Isometries between projection lattices of von Neumann algebras

We investigate surjective isometries between projection lattices of two von Neumann algebras. We show that such a mapping is characterized by means of Jordan *-isomorphisms. In particular, we prove that two von Neumann algebras without type I-1 direct summands are Jordan *-isomorphic if and only if their projection lattices are isometric. Our theorem extends the recent result for type I factors by G.P. Geher and P. Semrl, which is a generalization of Wigner's theorem. (C) 2018 Elsevier Inc. All rights reserved.