Isometry groups among topological groups (Q388800)

The main result is that every Polish group \(G\) is isomorphic to the isometry group of the Hilbert space \(l_2\) with respect to some complete compatible metric (i.e., a complete metric inducing the topology). If \(G\) is moreover compact, then \(l_2\) may be replaced with the Hilbert cube \(Q\), and if \(G\) is locally compact, with \(Q\) minus a point. This is a more concrete version of results due to \textit{S. Gao} and \textit{A. S. Kechris} [Mem. Am. Math. Soc. 766, 78 p. (2003; Zbl 1012.54038)], \textit{J. Melleray} [Proc. Am. Math. Soc. 136, No. 4, 1451--1455 (2008; Zbl 1134.54013)], and \textit{M. Malicki} and \textit{S. Solecki} [Math. Proc. Camb. Philos. Soc. 146, No. 1, 67--81 (2009; Zbl 1170.54013)], with a unified proof using simpler methods. Another result in a similar spirit says that again for a Polish group \(G\), there exists a separable real Banach space such that \(G\) is isomorphic to the group of linear isometries fixing some nonzero vector, endowed with the topology of pointwise convergence. All these results are generalized to (Raĭkov) complete topological groups of topological weight \(\beta \geq \aleph_0\) and a Hilbert space of dimension \(\beta\), respectively, a Banach space of topological weight \(\beta\).NEWLINENEWLINEA topological group \(G\) is said to be \({\mathcal G}_\delta\)-complete if it is complete in the group topology generated by its \({\mathcal G}_\delta\)-sets. This property is characterized in various ways and is shown to be pretty common; e.g., metrizable groups enjoy it. A topological group \(G\) is proved to be isomorphic to the full isometry group of some metric space if and only it is \({\mathcal G}_\delta\)-complete. The metric space may be chosen to be complete if and only if \(G\) itself is complete. The metric spaces can be chosen to have the same weight as \(G\). An important auxiliary result is that every closed subgroup of the isometry group of a metric space is isomorphic to the full isometry group of some other metric space.