ASYMPTOTIC STABILITY I: COMPLETELY POSITIVE MAPS

Abstract

We show that for every "locally finite" unit-preserving completely positive map P acting on a C*, there is a corresponding *-automorphism α of another unital C*-algebra such that the two sequences P, P2, P3, … and α, α2, α3, … have the same asymptotic behavior. The automorphism α 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 operator algebraic counterparts of the classical theory of Perron and Frobenius on the structure of square matrices with nonnegative entries.