Asymptotic Stability I: Completely Positive Maps

Details Asymptotic Stability I: Completely Positive Maps Asymptotic Stability I: Completely Positive Maps

2003-04-30

arxiv.org/abs/math/0304488v4

Mathematics

Operator Algebras

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Scientific paper

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.

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