  Generalizing von Neumann's result on type II$_1$ von Neumann algebras, we
characterize lattice isomorphisms between projection lattices of arbitrary von
Neumann algebras by means of ring isomorphisms between the algebras of locally
measurable operators. Moreover, we give a complete description of ring
isomorphisms of locally measurable operator algebras when the von Neumann
algebras are without type II direct summands.
