Topology Atlas Document # ppae-30


Compactifications of topological groups

Vladimir Uspenskij

Proceedings of the Ninth Prague Topological Symposium (2001) pp. 331-346

Every topological group G has some natural compactifications which can be a useful tool of studying G. We discuss the following constructions:

  1. the greatest ambit S(G) is the compactification corresponding to the algebra of all right uniformly continuous bounded functions on G;
  2. the Roelcke compactification R(G) corresponds to the algebra of functions which are both left and right uniformly continuous;
  3. 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 G-space X=MG is characterized by the following properties:

  1. X has no proper closed G-invariant subsets;
  2. for every compact G-space Y there exists a G-map X --> Y.
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.

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.