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Poisson boundaries of type III factors and the relative bicentralizer conjecture

A mix of human-written and AI-generated textHuman understanding: all partsmath.OA — Operator Algebras

Contributed by Shuoxing Zhou

Version 3 / Oct 06, 2026 / CC BY 4.0

Abstract

We develop Poisson boundary theory for general von Neumann algebras with separable preduals. As an application, we prove that, for every inclusion M⊂PM\subset P of countably decomposable von Neumann algebras with a normal faithful conditional expectation, M⊂c(P)M\subset c(P) has the weak Dixmier property, where c(P)c(P) is the continuous core of PP. As direct consequences, via the characterizations of Haagerup and Marrakchi, we have a completely new proof of Connes' bicentralizer conjecture and prove Marrakchi's relative bicentralizer conjecture. We also give a new proof of the solution to Kadison's problem on maximal abelian subalgebras for inclusions of von Neumann algebras with expectation.

Provenance statement

This version is posted to provide a public record of independently developed, AI-assisted work. Our result on the relative bicentralizer conjecture overlaps with OpenAI’s concurrent work, but the proofs use fundamentally different methods.

Tools used

OpenAI
ChatGPTVersion 6 Astra
Google
GeminiVersion 3.1 Pro

References

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Version history

  1. v1 · 44 pagesSubmitted by Shuoxing ZhouInitial depositSupersededOct 06, 2026
  2. v2 · 44 pagesSubmitted by Shuoxing ZhouSubmitted as a new versionSupersededOct 07, 2026
  3. v3 · 53 pagesSubmitted by Shuoxing ZhouSubmitted as a new versionCurrentNov 05, 2026