Result 290, Operator algebras

Relative bicentralizers and modular spectral recovery

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

Lean formalization Proof

The bigger picture

Why it matters

Can a complicated algebra of operators contain a more tractable subalgebra that preserves exactly what commutes with it in a larger setting? The manuscripts report such a construction, addressing a longstanding structural question in operator algebras.

What changes?

The main manuscript considers every inclusion N inside M of von Neumann algebras with separable preduals, assuming a faithful normal conditional expectation, a structure-preserving projection from M onto N. It reports an amenable, or structurally tractable, subalgebra P inside N that also admits such a projection. Inside M's continuous core, an enlargement encoding modular time evolution, exactly the same operators commute with P as with N. This is the claimed resolution of the relative bicentralizer conjecture under these assumptions.

What does that help mathematicians do?

The commutant equality means researchers could replace N by an amenable subalgebra when determining which operators in the continuous core commute with it, without losing any of that information. The summary also reports bicentralizer triviality for every type III1 factor with separable predual and every faithful normal state. Here triviality means that a particular family of approximate commutation tests singles out only scalar multiples of the identity, ruling out additional hidden structure of that kind.

Are there practical applications?

The immediate value is foundational: preserving a core commutant offers a controlled simplification for structural arguments. The companion manuscript also reports an application to spectral intertwining rigidity. Its recovery theorem assumes a faithful normal state with scalar centralizer: positive averaged squared norms for arbitrary bounded Hilbert-space operators on shrinking modular spectral bands are recovered on uniformly bounded algebra elements with shrinking spectral support.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

2 manuscripts

Expected amenable subalgebras preserving core commutants

September 23, 2026 19 pages

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

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:Expected-amenable-subalgebras-preserving-core-commutants-September-23-2026,
  author = {{OpenAI}},
  title = {{Expected amenable subalgebras preserving core commutants}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Expected-amenable-subalgebras-preserving-core-commutants-September-23-2026/Expected-amenable-subalgebras-preserving-core-commutants-September-23-2026.pdf}{OAI:Expected-amenable-subalgebras-preserving-core-commutants-September-23-2026}},
  year = {2026}
}

Bounded recovery for modular spectral averages

September 23, 2026 16 pages Main result formalized in Lean

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.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:Bounded-recovery-for-modular-spectral-averages-September-23-2026,
  author = {{OpenAI}},
  title = {{Bounded recovery for modular spectral averages}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Bounded-recovery-for-modular-spectral-averages-September-23-2026/Bounded-recovery-for-modular-spectral-averages-September-23-2026.pdf}{OAI:Bounded-recovery-for-modular-spectral-averages-September-23-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/290.md.

Relative bicentralizers and modular spectral recovery

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

For a faithful normal state with scalar centralizer in the stated standard-space representation, the formalized result converts positive modular spectral averages into uniformly bounded algebra elements. Given unit vectors in shrinking spectral bands around a real number ss and a positive limiting symmetric average for a bounded operator TT, it finds a subsequence of bounded algebra elements whose images under TT stay uniformly nonzero and whose spectral bands have four times the original widths. The absolute bicentralizer conjecture is not included.

Comparator links

Result Comparator statement
Bounded recovery from modular spectral averages BoundedRecovery.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 8 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.