Uniformly continuous functionals on the Fourier algebra of any locally compact group

Let G be any locally compact group. Let $VN\,(G)$ be the von Neumann algebra generated by the left regular representation of G. We study in this paper the closed subspace $UBC\mathop {(G)}\limits^ \wedge$ of $VN\, (G)$ consisting of the uniformly continuous functionals as defined by E. Granirer. When G is abelian, $UBC\mathop {(G)}\limits^ \wedge$ is precisely the bounded uniformly continuous functions on the dual group Ĝ. We prove among other things that if G is amenable, then the Banach algebra $UBC\mathop {(G)}\limits^ \wedge {\ast}$ (with the Arens product) contains a copy of the Fourier-Stieltjes algebra in its centre. Furthermore, $UBC\mathop {(G)}\limits^ \wedge {\ast}$ is commutative if and only if G is discrete. We characterize $W\mathop {(G)}\limits^ \wedge$, the weakly almost periodic functionals, as the largest subspace X of $VN\, (G)$ for which the Arens product makes sense on ${X^ {\ast} }$ and ${X^ {\ast} }$ is commutative. We also show that if G is amenable, then for certain subspaces Y of $VN\, (G)$ which are invariant under the action of the Fourier algebra $A\, (G)$, the algebra of bounded linear operators on Y commuting with the action of $A\, (G)$ is isometric and algebra isomorphic to ${X^ {\ast} }$ for some $X \subseteq UBC(\mathop {G)}\limits^ \wedge$.