On the structure of a cone of normal unbounded completely positive maps (Q1966188)

This paper deals with the special class of cones of unbounded completely positive maps, originally introduced by \textit{A. M. Chebotarev} [Math. Notes 52, No. 4, 1067-1077 (1992); translation from Mat. Zametki 52, No. 4, 112-127 (1992; Zbl 0820.47040)]. Let \({\mathcal H}\) be a separable Hilbert space and \({\mathcal F}\) be a topological vector space densely and continuously embedded in \({\mathcal H}\). \({\mathcal B}({\mathcal H})\) is the Banach algebra of all bounded linear operators in \({\mathcal H}\). Let \(CP_n({\mathcal H})\) denote the cone of all completely positive normal linear maps \(R:{\mathcal B}({\mathcal H})\to{\mathcal B}({\mathcal H})\). Since the family of pseudometrics \(\Theta_{A,B}\) \((A\subset{\mathcal F}, B\subset{\mathcal B}({\mathcal H}))\) depending on \({\mathcal F}\) induces the structure of a uniform topological space in \(CP_n({\mathcal H})\) (implying the existence of completion), \(CP_{n*}({\mathcal F})\) can be defined as the completion of \(CP_n({\mathcal H})\) with respect to the pseudometric \(\Theta_{A,B}\). The author proves constructive characterizations of \(CP_{n*}({\mathcal F})\) for the two cases, separately, where 1) \({\mathcal F}\subset{\mathcal H}\) is a Hilbert space, and 2) \({\mathcal F}\) is a countably Hilbert space, identified with a projective limit of Hilbert spaces \({\mathcal H}_n\). The key idea for the proof is based on the same methodology and techniques as the \textit{K. Kraus} theorem [Ann. Phys. 64, 311-335 (1971)], which provides a constructive characterization of the cone \(CP_n({\mathcal H})\). For related physical topics on the Lindblad equation with unbounded time-dependent coefficients, see \textit{A. M. Chebotarev}, \textit{J. García} and \textit{R. Quezada} [RIMS Kokyuroku 1035, 44-65 (1998)].