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 ${}^{\ast }$-isomorphisms. In particular, we prove that two von Neumann algebras without type I${}_1$ direct summands are Jordan ${}^{\ast }$-isomorphic if and only if their projection lattices are isometric. Our theorem extends the recent result for type I factors by G.P. Geh'er and P. \v{S}emrl, which is a generalization of Wigner’s theorem.