Characterizations of Normality

In this section, we prove the Theorem of Haagerup asserting that every normal weight on a W *-algebra is the pointwise least upper bound of the normal positive forms it majorizes.

Let be a C*-algebra. A weight on is a mapping with the properties

The set

is a face of, the set

is a left ideal of, and the set

is a facial subalgebra of, hence can be extended uniquely to a positive linear form, still denoted by, on the *-algebra

A family of weights on is called sufficient if

and is called separating if

In particular, the weight is called faithful if

Let be a weight on the C*-algebra. The formula

defines a prescalar product on with the properties:

Let be the Hilbert space associated with with the scalar product It follows that there exists a *-representation, uniquely determined, such that

where denotes the canonical mapping. The *-representation is called the GNS representation or the standard representation associated with We remark that

Let be a W *-algebra. A weight on is called normal if

for every norm-bounded increasing net, and lower w-semicontinuous if the convex sets

are w-closed. An important result concerning weights on W *-algebras is the following characterization of normality:

Theorem (U. Haagerup). Let 𝜑 be a weight on the W *-algebra. The following statements are equivalent:

(i) is normal;

(ii) is lower w-semicontinuous;

Later (2.10, 5.8) we shall see that is normal if and only if it is a sum of normal positive forms, in accordance with the definition used in ([L], 10.14).

In Sections 1.4–1.7, we present some general results that will be used in the proof of the theorem; Sections 1.6–1.12 contain the main steps of the proof.