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 α :N →N (i.e., a complete order automorphism α of a dual operator system N) that captures all of the asymptotic behavior of L, called the asymptotic 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 × 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 Nα. In general, we show the action of the asymptotic lift is trivial iff L is slowly oscillating in the sense that lim n→∞ ρ ◦ Ln+1 −ρ ◦ Ln = 0, ρ∈M∗. Hence α is often a nontrivial automorphism of N. The asymptotic lift of a variety of examples is calculated. © 2006 Elsevier Inc. All rights reserved.