Compact Semitopological Semigroups and Weakly Almost Periodic Functions

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“…Statements (a) and (b) are, respectively, statements (c) and (a) of Theorem III.8.2, p. 108, of [3], and are shown there to be equivalent. Statements (b), (c) and (d) are, respectively, statements (a), (b) and (d) of Theorem 1.1.8, p. 17, of [2] with K = {/: 5 E 5} and are shown there to be equivalent. D 2.4 Theorem.…”

“…is continuous, and (4) the pair (SS, 8S) is maximal with respect to these properties in the sense that if eb is a continuous homomorphism from S to a compact semigroup T and (T,eb) satisfies (1), (2) and (3) with <b replacing 8S and T replacing 8S, then there is a continuous homomorphism rj from 8S onto T such that r/ ° 8S = cb. If S is discrete, then (8S, 8S) is, up to isomorphism, the Stone-Cech compactification with semigroup structure as given in Theorem 2.2.…”

“…The following theorem characterizes weak almost periodic functions in terms of the (noncommutative) maximal left topological compactification (8S, 8S) of Theorem 2.4. (1) for all n E N, a" = limk^00f(tn + sk), (2) for all k € N, *t = lim^00/(r" + 5,), (3)lim^xû" = a, and…”

“…Let (T, eb) satisfy (1) and (2). We first show there is a continuous tj: 91 -» T such that tj o e = cb.…”

Semigroup Compactifications of Semidirect Products

“…Let S and T have (Hausdorff) topologies relative to which multiplication in S, T and 5© T (with the product topology) is separately continuous (in the terminology of [1], S, T and S(r)T are semitopological semigroups). In a previous paper the first author, generalizing some results of [10] and [14], showed that if T contains a dense subgroup then the almost periodic compactification (S(t)T)ap of S(r)T is a semidirect product A1© Y, where Y = TAP and A1 is a continuous homomorphic image of SAP [11].…”