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×nn×n matrix. In cases where M   is a von Neumann algebra, the asymptotic lift is shown to be a W∗W∗-dynamical system (N,Z)(N,Z), and we identify (N,Z)(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 NαNα. In general, we show the action of the asymptotic lift is trivial iff L is slowly oscillating in the sense thatlimn→∞‖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.