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,\dots and
\alpha, \alpha^2,\alpha^3,\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
one-parameter semigroups.
These results are counterparts of the classical
Perron-Frobenius theorem for operator algebras.