Noncommutative Poisson boundaries, ultraproducts and entropy

TL;DR

This paper develops a framework for noncommutative Poisson boundaries using ultraproducts of von Neumann algebras and applies it to establish key theorems on noncommutative entropy and boundary properties.

Contribution

It introduces a novel construction of noncommutative Poisson boundaries via ultraproducts and completes proofs of fundamental theorems in noncommutative entropy theory.

Findings

Constructed noncommutative Poisson boundaries using ultraproducts.

Completed proofs of Kaimanovich-Vershik's theorems on noncommutative entropy.

Established equivalences related to amenability and boundary triviality.

Abstract

We construct the noncommutative Poisson boundaries of tracial von Neumann algebras through the ultraproducts of von Neumann algebras. As an application of this result, we complete the proof of Kaimanovich-Vershik's fundamental theorems regarding noncommutative entropy. We also prove the Amenability-Trivial Boundary equivalence and Choquet-Deny-Type I equivalence for tracial von Neumann algebras.