  Non-commutative $L^p$-spaces are shown to constitute examples of a class of
Banach quasi *-algebras called CQ*-algebras. For $p\geq 2$ they are also proved
to possess a {\em sufficient} family of bounded positive sesquilinear forms
satisfying certain invariance properties. CQ *-algebras of measurable operators
over a finite von Neumann algebra are also constructed and it is proven that
any abstract CQ*-algebra $(\X,\Ao)$ possessing a sufficient family of bounded
positive tracial sesquilinear forms can be represented as a CQ*-algebra of this
type.
