  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.
