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

Parthasarathy, K. R. ; Sinha, K. B. (1999) Quantum Markov processes with a Christensen-Evans generator in a von Neumann algebra Bulletin of the London Mathematical Society, 31 (5). pp. 616-626. ISSN 0024-6093

Full text not available from this repository.

Official URL: http://blms.oxfordjournals.org/content/31/5/616

Related URL: http://dx.doi.org/10.1112/S0024609399005871


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-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 {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.

Item Type:Article
Source:Copyright of this article belongs to Oxford University Press.
ID Code:80973
Deposited On:02 Feb 2012 14:13
Last Modified:06 Jul 2012 06:15

Repository Staff Only: item control page