Weakly almost periodic functions and almost convergent functions on a group

Let G be a locally compact group, UC(G) the space of bounded uniformly continuous complex functions on G, C₀(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'q(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 Fq(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), W₀(G)C₀(G), F₀(G)|W₀(G) contains a linear isometric copy of l^∞. On the other hand, an example of a noncompact group G is given which satisfies the condition that C₀(G) = W₀(G).