Semiuniform semigroups and convolution

Semiuniform semigroups provide a natural setting for the convolution of generalized finite measures on semigroups. A semiuniform semigroup is said to be ambitable if each uniformly bounded uniformly equicontinuous set of functions on the semigroup is contained in an ambit. In the convolution algebras constructed over ambitable semigroups, topological centres have a tractable characterization. 1 Semiuniform products and semiuniform semigroups Functional analysis on topological and semitopological semigroups is well developed [1]. However, concrete semigroups often carry not only a topology but also a natural uniform structure. That suggests that it is worthwhile to investigate semigroups endowed with compatible uniform structures. An observation at the end of the author’s paper [9] points out that the main results in that paper, proved there for topological groups, hold more generally for semiuniform semigroups. Moreover, the definition and basic properties of ambitable topological groups [10], and their connection to topological centres in convolution algebras, are easily generalized to semiuniform semigroups. These generalizations are described in the current paper. Several theorems below are obtained by simple modifications of the proofs for topological groups in the author’s previous papers [9][10]. The modified proofs are included here for the sake of completeness. Most of the notation used here is defined in [9]. All uniform structures are assumed to be Hausdorff, and all linear spaces over the field R of reals. Following Isbell [4], we denote by X∗Y the semiuniform product of uniform spaces X and Y . By definition, a semigroup X with a uniform structure is a semiuniform semigroup if the semigroup operation (x, y) 7→ xy is uniformly continuous from the semiuniform product X∗X to X . In other words, X is a semiuniform semigroup if and only if • the set {x 7→ xy | y∈X} of mappings from X to X is uniformly equicontinuous; and • for each x ∈ X , the mapping y 7→ xy from X to X is uniformly continuous.