Nonseparability of quotient spaces of function algebras on topological semigroups

Abstract

Let S be a topological semigroup, C(S) the space of all bounded real-valued continuous functions on S. We define WUC(S) the subspace of C(S) consisting of all weakly uniformly continuous functions and WAP(S) the space of all weakly almost periodic functions in C(S). Among other results, for a large class of topological semigroups S, for which noncompact locally compact topological groups are a very special case, we prove that the quotient spaces WUC(S)/WAP(S) and, for nondiscrete S, C(S)/ WUC(S) are nonseparable. (The actual setting of these results is more general.) For locally compact topological groups, parts of our results answer affirmatively certain questions raised earlier by Ching Chou and E. E. Granirer. Introduction. Let S be a (Hausdorff jointly continuous) topological semigroup and m(S) the space of all bounded real-valued functions on S with the usual supnorm 11 11 s. For every functionf in m(S) and element x of S we define the functions xf and fx on S by xf(y) : f(xy) and fx(y) f(yx) for all y in S. In this paper we shall be concerned with the following closed subspaces of m(S): C(S) : = {f E m(S): f is continuous), LUC(S) := {f E C(S): the map x x f (x E S) is norm continuous), RUC(S) : = {f E C(S): the map x fx (x E S) is norm continuous), UC(S) := LUC(S) n RUC(S), LWUC(S) := {f E C(S): the map x x f (x E S) is weakly continuous), RWUC(S): = {f E C(S): the map x fx (x E S) is weakly continuous), WUC(S): = LWUC(S) n RWUC(S), WAP(S) := {f E C(S): the set {xf: x E S) is weakly relatively compact). These spaces of functions have appeared before in many publications (see e.g. [2, 3, 5, 6, 7, 8, 11 and 13]). In particular for a locally compact topological group G we have that LUC(G) (or UC(G)) is the usual space of left uniformly (or uniformly, respectively) continuous functions on G (see e.g. [10]). Received by the editors October 17, 1980 and, in revised form, March 27, 1981. 1980 Mathematics Subject Classification. Primary 43A60, 43A15; Secondary 22A20.