Almost periodic compactifications of product flows (Q1279807)

A semitopological flow \((S, X)\) consists of a semigroup \(S\) with a topology acting on a topological space \(X\) in such a way that the multiplication in \(S\) and the action \(S \times X \to X\) are both separately continuous. A compactification of \((S,X)\) is a flow \((S, X')\) in which \(X'\) is compact, the maps \(x' \mapsto sx'\) (\(s \in S\)) are continuous and there is a continuous map \(\varepsilon_X : X \to X'\) such that the \(S\)-action commutes with \(\varepsilon_X\) and \(\varepsilon_X (X)\) is dense in \(X'\). A bicompactification is a pair \((S', X')\) consisting of a semigroup \(S'\) with a compact topology acting algebraically on a compact space \(X'\) with a continuous homomorphism \(\varepsilon_S : S \to S'\) with \(\varepsilon_S (S)\) dense in \(S'\) such that (i) \((S, X')\) is a flow compactification of \((S, X),\) (ii) the action \(S' \times X' \to X'\) is continuous in the \(S'\)-variable, and (iii) \(S'\) is a semigroup compactification of \(S\) (which means that the multiplication \((s,t) \mapsto st\) in \(S'\) is continuous in \(s\) for every \(t\) in \(S',\) but is required to be continuous in \(t\) only when \(s\) is in \(\varepsilon_S (S)\). (Thus a bicompactification is not necessarily a semitopological flow.) In the paper a general theory of compactifications of flows is developed which provides natural extensions of theories of semigroup compactifications [as in the book Analysis on Semigroups by \textit{J. F. Berglund, H. D. Junghenn} and \textit{P. Milnes} (Wiley, New York, 1989; Zbl 0727.22001)] across a wide range from abstract results about universal properties to special results about compactifications of products. For example, if \({\mathcal Q}\) is a property of compactifications of flows which is inherited by subdirect products, the existence of one \({\mathcal Q}\)-compactification implies the existence of a universal \({\mathcal Q}\)-compactification. A new possibility here is to relate flow compactifications to semigroup compactifications. Flow compactifications include semigroup compactifications simply by considering the flow generated when a semigroup acts on itself, but there are more significant relationships. For example, the enveloping semigroup \(E(S, X')\) of a flow \((S, X')\) in which \(X'\) is compact is the closure in the pointwise topology of the set of maps \(s: X' \to X'\) with \(s \in S\); this is a compact semigroup. If \(E(S, X')\) has a property \({\mathcal P}\) and \(S\) has a universal \({\mathcal P}\)-compactification \(S^{\mathcal P},\) then \(E(S, X')\) is a quotient of \(S^{\mathcal P}\). It is proved that if \({\mathcal P}\) is a property inherited by factors then flows \((S, X)\) have universal \({\mathcal Q}\)-compactifications if \({\mathcal Q}\) is one of the equivalent properties: (i) \(E(S, X')\) has the property \({\mathcal P}\), (ii) \((S^{\mathcal P},X')\) is a bicompactification of \((S,X)\). The principal semigroup compactifications (the almost periodic, the weakly almost periodic, and the strongly almost periodic) all have analogues for flows. As for the semigroup case, there are approaches both through universal properties and through \(C^*\)-algebras of functions. Detailed results are given about relationships with quotients (including the flow generated by a group \(G\) on a homogeneous space \(G/H\) where \(H\) is a closed subgroup), and about minimally weakly almost periodic semigroups. The final sections consider compactifications of direct products of flows. The results here are too complex to describe in a review, but conditions are given under which the universal \({\mathcal Q}\)-compactification of \((S \times T, X \times Y)\) is \((S \times T, X^{\mathcal Q} \times Y^{\mathcal Q}),\) and, for a general family \((S_i, X_i)\) of flows, under which the universal \({\mathcal Q}\)-compactification of \((\prod_i S_i , \prod_i X_i)\) is \((\prod_i S_i , \prod_i X_i^{\mathcal Q}.)\).