Shuoxing Zhou

math.OA — Operator Algebras

Poisson boundaries of type III factors and the relative bicentralizer conjecture

Contributed by Shuoxing Zhou

We develop the Poisson boundary theory for general von Neumann algebras with separable preduals. As applications, we generalize and give new proofs of the results on the weak Dixmier property for inclusions with expectations or operator-valued weights \cite{Pop99,Mar19,Iso25}. We also prove that every continuous factor admits irreducible embeddings into the hyperfinite factors of type $\mathrm{III}_\lambda$, $0<\lambda\leq1$, and into some hyperfinite factor of type $\mathrm{III}_0$. Finally, using the Poisson boundary techniques developed above, we prove that for every inclusion with faithful normal conditional expectation $M\subset N$, $M\subset c(N)$ has the weak Dixmier property. By Marrakchi's characterization \cite[Theorem~D]{Ma25}, this gives the relative bicentralizer conjecture for von Neumann algebras with separable preduals.