Trace representation of weights on partial O\(^{*}\)-algebras (Q1826765)

Many important examples of states \(f\) in quantum physics are trace functionals, i.e., \(f(X)=\text{tr }\overline{TX}\) with a certain trace operator \(T\). Along these lines, the following quantum moment problem (QMP) is of some interest: under which conditions is every strongly positive linear functional defined on a (partial) \(O^*\)-algebra a trace functional? If the answer is affirmative, a (partial) \(O^*\)-algebra is called QMP-solvable. It is known: if (i) \({\mathcal M}\) is a self-adjoint \(O^*\)-algebra which contains the inverse of a compact operator, or if (ii) \({\mathcal M}\) is an \(O^*\)-algebra whose domain \({\mathcal D}\), endowed with the graph topology \(t_{\mathcal M}\), is a Fréchet-Montel space, then the \(O^*\)-algebra \({\mathcal M}\) is QMP-solvable [see \textit{K. Schmüdgen}, ``Unbounded operator algebras and representation theory'' (Oper. Theory, Adv. Appl. 37) (1990; Zbl 0697.47048)]. These results were generalized from the case of positive linear functionals \(f\) to weights \(\varphi\) on \(O^*\)-algebras in [\textit{A. Inoue} and \textit{K.-D. Kürsten}, J. Math. Anal. Appl. 247, 136--155 (2000; Zbl 0969.47051)]. The paper under review is aimed at a generalization of the Inoue-Kürsten's result to the case of partial \(O^*\)-algebras. It is shown that if the partial \(O^*\)-algebra \({\mathcal M}\) contains the inverse \(N\) of some positive compact operator such that \(N\square N\) is defined, then every weight \(\varphi\) on \({\mathcal M}_+=\{X\in{\mathcal M}\,| \, X\geq 0 \;\text{ iff }\; (X\xi ,\xi )\geq 0, \forall\, \xi\in{\mathcal D}\}\) satisfying \(\varphi (N\square N)<\infty\) is trace weighted by some positive trace operator \(\Omega\) (i.e., \(\varphi (X^\dagger\square X)=\text{ tr}\,(\Omega X^\dagger)^*\overline{\Omega X^\dagger}\) for certain \(X\in{\mathcal M}\)). Furthermore, trace representations of uniformly continuous linear functionals and of regular weights are considered.