Quasi *-algebras of measurable operators

TL;DR

This paper introduces CQ*-algebras, a class of Banach quasi *-algebras exemplified by non-commutative L^p spaces and measurable operators, establishing their properties and representation theory.

Contribution

It characterizes CQ*-algebras, demonstrates their structure in non-commutative L^p spaces, and provides a representation theorem for abstract CQ*-algebras with sufficient positive forms.

Findings

Non-commutative L^p spaces are CQ*-algebras.

CQ*-algebras of measurable operators are constructed.

Representation theorem for CQ*-algebras with positive sesquilinear forms.

Abstract

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.