Quantum Markov Processes with a Christensen-Evans Generator in a Von Neumann Algebra |
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Authors: | Parthasarathy K R; Sinha K B |
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Institution: | Indian Statistical Institute, Delhi Centre 7 S.J.S. Sansanwal Marg, New Delhi 110016, India |
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Abstract: | Let A be a unital von Neumann algebra of operators on a complexseparable Hilbert space H0, and let {Tt, t 0} be a uniformlycontinuous quantum dynamical semigroup of completely positiveunital maps on A. The infinitesimal generator L of {Tt} is abounded linear operator on the Banach space A. For any Hilbertspace K, denote by B(K) the von Neumann algebra of all boundedoperators on K. Christensen and Evans 3] have shown that Lhas the form formula] where is a representation of A in B(K) for some Hilbert spaceK, R: H0 K is a bounded operator satisfying the minimalitycondition that the set {(RX(X)R)u, uH0, XA} is totalin K, and K0 is a fixed element of A. The unitality of {Tt}implies that L(1) = 0, and consequently K0=iHR*R, whereH is a hermitian element of A. Thus (1.1) can be expressed as formula] We say that the quadruple (K, , R, H) constitutes the set ofChristensenEvans (CE) parameters which determine theCE generator L of the semigroup {Tt}. It is quite possible thatanother set (K', ', R', H') of CE parameters may determine thesame generator L. In such a case, we say that these two setsof CE parameters are equivalent. In Section 2 we study thisequivalence relation in some detail. 1991 Mathematics SubjectClassification 81S25, 60J25. |
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