Decomposition of an order isomorphism between matrix-ordered Hilbert spaces

Abstract

The purpose of this note is to show that any order isomorphism between noncommutative L 2 L^{2} -spaces associated with von Neumann algebras is decomposed into a sum of a completely positive map and a completely co-positive map. The result is an L 2 L^{2} version of a theorem of Kadison for a Jordan isomorphism on operator algebras.