The eigenvectors of semigroups of positive maps on von Neumann algebras

TL;DR

This paper characterizes eigenvectors of ergodic semigroups of positive maps on von Neumann algebras and demonstrates that positivity alone does not guarantee Frobenius theory applicability.

Contribution

It provides a description of eigenvectors for ergodic semigroups and shows limitations of Frobenius theory under mere positivity of maps.

Findings

Eigenvectors of ergodic semigroups are characterized.

Positivity alone is insufficient for Frobenius theory to hold.

Examples illustrate the limitations of positivity in this context.

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.