Structure theorem of the generator of a norm continuous completely positive semigroup: an alternative proof using Bures distance
type: Item
instance of: scholarly article
title: Structure theorem of the generator of a norm continuous completely positive semigroup: an alternative proof using Bures distance
zbMATH Open document ID: 1406.46056
author: Mithun Mukherjee
published in: Positivity
publication date: 2018-04-19T00:00:00Z
review text: 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)].
reviewed by: Michael Skeide
Mathematics Subject Classification ID: 46L57
Mathematics Subject Classification ID: 46L55
Mathematics Subject Classification ID: 46L53
Mathematics Subject Classification ID: 46L08
zbMATH DE Number: 6861648
zbMATH Keywords: \(C^*\)-algebras
zbMATH Keywords: von Neumann algebras
zbMATH Keywords: quantum dynamical semigroups
zbMATH Keywords: Bures distance
MaRDI profile type: Publication
full work available at URL: s11117-017-0494-9
OpenAlex ID: W2610862216
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DOI: 10.1007/S11117-017-0494-9
description: scientific article; zbMATH DE number 6861648