QUANTUM MARKOV PROCESSES WITH A CHRISTENSEN–EVANS GENERATOR IN A VON NEUMANN ALGEBRA

Abstract

Let [Ascr ] be a unital von Neumann algebra of operators on a complex separable Hilbert space ([Hscr ]₀, and let {T_t, t [ges ] 0} be a uniformly continuous quantum dynamical semigroup of completely positive unital maps on [Ascr ]. The infinitesimal generator [Lscr ] of {T_t} is a bounded linear operator on the Banach space [Ascr ]. For any Hilbert space [Kscr ], denote by [Bscr ]([Kscr ]) the von Neumann algebra of all bounded operators on [Kscr ]. Christensen and Evans [3] have shown that [Lscr ] has the form

formula here

where π is a representation of [Ascr ] in [Bscr ]([Kscr ]) for some Hilbert space [Kscr ], R: [Hscr ]₀ → [Kscr ] is a bounded operator satisfying the ‘minimality’ condition that the set {(RX−π(X)R)u, u∈[Hscr ]₀, X∈[Ascr ]} is total in [Kscr ], and K₀ is a fixed element of [Ascr ]. The unitality of {T_t} implies that [Lscr ](1) = 0, and consequently K₀ = iH−½R*R, where H is a hermitian element of [Ascr ]. Thus (1.1) can be expressed as

formula here

We say that the quadruple ([Kscr ], π, R, H) constitutes the set of Christensen–Evans (CE) parameters which determine the CE generator [Lscr ] of the semigroup {T_t}. It is quite possible that another set ([Kscr ]′, π′, R′, H′) of CE parameters may determine the same generator [Lscr ]. 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.