Semigroup compactifications by generalized distal functions and a fixed point theorem

R D Pandian

doi:10.1155/s0161171291000285

Abstract

The notion of Semigroup compactification which is in a sense, a generalization of the classical Bohr (almost periodic) compactification of the usual additive realsR, has been studied by J. F. Berglund et. al. [2]. Their approach to the theory of semigroup compactification is based on the Gelfand-Naimark theory of commutativeC*algebras, where the spectra of admissibleC*-algebras, are the semigroup compactifications. H. D. Junghenn's extensive study of distal functions is from the point of view of semigroup compactifications [5]. In this paper, extending Junghenn's work, we generalize the notion of distal flows and distal functions on an arbitrary semitopological semigroupS, and show that these function spaces are admissibleC*- subalgebras ofC(S). We then characterize their spectra (semigroup compactifications) in terms of the universal mapping properties these compactifications enjoy. In our work, as it is in Junghenn's, the Ellis semigroup plays an important role. Also, relating the existence of left invariant means on these algebras to the existence of fixed points of certain affine flows, we prove the related fixed point theorem.

Highlights

The notion of "Semigroup compactification" which is in a sense, a generalization of the classical Bohr compactlflcation of the usual additive reals R, has been studied by J.F
Relating the existence of left Invariant means on these algebras to the existence of fixed points of certain affine flows, we prove the related fixed point theorem
We prove the existence of an u in n fiX

Summary

For any q

Z, q n rp where r g Z, and q(x) (rpn)(x) r(p n (x)) r(p n (y)) rpn(y) q(y). the flow is n-distal. Since Z N E(S, X) n is left simple, Z Zqo. Let g Z. As E(S, Z) n is right topological, it follows that pd p for all p E E(S, Z) n proving that E(S, Z) n is left simple. A) Let (X,a) denote the canonical LMC-compactificaion of S. The fact that X is the set of all multiplicatlve means proves that Dn(s) is an algebra As uev(1) uv(1), Dn(S) contains all the constant functions Let w MM(Dn(S)) and f Dn (S). Let (X,a) denote the canonical LMC-compactiflcatlon of S. Let X be a compact topological space, and g I, indexed by a directed set I, be a family of topological spaces.

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