The infinite unitary and related groups are algebraically determined Polish groups

Let G be a Polish group. G is said to be an algebraically determined Polish group if for any Polish group H and algebraic isomorphism φ:H→G we have that φ is a topological isomorphism. Let H be a separable infinite dimensional complex Hilbert space. The purpose of this paper is to prove that the unitary group and the complex isometry group of H are algebraically determined Polish groups. Similar results hold for most (but not all) of the finite dimensional complex isometry groups but are false for the finite dimensional unitary groups.