Orthomorphisms of a commutative W*-algebra

Abstract

AbstractIf M is a commutative W*-algebra of operators and if ReM is the Dedekind complete Riesz space of self-adjoint elements of M, then it is shown that the set of densely defined self-adjoint transformations affiliated with ReM is a Dedekind complete, laterally complete Riesz algebra containing ReM as an order dense ideal. The Riesz algebra of densely defined orthomorphisms on ReM is shown to coincide with , and via the vector lattice Randon-Nikodym theorem of Luxemburg and Schep, it is shown that the lateral completion of ReM may be identified with the extended order dual of ReM.