Transition probability preserving maps on a Grassmann space in a semifinite factor (Q2306216)

Suppose that \(\mathcal{H}\) is a Hilbert space and \(\mathrm{Tr}\) is the canonical trace on \(B(\mathcal{H})\). Let \(\mathscr{P}_1\) be the Grassmann space of rank-one projections in \(B(\mathcal{H})\). Wigner's theorem states that if \(\varphi: \mathscr{P}_1 \to \mathscr{P}_1\) is a surjective map preserving the transition probability, i.e., \(\mathrm{Tr}(PQ) = \mathrm{Tr}(\varphi(P)\, \varphi(Q))\) for all \(P, Q \in \mathscr{P}_1\), then there is either a \(*\)-isomorphism or a \(*\)-anti-isomorphism \(\sigma: B(\mathcal{H}) \to B(\mathcal{H})\) such that \(\varphi(P) = \sigma(P)\) for every \(P \in \mathscr{P}_1\). The main aim of the paper under review is to generalize Wigner's theorem by providing a characterization of surjective transition probability preserving transformations on the Grassmann space of infinite projections in an infinite semifinite factor.