# 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.