The close relationship between the notions of positive
forms and representations
for a C*-algebra A is one of the most basic
facts in the subject. In particular the weak
containment of representations is well understood in terms of positive
forms: given
a representation π of A in a Hilbert space H and
a positive form φ on A, its associated
representation π φ is weakly contained in π (that is,
ker π φ ⊃ ker π) if and only if φ
belongs to the weak* closure of the cone of all finite sums of
coefficients of π. Among
the results on the subject, let us recall the following ones. Suppose
that A is concretely
represented in H. Then every positive form φ on A
is the weak* limit of forms of the type
x [map ] [sum ]ki=1
〈ξi, xξi〉
with the ξi in H; moreover if A
is a von Neumann subalgebra
of ℒ(H) and φ is normal, there exists a sequence
(ξi)i [ges ] 1 in H
such that φ (x) = [sum ]i [ges ] 1
〈ξi, xξi〉
for all x.