Trace representation of weights on partial O*-algebras

The purpose of this paper is to investigate when a weight <f>ϕ</f> on a partial O&ast;-algebra <f>M</f> is a trace weighted by a positive self-adjoint operator <f>Ω</f>, that is, <f>ϕ(X†□X)=tr(X†&ast;Ω)&ast;<ovl type="bar" STYLE="S">(X†&ast;Ω)</ovl></f> whenever <f>X∈M</f> s.t. <f>X†∈L(X)</f> and <f>ϕ(X†□X)<∞</f>. It is shown that if <f>M</f> contains the inverse <f>N</f> of a positive compact operator such that the weak multiplication <f>N□N</f> is defined, then every weight <f>ϕ</f> on <f>M+</f> satisfying <f>ϕ(N□N)<∞</f> is a trace weighted by some positive trace operator. [Copyright &y& Elsevier]