Trace representation of weights on partial O*-algebras

The purpose of this paper is to investigate when a weight ϕ on a partial O∗-algebra M is a trace weighted by a positive self-adjoint operator Ω, that is, ϕ(X† X) = tr(X†∗Ω)∗(X†∗Ω) whenever X ∈Ms.t. X† ∈ L(X) and ϕ(X† X) <∞. It is shown that ifMcontains the inverse N of a positive compact operator such that the weak multiplication N N is defined, then every weight ϕ on M+ satisfying ϕ(N N)<∞is a trace weighted by some positive trace operator.  2003 Elsevier Inc. All rights reserved