Quantum Markov processes with a Christensen-Evans generator in a von Neumann algebra

Abstract

Let A be a unital von Neumann algebra of operators on a complex separable Hilbert space H₀, and let {T_t, t ≥ 0} be a uniformly continuous quantum dynamical semigroup of completely positive unital maps on A. The infinitesimal generator L of {T_t} 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 {T_t} implies that L(1) = 0, and consequently K₀=iH-½RR, 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 {T_t}. 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.