Semigroup Compactifications by Generalized Distal Functions

The notion of "Semigroup compactification" which is in a sense, a generalization of the classical Bohr (almost periodic) compactlflcation of the usual additive reals R, 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 commutative Calgebras, where the spectra of admissible C*-aigebras, are the semigroup compactifications. H.D. Junghenn’s extensive study of distal functions is from the point of view of semigroup compactiflcations [5]. In this paper, extending Junghenn’s work, we generalize the notion of distal flows and distal functions on an arbitrary semitopological semigroup S, and show that these function spaces are admissible C*subalgebras of C(S). We then characterize their spectra (semigroup compactifications) in terms of the universal mapping properties these compactlficatlons enjoy. In our work, as it is in Junghenn’s, the Ellis semlgroup 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.