Uniformly Continuous Functionals on the Fourier Algebra of Any Locally Compact Group

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