Doob equivalence and non-commutative peaking for Markov chains

Xinxin Chen,Yifan Zhang,Langwen Hui,Christopher Linden,Adam Dor-On

doi:10.4171/jncg/444

Abstract

In this paper we show how questions about operator algebras constructed from stochastic matrices motivate new results in the study of harmonic functions on Markov chains. More precisely, we characterize coincidence of conditional probabilities in terms of (generalized) Doob transforms, which then leads to a stronger classification result for the associated operator algebras in terms of spectral radius and strong Liouville property. Furthermore, we characterize the non-commutative peak points of the associated operator algebra in a way that allows one to determine them from inspecting the matrix. This leads to a concrete analogue of the maximum modulus principle for computing the norm of operators in the ampliated operator algebras.

Highlights

The theory of Markov chains has applications in diverse areas of research such as group theory, dynamical systems, electrical networks and information theory
For the basic theory of Probability, Markov chains, random walks and their applications we refer the reader to [16, 18, 24]. These days, connections with operator algebras seem to manifest mostly in quantum information theory, where Markov chains are generalized to quantum channels
In this paper we resolve problems related to Markov chains motivated from studying operator algebras associated to stochastic matrices as in [14, 15]

Summary

Introduction

The theory of Markov chains has applications in diverse areas of research such as group theory, dynamical systems, electrical networks and information theory. Our characterization of Doob equivalence solves this problem and allows us to show (see Corollary 3.8) that if P and Q are irreducible stochastic matrices with ρ(P ) = ρ(Q) such that either P or Q are strong Liouville, the isometric isomorphism of T+(P ) and T+(Q) implies conjugacy of P and Q. This result is optimal in the sense that Example 3.6 shows we cannot weaken recurrence in [14, Theorem 3.11] to amenability without assuming strong Liouville property for one of the matrices.

Preliminaries

Doob equivalence

Findings

Non-commutative peaking