Reflexively representable but not Hilbert representable compact flows and semitopological semigroups

Michael Megrelishvili¹

¹ Department of Mathematics Bar-Ilan University 52900 Ramat-Gan, Israel

Colloquium Mathematicum, Tome 110 (2008) no. 2, pp. 383-407.

Résumé

We show that for many natural topological groups $G$ (including the group ${\mathbb Z}$ of integers) there exist compact metric $G$-spaces (cascades for $G={\mathbb Z}$) which are reflexively representable but not Hilbert representable. This answers a question of T. Downarowicz. The proof is based on a classical example of W. Rudin and its generalizations. A~crucial step in the proof is our recent result which states that every weakly almost periodic function on a compact $G$-flow $X$ comes from a $G$-representation of $X$ on reflexive spaces. We also show that there exists a monothetic compact metrizable semitopological semigroup $S$ which does not admit an embedding into the semitopological compact semigroup ${\mit\Theta}(H)$ of all contractive linear operators on a Hilbert space $H$ (though $S$ admits an embedding into the compact semigroup ${\mit\Theta}(V)$ for certain reflexive $V$).

Détail

DOI : 10.4064/cm110-2-5

Keywords: many natural topological groups including group mathbb integers there exist compact metric g spaces cascades mathbb which reflexively representable hilbert representable answers question downarowicz proof based classical example rudin its generalizations crucial step proof recent result which states every weakly almost periodic function compact g flow comes g representation reflexive spaces there exists monothetic compact metrizable semitopological semigroup which does admit embedding semitopological compact semigroup mit theta contractive linear operators hilbert space though admits embedding compact semigroup mit theta certain reflexive