Jordan homomorphisms between algebras of measurable operators

Abstract

We prove that every self-adjoint algebra homomorphism between algebras of measurable operators is continuous and can be expressed as a sum of a self-adjoint algebra homomorphism as well as a self-adjoint algebra anti-homomorphism. We also provide sufficient conditions under which a surjective Jordan homomorphism between algebras of measurable operators is either an algebra homomorphism or an algebra anti-homomorphism.