Trace Representation of Weights on Algebras of Unbounded Operators

Abstract

The purpose of this paper is to study when a weight ϕ on an O*-algebra M on a dense subspace D in a Hilbert space H is a trace weighted by a positive self-adjoint operator, that is, when there exists a positive self-adjoint operator Ω such that ϕ(X†X)=tr(X†*Ω)*X†*Ω whenever ϕ(X†X)<∞. We show that if M is QMP-solvable (that is, every strongly positive linear functional on M is a trace weighted by a positive trace class operator), then every sequentially monotonously regular weight ϕ on the positive cone M+≡{X∈M;X≥0} (ϕ=supnfn for some increasing sequence {fn} of strongly positive linear functionals on M) with the property that {X†ξ; ξ∈D, X∈M, and ϕ(X†X)<∞} is total in H is a trace weighted by a positive self-adjoint operator. More general results are obtained for an O*-algebra M containing the algebra F(D) generated by the set of all symmetric finite rank operators A satisfying AD⊂D. We also show that if ϕ is a weight on M+ and there exists an element N of M+ such that ϕ(N2)<∞ and N−1 is a bounded compact operator on H, then there exists a positive trace class operator T on H such that ϕ(X†X)=tr(T1/2X†)*T1/2X† whenever ϕ(X†N2X)<∞.