Result 288, Operator algebras

Kadison's similarity conjecture

Proves Kadison's similarity conjecture: every bounded complex-linear unital algebra homomorphism from a unital complex C∗-algebra to operators on a Hilbert space becomes a ∗*-homomorphism after conjugation by a bounded invertible operator.

Lean formalization Proof

The bigger picture

Why it matters

An operator representation can preserve multiplication while failing to preserve adjoints, the conjugate-transpose operation familiar from matrices. The unreviewed manuscript claims that every bounded representation of the stated kind can be corrected by a reversible change of coordinates.

What changes?

The claimed theorem covers every bounded, complex-linear, identity-preserving algebra homomorphism from a unital complex C*-algebra into the bounded operators on any complex Hilbert space, including infinite-dimensional spaces. A C*-algebra is a complete normed algebra with an adjoint operation tied to its norm. The claim is that conjugating the homomorphism's outputs by a bounded invertible operator makes it preserve adjoints as well as multiplication. No additional assumptions on the algebra or Hilbert space are stated.

What does that help mathematicians do?

If established, this would rule out bounded homomorphisms in this setting whose failure to preserve adjoints cannot be removed by such a coordinate change. Researchers could therefore study them through adjoint-preserving representations. The abstract also reports a single hyperreflexivity constant for all unital von Neumann algebras on arbitrary complex Hilbert spaces: a uniform bound controlling distance to an algebra through how much an operator fails to preserve its invariant subspaces.

Are there practical applications?

The immediate value is foundational in operator algebra theory. The claimed similarity theorem would let researchers bring adjoint-preserving representation methods to bounded homomorphisms after changing coordinates. The uniform hyperreflexivity claim would additionally give approximation estimates independent of the particular algebra and Hilbert space. The supplied sources do not describe a computational procedure or practical deployment.

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.

Manuscript

Kadison's similarity theorem through uniform derivation estimates

September 23, 2026 31 pages

We prove that every bounded complex-linear unital algebra homomorphism from a unital complex C∗-algebra into the bounded operators on an arbitrary Hilbert space is similar to a ∗*-homomorphism. This resolves Kadison's similarity conjecture positively. We also obtain one universal hyperreflexivity constant for all unital von Neumann algebras on arbitrary complex Hilbert spaces.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:Kadisons-similarity-theorem-through-uniform-derivation-estimates-September-23-2026,
  author = {{OpenAI}},
  title = {{Kadison's similarity theorem through uniform derivation estimates}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Kadisons-similarity-theorem-through-uniform-derivation-estimates-September-23-2026/paper.pdf}{OAI:Kadisons-similarity-theorem-through-uniform-derivation-estimates-September-23-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/288.md.

Kadison's similarity conjecture

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

Scope

Kadison's similarity problem asks whether every bounded unital homomorphism from a unital complex C∗C^*-algebra into operators on a Hilbert space is similar to a ∗*-homomorphism. The formalization proves this for arbitrary Hilbert spaces. It also gives one universal hyperreflexivity constant for all unital von Neumann algebras.

The linked commutator estimate is uniform over all finite matrix amplifications: its constant is independent of the algebra, Hilbert space, and amplification size. This is the quantitative derivation estimate supporting the similarity result.

Comparator links

Result Comparator statement
Kadison's similarity theorem and uniform hyperreflexivity KadisonSimilarity.lean
Uniform amplified commutator estimate UniformCommutator.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 9 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.