Algebraically determined semidirect products

Abstract

Let G be a Polish (i.e., complete separable metric topological) group. Define G to be an algebraically determined Polish group if given any Polish group L and an algebraic isomorphism φ:L↦G, then φ is a topological isomorphism. The purpose of this paper is to prove a general theorem that gives useful sufficient conditions for a semidirect product of two Polish groups to be algebraically determined. This general theorem will provide a flowchart or recipe for proving that some special semidirect products are algebraically determined. For example, it may be used to prove that the natural semidirect product H⋊G, where H is the additive group of a separable Hilbert space and G is a Polish group of unitaries on H acting transitively on the unit sphere with −I∈G, is algebraically determined. An example of such a G is the unitary group of a separable irreducible C⁎-algebra with identity on H. Not all nontrivial semidirect products of Polish groups are algebraically determined, for it is known that the Heisenberg group H3(R) is a semidirect product of the form R2⋊θR1 and is not an algebraically determined Polish group.