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Noncommutative Poisson boundaries, ultraproducts and entropy

Published 22 Jun 2023 in math.OA, math.DS, and math.GR | (2306.12763v3)

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.

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