Compactifications of semigroups and semigroup actions
by
Michael Megrelishvili
Bar-Ilan University, Israel
Coauthors: Vladimir Uspenskij (Ohio University)

Let's say that an action of a topological semigroup S on X is compactifiable if this action is a restriction of a jointly continuous action of S on a Hausdorff compact space Y. A topological semigroup is compactifiable if the left action of S on itself is compactifiable.

It is well known and important that every Hausdorff topological group is compactifiable (in the sense defined above). This result cannot be extended to the class of Tychonoff (even, locally compact Polish) topological semigroups. However, we show that many natural constructions in Topology and Analysis lead to compactifiable semigroups and actions.

We prove that the semigroup C(K, K) of all continuous selfmaps on the Hilbert cube K is a universal second countable compactifiable semigroup (semigroup version of Uspenskij's theorem). Moreover, the Hilbert cube K under the action of C(K, K) is universal in the realm of all compactifiable S-flows (with compactifiable S) X where both X and S are second countable.

Date received: June 4, 2003