CQ *-algebras of measurable operators

We study, from a quite general point of view, a CQ*-algebra (X, 𝖀₀) possessing a sufficient family of bounded positive tracial sesquilinear forms. Non-commutative L²-spaces are shown to constitute examples of a class of CQ*-algebras and any abstract CQ*-algebra (X, 𝖀₀) possessing a sufficient family of bounded positive tracial sesquilinear forms can be represented as a direct sum of non-commutative L²-spaces.