Heneri A. M. Dzinotyiweyi

Nonseparability of quotient spaces of function algebras on topological semigroups

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.

Algebras of measures on C-distinguished topological semigroups

Let S be a (jointly continuous) topological semigroup, C ( S ) the set of all bounded complex-valued continuous functions on S and M ( S ) the set of all bounded complex-valued Radon measures on S . Let ( S ) (or ( S )) be the set of all µ ∈ M ( S ) such that x → │µ│ ( x -1 C ) (or x → │µ│( Cx -1 ), respectively) is a continuous mapping of S into ℝ, for every compact set C ⊆ S , and . (Here │µ│ denotes the measure arising from the total variation of µ and the sets x -1 C and Cx -1 are as defined in Section 2.) When S is locally compact the set M a ( S ) was studied by A. C. and J. W. Baker in (1) and (2), by Sleijpen in (14), (15) and (16) and by us in (3). In this paper we show that some of the results of (1), (2), (14) and (15) remain valid for certain non-locally compact S and raise some new problems for such S .

The cardinality of the set of left invariant means on topological semigroups

For a very large class of topological semigroups, we establish lower and upper bounds for the cardinality of the set of left invariant means on the space of left uniformly continuous functions. In certain cases we show that such a cardinality is exactly , where b is the smallest cardinality of the covering of the underlying topological semigroup by compact sets.

Non-archimedean harmonic analysis on topological semigroups

Abstract Let K be a complete non-Archimedean valued field, S a commutative topological semigroup (not necessarily locally compact). We study the convolution Banach algebra M(S) of all tight K-valued measures on S and its ideal Ma(S) consisting of all elements μ of M(S) for which the shift map x ↦ x ∗μ is continuous (the analogue of the group algebra). In particular, we show that under reasonable conditions the Fourier-Stieltjes transform is isometric. Furthermore, we describe the images of M(S) and Ma(S) under the Fourier-Stieltjes transform. The results obtained are known for groups but new for topological (semi) lattices.

Almost convergent and weakly almost periodic functions on a semigroup

Let S be a topological semigroup, UC(S) the set of all bounded uniformly continuous functions on S, WAP(S) the set of all (bounded) weakly almost periodic functions on S, EO(S) := {f E UC(S): m(Ifj) = 0 for each left and right invariant mean m on UC(S)) and WO(S) := {f E WAP(S): m(Ifj) = 0 for each left and right invariant mean m on WAP(S)). Among other results, for a large class of noncompact locally compact topological semigroups S, we show that the quotient space EO(S)/ WO(S) contains a linear isometric copy of l? and so is nonseparable.