The notion of “Semigroup compactification” which is in a sense, a generalization of
the classical Bohr (almost periodic) compactification 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 C* algebras,
where the spectra of admissible C*-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 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 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.
