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Does the isometry group of a real separable infinite-dimensional Hilbert space have two connected components? Or, conversely, is the there even a Kuiper's theorem in the real case?

How does the isometry group of a real infinite-dimensional Hilbert separable look like?

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It is contractible, according to Kuiper himself who also writes:

Putnam and Wintner [5, 6] proved with the help of spectral resolutions that $U_{\mathbf R}$ and hence $GL_{\mathbf R}$ are connected.

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