transactions of the american mathematical society Volume 2X4, Number I. Julv 1984 FILTERS AND THE WEAKALMOST PERIODIC COMPACTIFICATION OF A DISCRETE SEMIGROUP BY JOHN F. BERGLUND AND NEIL HINDMAN1 Abstract. The weak almost periodic compactification of a semigroup is a compact semitopological semigroup with certain universal properties relative to the original semigroup. It is not, in general, a topological compactification. In this paper an internal construction of the weak almost periodic compactification of a discrete semigroup is constructed as a space of filters, and it is shown that for discrete semigroups, the compactification is usually topological. Other results obtained on the way to the main one include descriptions of weak almost periodic functions on closed subsemigroups of topological groups, conditions for functions on the additive natural numbers or on the integers to be weak almost periodic, and an example to show that the weak almost periodic compactification of the natural numbers is not the closure of the natural numbers in the weak almost periodic compactification of the integers. 1. Introduction. A semitopological semigroup is a triple (S, +, 5") such that (S, +) is a semigroup, (S,^) is a Hausdorff topological space, and + is separately continuous. While topological semigroups (those for which + is jointly continuous) have received most of the attention over the years, it is semitopological semigroups which arise naturally in such subjects as abstract harmonic analysis and functional analysis. For example, semitopological semigroups occur as the structure space of a measure algebra, as the weak operator closure of certain semigroups of operators, as the enveloping semigroup of certain flows, and as the weak almost periodic com- pactification of topological groups. We write our semigroups additively, by the way, since we shall be concerned mostly with the semigroup (N, + ) of natural numbers and the group (Z, + ) of integers. Define a function / in C(S), the set of bounded continuous complex-valued functions on S, to be weak almost periodic on S provided lim lim f(s„ + tk) = lim lim f(s„ + tk) n — ao /<->oo k-> oo n —oo whenever (sn)™=x and (tk)f=x are sequences in 5"and all limits involved exist. (For other characterizations see Theorem 2.3.) The set W(S) = {/ G C(S) |/is weak almost periodic} is a sub-C*-algebraof C(S). Received by the editors May 17, 1982and, in revised form, September 16, 1982 and May 2, 1983. 1980 Mathematics Subject Classification. Primary 22A15; Secondary43A60. 1 This author gratefully acknowledges support from the National Science Foundation under grants MCS 78-02330 and MCS 81-00733. ©1984 American Mathematical Society 0002-9947/84 $1.00 + $.25 per page 1 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use