Trace representation of weights on algebras of unbounded operators (Q1576972)

The authors study the problem of the existence of a positive self-adjoint operator \(\Omega\) in a Hilbert space \({\mathcal H}\) such that \(\varphi(X^\dag X)= \text{tr}(X^{\dag*}\Omega)^* \overline{X^{\dag*}\Omega}\) whenever \(\varphi(X^\dag X)< \infty\), where \(\varphi\) is a weight on a \({\mathcal O}^*\)-algebra \({\mathcal M}\) on a dense subspace \({\mathcal D}\) in \({\mathcal H}\); in other words, when the weight \(\varphi\) is a trace weighted by a positive self-adjoint operator (here \(X^\dag\) is the restriction of \(X^*\) to \({\mathcal D}\)). The authors give an answer to this problem when assuming several technical conditions on both \({\mathcal M}\) and \(\varphi\). More general results are obtained when \({\mathcal M}\) is an \({\mathcal O}^*\)-algebra containing the algebra generated by the set of all symmetric finite rank operators \(A\) satisfying \(A{\mathcal D}\subset{\mathcal D}\).