CQ -algebras of measurable operators

Abstract

We study, from a quite general point of view, a CQ*-algebra (X, A0) possessing a sufficient family of bounded positive tracial sesquilinear forms. Non-commutative L^2 -spaces are shown to constitute examples of a class of CQ*-algebras and any abstract CQ*-algebra (X, A0) possessing a sufficient family of bounded positive tracial sesquilinear forms can be represented as a direct sum of non-commutative L^2-spaces.