Weakly almost periodic functions and Fourier-Stieltjes algebras of locally compact groups

Abstract:

A noncompact locally compact group $G$ is called an Eberlein group if $W(G) = B{(G)^ - }$ where $W(G)$ is the algebra of continuous weakly almost periodic functions on $G$ and $B{(G)^ - }$ is the uniform closure of the Fourier-Stieltjes algebra of $G$. We show that if $G$ is a noncompact $[IN]$-group or a noncompact nilpotent group then $W(G)/B{(G)^ - }$ contains a linear isometric copy of ${l^\infty }$. In particular, $G$ is not an Eberlein group. On the other hand, finite direct products of Euclidean motion groups and, by a result of W. Veech, noncompact semisimple analytic groups with finite centers are Eberlein groups.