  We show that a map between projection lattices of semi-finite von Neumann
algebras can be extended to a Jordan $*$-homomorphism between the von Neumann
algebras if this map is defined in terms of the support projections of images
(under the linear map) of projections and the images of orthogonal projections
have orthogonal support projections. This has numerous fundamental applications
in the study of isometries and composition operators on quantum symmetric
spaces and is of independent interest, since it provides a partial
generalization of Dye's Theorem without the requirement that the initial von
Neumann algebra be free of type $I_2$ summands.
