The asymptotic lift of a completely positive map

Abstract. 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. 1. introduction Throughout this paper we use the term UCP map to denote a normal unit-preserving completely positive map L: M1 → M2 of one dual operator system into another. While we are primarily concerned with the dynamical properties of UCP maps L: M → M that act on a von Neumann algebra M, it is appropriate to broaden that category to include UCP self-maps of more general dual operator systems. Stochastic n × n matrices P = (pij) describe the transition probabilities of n-state Markov chains. The asymptotic properties of the sequence of powers of the transition matrix govern the long-term statistical behavior of the process after initial transient fluctuations have died out ([Doo53], pp. 170–185). A stochastic n × n matrix P = (pij) gives rise to a UCP map of the commutative von Neumann algebra Cn by way of (Px)i = n∑ j=1 pijxj, 1 ≤ i ≤ n, x = (x1,..., xn) ∈ Cn