Almost periodic compactifications of semidirect products of flows (Q1865688)

The pair \((S,X)\), where \(S\) is a semitopological semigroup acting (by a separately continuous left action) on a topological space \(X\), is called a flow. Let \(S\widetilde\times T:=S\times_\varphi T\) be a semidirect product of semitopological semigroups \(S\) and \(T\), where \(\varphi\colon T\to S^S\) is a homomorphism from \(T\) into the continuous homomorphisms on \(S\to S\). If \((S,X),(T,X)\) and \((T,Y)\) are flows linked by the identity \[ t(sx)=(\varphi(t)(s))(tx),\;\forall (s,t,x)\in S\times T\times X, \] then under a suitable continuity condition one can define in a natural manner the semidirect product flow \((S\widetilde\times T,X\times Y)\). The author shows in the present paper that under certain conditions (concerning compact generation) the almost periodic (AP) compactification of this semidirect product flow is \(X^{AP}\times Y^{AP}\), where \(X^{AP}\) is the almost periodic compactification of a related flow \((S\widetilde \times T,X)\) and \(Y^{AP}\) is the AP-compactification of \((T,Y)\). Similar results are proved for the strongly almost periodic (SAP) compactification. Both of the results on the AP-compactification and on the SAP-compactification are obtained as consequences of a more general theorem on compactifications of the semidirect product flow \((S\widetilde\times T, X\times Y)\). The paper includes applications to flows constructed from quotients of semigroups by left congruences. The results of the paper are generalizations of theorems proved by the author for direct products of flows [see \textit{H. D. Junghenn}, Semigroup Forum 58, No. 2, 296-312 (1999; Zbl 0914.22003)] and also of theorems proved for semidirect products of semigroups [see \textit{H. D. Junghenn} and \textit{B. Lerner}, Trans. Am. Math. Soc. 265, 393-404 (1981; Zbl 0469.22003) and \textit{P. Milnes}, Colloq. Math. 44, 125-136 (1981; Zbl 0482.43005)].