Trace Representation of Weights on Algebras of Unbounded Operators

The purpose of this paper is to study when a weight w on an OU-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 V such that w X†X.str X†UV.UX †UV whenever w 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 w on the positive cone Mqʹ XgM; XG04 w ssupn fn for some increasing sequence fn4 of strongly positive linear functionals on M. with the property that X †j ; j gD, XgM, and w X†X.-`4is total in H is a trace weighted by a positive self-adjoint operator. More general results are obtained for an OU-algebra M containing the algebra F D. generated by the set of all symmetric finite rank operators A satisfying A D;D. We also show that if w is a weight on Mqand there exists an element N of Mqsuch that w N2 .-` and Ny1 is a bounded compact operator on H, then there exists a positive trace class operator T on H such that w X†X.str T1r2X†.UT1r2X† whenever w X†N2X. -`.