Weakly almost periodic functions and almost convergent functions on a group

Abstract:

Let $G$ be a locally compact group, $UC(G)$ the space of bounded uniformly continuous complex functions on $G,{C_0}(G)$ the subspace of $UC(G)$ consisting of functions vanishing at infinity. Let $W(G)$ be the space of weakly almost periodic functions on $G$ and ${W_0}(G)$ the space of functions in $W(G)$ such that their absolute values have zero invariant mean. If $G$ is amenable let $F(G)$ be the space of almost convergent functions in $UC(G)$ and ${F_0}(G)$ the space of functions in $F(G)$ such that their absolute values are almost convergent to zero. The inclusive relations among the above-mentioned spaces are studied. It is shown that if $G$ is noncompact and satisfies certain conditions, e.g. $G$ is nilpotent, then each of the quotient Banach spaces $UC(G)/W(G),{W_0}(G)/{C_0}(G),{F_0}(G)/{W_0}(G)$ contains a linear isometric copy of ${l^\infty }$. On the other hand, an example of a noncompact group $G$ is given which satisfies the condition that ${C_0}(G) = {W_0}(G)$.