The eigenvectors of semigroups of positive maps on von Neumann algebras

Abstract

The eigenvectors of an ergodic semigroup of linear normal positive unital maps on a von Neumann algebra are described. Moreover, it is shown by means of examples, that mere positivity of the maps in question is not sufficient for Frobenius theory as in S. Albeverio and R. H\{o}egh-Krohn, \emph{Frobenius theory of positive maps of von Neumann algebras}, Comm. Math. Phys. \textbf{64} (1978), 83--94, to hold.

The full version will appear in Ergodic Theory and Dynamical Systems (2009). A preliminary (and somewhat more elaborate) version was published at arXiv under the title "Frobenius theory fails for semigroups of positive maps on von Neumann algebras"