Asymptotic stability of completely positive maps

Abstract: We show that for every "locally finite" unit-preserving completely positive map $P$ acting on a $C^*$-algebra, there is a *-automorphism $\alpha$ of another $C^*$-algebra such that the two sequences $P, P2, P3,\dots$ and $\alpha, \alpha2,\alpha3,\dots$ have the same {\em asymptotic} behavior. The automorphism $\alpha$ is uniquely determined by $P$ up to conjugacy. Similar results hold for normal completely positive maps on von Neumann algebras, as well as for semigroups.

These developments are associated with our work on a new asymptotic spectral invariant for *-automorphisms of C*-algebras and von Neumann algebras. Time permitting, we will briefly discuss the application to dynamics.