We show that for many natural topological groups G (including the group ℤ of integers) there exist compact metric G-spaces (cascades for G = 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 Θ(H) of all contractive linear operators on a Hilbert space H (though S admits an embedding into the compact semigroup Θ(V) for certain reflexive V).
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm110-2-5,
author = {Michael Megrelishvili},
title = {Reflexively representable but not Hilbert representable compact flows and semitopological semigroups},
journal = {Colloquium Mathematicae},
volume = {111},
year = {2008},
pages = {383-407},
zbl = {1158.37008},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm110-2-5}
}
Michael Megrelishvili. Reflexively representable but not Hilbert representable compact flows and semitopological semigroups. Colloquium Mathematicae, Tome 111 (2008) pp. 383-407. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm110-2-5/