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Abstract

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 $V$).