Remark: Family 290. Connes' bicentralizer conjecture and relative bicentralizers

Proves Connes' bicentralizer conjecture for every type ${\mathrm{III}}_1$ factor with separable predual and every faithful normal state. More generally, for every inclusion $N\subset M$ of von Neumann algebras with separable preduals admitting a faithful normal conditional expectation, constructs an amenable expected subalgebra $P\subset N$ with $P^{\prime }\cap c(M)=N^{\prime }\cap c(M)$, resolving the relative bicentralizer conjecture.

The authors are OpenAI. This companion does not verify the proof. The claim is not a theorem of the Enter series and not a premise of any registered sheet. It does not close O.1. O.1 stays NOT_CLOSED. It is not a de Sitter state and not the QdS external-leg scheme. No quantum coefficient is admitted. No compilation is asserted.