Dye's theorem for tripotents in von Neumann algebras and JBW* triples

Abstract: We study morphisms of the generalized quantum logic of tripotents in JBW*-triples and von Neumann algebras. Especially, we establish generalization of celebrated Dye's theorem on orthoisomorphisms between von Neumann lattices to this new context. We show one-to-one correspondence between maps on tripotents preserving orthogonality, orthogonal suprema, and reflection $u\to -u$, on one side, and their extensions to maps that are real linear on sets of elements with bounded range tripotents on the other side. In a more general description we show that quantum logic morphisms on tripotent structures are given by a family of Jordan -homomorphisms on 2-Peirce subspaces. By examples we exhibit new phenomena for tripotent morphisms that have no analogy for projection lattices and demonstrated that the above mention tripotent versions of Dye's theorem cannot be improved. On the other hand, in a special case of JBW$^$-algebras we can generalize Dye's result directly. Besides we show that structure of tripotents in C$^*$-algebras determines their projection poset and is a complete Jordan invariant for von Neumann algebras.