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 a generalization of celebrated Dye’s theorem on orthoisomorphisms between von Neumann lattices to this new context. We show the existence of a one-to-one correspondence between the following maps: (1) quantum logic morphisms between the posets of tripotents preserving reflection \(u\rightarrow - u\) (2) maps between triples that preserve tripotents and are real linear on sets of elements with bounded range tripotents. In a more general description we show that quantum logic morphisms on structure of tripotents are given by a family of Jordan *-homomorphisms on Peirce 2-subspaces. By examples we demonstrate optimality of the results. Besides we show that the set of partial isometries with its partial order and orthogonality relation is a complete Jordan invariant for von Neumann algebras.