Extension of projection mappings

Abstract

We provide sufficient conditions for a map between projection lattices of semi-finite von Neumann algebras to be extended to a Jordan *-homomorphism. These conditions are sufficiently general for the procedures developed herein to have numerous fundamental applications in the study of isometries and composition operators on quantum symmetric spaces. Furthermore, these techniques are of independent interest, since they provide a partial generalization of Dye’s theorem without the requirement that the initial von Neumann algebra be free of type I ₂ summands.