let K be an ultraspherical hypergroup associated to a locally compact group G and a spherical projector π and let VN(K) denote the dual of the Fourier algebra A(K) corresponding to K. In this note, invariant means on VN(K) are defined and studied. We show that the set of invariant means on VN(K) is nonempty. Also, we prove that, if H is an open subhypergroup of K, then the number of invariant means on VN(H) is equal to the number of invariant means on VN(K). We also show that a unique topological...