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group extension; semidirect product; topological group; semitopological semigroup; right topological semigroup; compactification; almost periodic; weakly almost periodic; strongly almost periodic
Let $N$ and $K$ be groups and let $G$ be an extension of $N$ by $K$. Given a property $\mathcal P$ of group compactifications, one can ask whether there exist compactifications $N^{\prime }$ and $K^{\prime }$ of $N$ and $K$ such that the universal $\mathcal P$-compactification of $G$ is canonically isomorphic to an extension of $N^{\prime }$ by $K^{\prime }$. We prove a theorem which gives necessary and sufficient conditions for this to occur for general properties $\mathcal P$ and then apply this result to the almost periodic and weakly almost periodic compactifications of $G$.
[1] J. F.  Berglund, H. D.  Junghenn and P.  Milnes: Analysis on Semigroups: Function Spaces, Compactifications, Representations. Wiley, New York, 1989. MR 0999922
[2] K.  de  Leeuw and I.  Glicksberg: Almost periodic functions on semigroups. Acta Math. 105 (1961), 99–140. DOI 10.1007/BF02559536 | MR 0131785
[3] H. D.  Junghenn and B.  Lerner: Semigroup compactifications of semidirect products. Trans. Amer. Math. Soc. 265 (1981), 393–404. DOI 10.1090/S0002-9947-1981-0610956-2 | MR 0610956
[4] H. D.  Junghenn and P.  Milnes: Distal compactifications of group extensions. Rocky Mountain Math.  J. 29 (1999), 209–227. DOI 10.1216/rmjm/1181071687 | MR 1687663
[5] M.  Landstad: On the Bohr compactification of a transformation group. Math.  Z. 127 (1972), 167–178. DOI 10.1007/BF01112609 | MR 0310853 | Zbl 0236.54030
[6] A. T.  Lau, P.  Milnes and J. S.  Pym: Compactifications of locally compact groups and quotients. Math. Proc. Camb. Phil. Soc. 116 (1994), 451–463. DOI 10.1017/S030500410007273X | MR 1291752
[7] P.  Milnes: Almost periodic compactifications of direct and semidirect products. Coll. Math. 44 (1981), 125–136. MR 0633106 | Zbl 0482.43005
[8] M.  Rieffel: $C^{\star }$-algebras associated with irrational rotation algebras. Pacific J.  Math. 93 (1981), 415–429. DOI 10.2140/pjm.1981.93.415 | MR 0623572 | Zbl 0499.46039
[9] O.  Schreier: Über die Erweiterung von Gruppen I. Monatsh. Math. Phys. 34 (1926), 165–180. DOI 10.1007/BF01694897 | MR 1549403
[10] W.  R.  Scott: Group Theory. Prentice Hall, Englewood Cliffs, N.  J., 1964. MR 0167513 | Zbl 0126.04504
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