for a separable Hilbert space $H$ it is known that the unitary group $U(H)$ is contractible, both for the norm topology (Kuiper's theorem) and for the strong operator topology (Dixmier and Douady, 1963). Note that the strong operator topology coincides with the compact open topology in this case.

Does somebody know whether contractibility of the unitary group of at least some special pre Hilbert spaces equipped with the strong operator topology or the compact open topology still holds?

The example I have in mind is the underlying algebraic Fock space $\bigwedge PH \oplus (PH^\perp)^*$ of the Fock space.

Thanks in advance, StW


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