Year: 2013
Type: article
Abstract: It is shown that a topological group G is topologically isomorphic to the isometry group of a (complete) metric space if and only if G coincides with its Ᏻ δ -closure in the Raȋkov completion of G (resp. if G is Raȋkov-complete).It is also shown that for every Polish (resp.compact Polish; locally compact Polish) group G there is a complete (resp.proper) metric d on X inducing the topology of X such that G is isomorphic to Iso(X, d), where X = ℓ 2 (resp.X = [0, 1] ω ; X = [0, 1] ω \ {point}).It is demonstrated that there are a separable Banach space E and a nonzero vector e ∈ E such that G is isomorphic to the group of all (linear) isometries of E which leave the point e fixed.Similar results are proved for arbitrary Raȋkov-complete topological groups.
Source: Pacific Journal of Mathematics
Author: Piotr Niemiec
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: bronze
OpenAlex ID: W2130198770