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Compactifications of topological groups | | Vladimir Uspenskij
; | | Date: |
10 Apr 2002 | | Journal: | Proceedings of the Ninth Prague Topological Symposium, (Prague, 2001), pp. 331--346, Topology Atlas, Toronto, 2002 | | Subject: | General Topology MSC-class: 22A05 (Primary) 22A15, 22F05, 54D35, 54H15, 57S05 (Secondary) | math.GN | | Abstract: | 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 envelopping compact semitopological semigroup of $G$ (`semitopological’ means that the multiplication is separately continuous). The universal minimal compact $G$-space $X=M_G$ 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 $X o Y$. 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. | | Source: | arXiv, math.GN/0204144 | | Services: | Forum | Review | PDF | Favorites | |
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