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.

Remark 5.7 has been added to show Choquet-Deny property of groups can not be perfectly extended to tracial von Neumann algebras

References in corpus (2)

• Choquet-Deny groups and the infinite conjugacy class property
• Entropy, Ultralimits and the Poisson boundary

Cited by in corpus (2)

• Hypertrace and entropy gap characterizations of property (T) for ${\mathrm{II}}_1$ factors
• Operator algebraic characterization of the noncommutative Poisson boundary