Universal compactifications of semitopological semigroups with respect to the properties satisfying the varieties of semigroups and groups are studied through two function algebras.

https://downloads.hindawi.com/journals/ijmms/2001/970689.pdf

Some algebraic universal semigroup compactifications

The universal abelian, band, and semilattice compactifications of a semitopological semigroup are characterized in terms of three function algebras. Some relationships among these function algebras and some well-known ones, from the universal compactification point of view, are also discussed.

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The universal semilattice compactification of a semigroup

Among other things, a function algebra on a semitopological semigroup whose associated semigroup compactification is universal with respect to the property of being a rectangular group is studied. Mathematics Subject Classification: 43A60, 22A20, 20M10

Rectangular group compactification of a semigroup

The purpose of this paper is to introduce an algebra of functions on a semitopological semigroup and to study these functions from the point of view of universal semigroup compactification. We show that the corresponding semigroup compactification of this algebra is universal with respect to the property of being a nilpotent group. The general approach to the theory of semigroup compactification is based on the Gelfand-Naimark theory of C∗-algebras of functions. There are many papers which deal with the characterization of certain universal semigroup compactifications in terms of function algebras. Seminal work in this context was done by K. de Leeuw and I. Glicksberg [3]. The universal group compactification is given by Junghenn [6], in terms of some types of distal functions. This paper deals with the construction of a function algebra on a semitopological semigroup, used to characterize the universal nilpotent group compactification of it, and investigating some of its properties. First we recall some preliminaries. Throughout this paper, S shall be a semitopological semigroup, unless otherwise mentioned. For notation and terminology we shall follow Berglund et al. [2] as far as possible. Thus a semigroup compactification of S is a pair (ψ,X), where X is a compact, Hausdorff, right topological semigroup and ψ : S → X is a continuous homomorphism with dense image such that for each s ∈ S, the mapping x 7→ ψ(s)x : X → X is continuous. The C∗-algebra of all continuous bounded complex-valued functions on a topological space Y is denoted by C(Y ). For C(S) left and right translations Ls and Rt are defined for all s, t ∈ S and f ∈ C(S) by (Lsf)(t) = f(st) = (Rtf)(s). A left translation invariant C∗-subalgebra F of C(S) (i.e. Lsf ∈ F for all s ∈ S and f ∈ F ) containing the constant functions is called m-admissible if the function s 7→ (Tμf)(s) = μ(Lsf) is in F for all f ∈ F and μ ∈ S (=the spectrum of F ); then the product of μ, ν ∈ S can be defined by μν = μ ◦ Tν and the Gelfand topology on S makes ( , S ) a semigroup compactification (called the F -compactification) of S, where : S → S is the evaluation mapping. The reader is referred to sections 3.1 and 3.3 of [2] for the one-to-one correspondence between compactifications of S and m-admissible subalgebras of C(S), and also for a discussion of universal P -compactifications, whose existence for a wide variety of properties P is given in terms of subdirect products. Received by the editors January 16, 1996. 1991 Mathematics Subject Classification. Primary 22A20, 43A60.

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H. R. Ebrahimi-Vishki

Papers

The universal nilpotent group compactification of a semigroup