On non-commutative transfer operators and Radon-Nikodym derivatives

Abstract

We study relations between non-commutative Ruelle transfer operators over the C${}^{\ast }$-algebra $B(\mathcal{H})$ of linear bounded operators over separable Hilbert spaces $\mathcal{H}$ (infinite-dimensional) and other completely positive maps. Transfer operators possess a simple description in terms of the so called non-commutative Radon-Nikodym derivatives. We describe the problem of existence of a largest positive eigenvalue associated to a positive eigenfunction and uniform convergence of sequences of iterates of transfer operators over $B(\mathcal{H})$. Part of the proof related to the Ruelle-Perron-Frobenius theorem is obtained by adapting results from quantum spin chain analysis.

arXiv commentThis paper has been withdrawn by the author due to gaps in the main argument, therefore this work is incomplete at the moment