Relative bicentralizers and modular spectral recovery

What OpenAI claims

Proves Connes' bicentralizer conjecture for every type III1 factor with separable predual and every faithful normal state. More generally, for every inclusion N ⊂ M of von Neumann algebras with separable preduals admitting a faithful normal conditional expectation, constructs an amenable expected subalgebra P ⊂ N with P'∩ c(M)=N'∩ c(M), resolving the relative bicentralizer conjecture. (Lean)

Papers (2)

1. Expected amenable subalgebras preserving core commutants

   23 September 2026 · Preprint, not peer reviewed · PDF

   We prove that every inclusion N ⊂ M of von Neumann algebras with separable preduals and a faithful normal conditional expectation contains an expected amenable subalgebra P ⊂ N such that P'∩ c(M)=N'∩ c(M), where c(M) denotes the continuous core of M. This resolves the relative bicentralizer conjecture in this setting.

2. Bounded recovery for modular spectral averages

   23 September 2026 · Preprint, not peer reviewed · PDF

   We prove a bounded spectral recovery theorem for a von Neumann algebra with a faithful normal state whose centralizer consists only of scalars. A positive averaged squared norm for an arbitrary bounded Hilbert-space operator on shrinking modular spectral bands can be recovered on uniformly bounded algebra elements with shrinking spectral support. We apply this theorem to spectral intertwining rigidity and obtain an alternative proof of bicentralizer triviality for type III1 factors with separable predual.

Status

Preprint. The papers have not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.