Abstract We prove an optimal version of Uhlhorn’s generalization of Wigner’s unitary–antiunitary theorem. The main tool in our proof is Gleason’s theorem.
It is well known that transformations of $ \mathbb{C}^n $ preserving the standard inner product are unitary transformations. In this paper, all bijective transformations of isotropic sets of $ \mathbb{C}P^{n} $ preserving $ H $-orthogonality in both directions, called $ H $-orthogonal transformations, have been determined. This is a generalization of Uhlhorn's version of Wigner's unitary-antiunitary theorem. The group of $ H $-orthogonal transformations on some other sets of $ \mathbb{C}P^{n} $ were also determined.