I have now expanded a fair bit, written out the actual statement (starting in a new section “Preliminaries” here) and also adding an “Idea”-section (here).

]]>The statement in the entry was missing the condition that the map sends lines to lines, i.e. that it is a map of projective spaces. I have made a quick edit, but no time for more for the moment.

]]>Is this theorem true at all? Let the function $f$ map $(z_1,z_2,\dots)$ to $(\overline z_1, z_2,\dots)$ where the coordinates refer to a Hilbert basis. This is a surjective norm-preserving transformation but isn’t unitary or anti-unitary even up to phase.

]]>.

]]>and added this one (thanks to David R.):

- C. S. Sharma and D. F. Almeida,
*Additive isometries on a quaternionic Hilbert space*, Journal of Mathematical Physics 31, 1035 (1990) (doi:10.1063/1.528779)

added pointer to

- Valentine Bargman,
*Note on Wigner’s theorem on symmetry transformations*, Journal of Mathematical Physics 5.7 (1964): 862-868 (doi:10.1063/1.1704188)