Asymptotic Stability I: Completely Positive Maps

We show that for every "locally finite" unit-preserving completely positive map P acting on a C*-algebra, there is a corresponding *-automorphism \alpha of another unital C*-algebra such that the two sequences P, P^2,P^3,... and \alpha, \alpha^2,\alpha^3,... 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 one-parameter semigroups. These results can be viewed as operator algebraic counterparts of the classical Perron-Frobenius theorem on the structure of square matrices with nonnegative entries.