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 ${\mathrm{I}}_1$ 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. Gehér and P. ?emrl, which is a generalization of Wigner's theorem.