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K. R. Parthasarathy, K. B. Sinha, Quantum Markov Processes with a Christensen–Evans Generator in a Von Neumann Algebra, Bulletin of the London Mathematical Society, Volume 31, Issue 5, September 1999, Pages 616–626, https://doi.org/10.1112/S0024609399005871
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Abstract
Let A be a unital von Neumann algebra of operators on a complex separable Hilbert space H0, and let {Tt, t ≥ 0} be a uniformly continuous quantum dynamical semigroup of completely positive unital maps on A. The infinitesimal generator L of {Tt} is a bounded linear operator on the Banach space A. For any Hilbert space K, denote by B(K) the von Neumann algebra of all bounded operators on K. Christensen and Evans [3] have shown that L has the form
[formula]
where π is a representation of A in B(K) for some Hilbert space K, R: H0 → K is a bounded operator satisfying the ‘minimality’ condition that the set {(RX−π(X)R)u, u∈H0, X∈A} is total in K, and K0 is a fixed element of A. The unitality of {Tt} implies that L(1) = 0, and consequently K0=iH−½R*R, where H is a hermitian element of A. Thus (1.1) can be expressed as
[formula]
We say that the quadruple (K, π, R, H) constitutes the set of Christensen–Evans (CE) parameters which determine the CE generator L of the semigroup {Tt}. It is quite possible that another set (K′, π′, R′, H′) of CE parameters may determine the same generator L. In such a case, we say that these two sets of CE parameters are equivalent. In Section 2 we study this equivalence relation in some detail. 1991 Mathematics Subject Classification 81S25, 60J25.
