On non-commutative transfer operators and Radon-Nikodym derivatives

TL;DR

This paper explores the properties of non-commutative Ruelle transfer operators on operator algebras, focusing on eigenvalues, eigenfunctions, and convergence, with applications to quantum spin chains.

Contribution

It introduces a framework connecting non-commutative transfer operators with Radon-Nikodym derivatives and extends the Ruelle-Perron-Frobenius theorem to infinite-dimensional settings.

Findings

Existence of a largest positive eigenvalue under certain conditions

Uniform convergence of transfer operator iterates established

Connection between transfer operators and quantum spin chain analysis

Abstract

We study relations between non-commutative Ruelle transfer operators over the C$^*$-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.