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

Abstract:

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 $UBC(\hat {G})$ of $VN(G)$ consisting of the uniformly continuous functionals as defined by E. Granirer. When $G$ is abelian, $UBC(\hat {G})$ is precisely the bounded uniformly continuous functions on the dual group $\hat {G}$. We prove among other things that if $G$ is amenable, then the Banach algebra $UBC(\hat {G})^\ast$ (with the Arens product) contains a copy of the Fourier-Stieltjes algebra in its centre. Furthermore, $UBC(\hat {G})^\ast$ is commutative if and only if $G$ is discrete. We characterize $W(\hat {G})$, the weakly almost periodic functionals, as the largest subspace $X$ of $VN(G)$ for which the Arens product makes sense on $X^*$ and $X^*$ 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 $X^*$ for some $X \subseteq UBC(\hat {G})$.