Topology Atlas
Document # ppae30
Compactifications of topological groups
Vladimir Uspenskij
Proceedings of the Ninth Prague Topological Symposium
(2001)
pp. 331346
Every topological group G has some natural compactifications which can
be a useful tool of studying G.
We discuss the following constructions:
 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).
The universal minimal compact Gspace X=M_{G} is characterized by
the following properties:
 X has no proper closed Ginvariant subsets;

for every compact Gspace Y there exists a Gmap X > Y.
A group G is extremely amenable, or has the fixed point on
compacta property, if M_{G} is a singleton.
We discuss some results and questions by V. Pestov and E. Glasner on
extremely amenable groups.
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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Copyright © 2002
Charles University and
Topology Atlas.
Published April 2002.