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Noncommutative Poisson boundaries, ultraproducts and entropy

Shuoxing Zhou

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.

Noncommutative Poisson boundaries, ultraproducts and entropy

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.
Paper Structure (5 sections, 17 theorems, 166 equations)

This paper contains 5 sections, 17 theorems, 166 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Entropy of noncommutative Poisson boundaries
  4. Trivial boundaries of amenable von Neumann algebras
  5. Choquet-Deny property of von Neumann algebras

Key Result

Theorem A

Let $\varphi\in \mathcal{S}_\tau(B(L^2(M,\tau))$ be a normal regular strongly generating hyperstate such that $H(\varphi)<+\infty$. Take an increasing sequence $\{a_n\}\subset\mathbb{R}$ satisfying $\frac{1}{2}\leq a_n< 1$ and $\lim_n\limits a_n=1$, and an ultrafilter $\omega\in \beta \mathbb{N}\set Then the $\varphi$-Poisson boundary with canonical hyperstate $(\mathcal{B}_\varphi,\zeta)$ can be

Theorems & Definitions (37)

  • Theorem A
  • Theorem B
  • Theorem C
  • Theorem 3.1
  • proof
  • Lemma 3.2
  • proof
  • Theorem 3.3
  • proof
  • Definition 3.4
  • ...and 27 more