The asymptotic lift of a completely positive map

Starting with a unit-preserving normal completely positive map L: M -> M acting on a von Neumann algebra-or more generally a dual operator system-we show that there is a unique reversible system alpha: N -> N (i.e., a complete order automorphism of a dual operator system N) that captures all of the asymptotic behavior of L, called the asympiotic lift of L. This provides a noncommutative generalization of the Frobenius theorems that describe the asymptotic behavior of the sequence of powers of a stochastic n x n matrix. In cases where M is a von Neumann algebra, the asymptotic lift is shown to be a W*dynamical system (N, Z), and we identify (N, Z) as the tail flow of the minimal dilation of L. We are also able to identify the Poisson boundary of L as the fixed algebra N-alpha. In general, we show the action of the asymptotic lift is trivial iff L is slowly oscillating in the sense that [GRAPHICS] Hence alpha is often a nontrivial automorphism of N. The asymptotic lift of a variety of examples is calculated. (c) 2006 Elsevier Inc. All rights reserved.