Random walks on Bratteli diagrams

TL;DR

This paper explores the relationship between hyperfinite von Neumann algebras and Poisson boundaries of time-dependent random walks, providing detailed proofs of key theorems linking operator algebras and probability theory.

Contribution

It offers a detailed explanation and proof of Connes and Woods' theorems connecting hyperfinite von Neumann algebras with Poisson boundaries, including models for conditional expectations.

Findings

Construction of a large class of states on hyperfinite von Neumann algebras

Ergodic decomposition of Markov measures via harmonic functions

Model for conditional expectations on finite dimensional C*-algebras

Abstract

In the eighties, A. Connes and E. J. Woods made a connection between hyperfinite von Neumann algebras and Poisson boundaries of time dependent random walks. The present paper explains this connection and gives a detailed proof of two theorems quoted there: the construction of a large class of states on a hyperfinite von Neumann algebra (due to A. Connes) and the ergodic decomposition of a Markov measure via harmonic functions (a classical result in probability theory). The crux of the first theorem is a model for conditional expectations on finite dimensional C*-algebras. The proof of the second theorem hinges on the notion of cotransition probability.