Structure theorem of the generator of a norm continuous completely positive semigroup: an alternative proof using Bures distance
A bounded linear map \(L\) on a \(C^*\)-algebra \(A\) generates a (uniformly continuous) CP-semigroup if and only if \(L\) is Hermitian (that is, \(L(a^*)=L(a)^*\)) and conditionally completely positive (that is, \[ \sum_{i,j}b_i^*L(a_i^*a_j)b_j ~\geq ~0 \] (finite sum), whenever \(\sum_ia_ib_i=0\)). See, for instance, [\textit{D. E. Evans} and \textit{J. T. Lewis}, Dilations of irreversible evolutions in algebraic quantum theory. Commun. Dublin Inst. Advanced Stud., Ser. A 24 (1977; Zbl 0365.46059)]. One of the main results of \textit{E. Christensen} and \textit{D. E. Evans} [J. Lond. Math. Soc., II. Ser. 20, 358--368 (1979; Zbl 0448.46040)] is that any such \(L\) has the form \[ L(a) ~=~ \Phi(a)+B^*a+aB, \] where \(\Phi\) is a CP-map into the strong closure \(A''\) of \(A\) acting nondegenerately on a Hilbert space \(H\) and where \(B\in A''\). In the present note, the author presents a new proof of the intermediate result that \(L\) has the stated form for a CP-map \(\Phi\) into \(B(H)\supset A\) and for a \(B\in B(H)\). This proof is based on earlier results on the Bures distance of CP-maps in [\textit{B. V. R. Bhat} and \textit{K. Sumesh}, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 16, No. 4, Article ID 1350031, 22 p. (2013; Zbl 1295.46043)].
scientific article; zbMATH DE number 6861648