Document # ppae-30
Compactifications of topological groups
Proceedings of the Ninth Prague Topological Symposium
Every topological group G has some natural compactifications which can
be a useful tool of studying G.
We discuss the following constructions:
The universal minimal compact G-space X=MG is characterized by
the following properties:
- the greatest ambit S(G) is the compactification corresponding
to the algebra of all right uniformly continuous bounded functions on G;
the Roelcke compactification R(G) corresponds to the algebra
of functions which are both left and right uniformly continuous;
the weakly almost periodic compactification W(G) is the
enveloping compact semitopological semigroup of G (`semitopological'
means that the multiplication is separately continuous).
A group G is extremely amenable, or has the fixed point on
compacta property, if MG is a singleton.
We discuss some results and questions by V. Pestov and E. Glasner on
extremely amenable groups.
- X has no proper closed G-invariant subsets;
for every compact G-space Y there exists a G-map X --> Y.
The Roelcke compactifications were used by M. Megrelishvili to prove that
W(G) can be a singleton. They can be used to prove that certain groups
are minimal. A topological group is minimal if it does not admit a
strictly coarser Hausdorff group topology.
Mathematics Subject Classification. 22A05 (22A15 22F05 54D35 54H15 57S05).
Keywords. topological groups, compactifications, universal minimal
compact $G$-space, extremely amenable group, Roelcke compactification.
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Comments. This article is in final form.
Copyright © 2002
Charles University and
Published April 2002.