Automorphisms of spectral lattices of positive contractions on von Neumann algebras

Abstract

We show that any spectral lattice orthoautomorphism of the structure of positive contractions on a von Neumann algebra, endowed with the spectral order and orthogonality relation, that preserves scalar operators is a composition of function calculus with natural transformation of spectral resolutions given by an orthoautomorphism of the projection lattice. In case of von Neumann algebras without Type I 2 direct summand any such a map is a composition of function calculus with Jordan * -automorphism. This result is a parallel to famous Dye's theorem and generalizes so far known results on preservers of the spectral order on matrices and operators. Moreover general spectral lattice automorphism are studied.