SOME REMARKS ON THE CONE OF COMPLETELY POSITIVE MAPS BETWEEN VON NEUMANN ALGEBRAS

Abstract

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 ]^k_i=1 〈ξ_i, xξ_i〉 with the ξ_i in H; moreover if A is a von Neumann subalgebra of ℒ(H) and φ is normal, there exists a sequence (ξ_i)_{i [ges ] 1} in H such that φ (x) = [sum ]_{i [ges ] 1} 〈ξ_i, xξ_i〉 for all x.