Compactifications of topological groups
By V. Uspenskij. 2002
65 citations recorded by the source corpus.
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 $X\to Y$. 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.
Subfields: Computational Theory and Mathematics, Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/8da9491139b1401f46375b1f5d66760f698daf3d
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In Geometry and Topology
In Mathematical Physics
Citations in the corpus
- On subgroups of minimal topological groups (2000, background): “There is another universal topological group with a countable base, namely the group Iso(U) of all self-isometries of the Urysohn universal metric space U [47], [27, Theorem 2.3.1], [ 50, Theorem 6.1 ].”
- The Urysohn universal metric space is homeomorphic to a Hilbert space (2003, methodology): “Katětov’s construction was used in [10, 11, 12] to prove that the topological group Is (U) of all isometries of U is universal, in the sense that it contains an isomorphic copy of every topological group with a countable base.”