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.