Result 299, Operator algebras

The Kirchberg–Rørdam character criterion and infinite tensor-power Jiang–Su stability

A nonzero unital separable complex C∗-algebra is Jiang–Su stable exactly when its norm central-sequence algebra has no characters, for every free ultrafilter. This answers the Kirchberg–Rørdam character question. Also, the infinite minimal tensor power of every such algebra without characters is Jiang–Su stable, answering the Dadarlat–Toms question.

Lean formalization Proof

The bigger picture

Why it matters

An operator algebra can absorb a fixed building block, the Jiang-Su algebra, without changing its structure. The unreviewed manuscript reports that this absorption is completely detected by the absence of certain scalar-valued representations.

What changes?

For every nonzero unital separable complex C*-algebra, an operator algebra with an identity and a countable dense subset, the reported criterion equates Jiang-Su stability with having no characters in its norm central-sequence algebra, for every free ultrafilter on the natural numbers. Central sequences asymptotically commute with each fixed element; the ultrafilter specifies the limiting construction. A character is a nonzero homomorphism to the complex numbers. Stability means tensoring with the Jiang-Su algebra gives an isomorphic algebra.

What does that help mathematicians do?

The manuscript also reports that the infinite minimal tensor power, formed by repeatedly tensoring copies using the minimal tensor product, is Jiang-Su stable whenever the original nonzero unital separable complex C*-algebra has no characters. It also contains a unital copy of the Jiang-Su algebra. This supplies a construction with guaranteed absorption under a condition on the starting algebra, without claiming that the starting algebra itself absorbs.

Are there practical applications?

The immediate value is foundational. To rule out Jiang-Su stability, a researcher can seek a character in a norm central-sequence algebra for a free ultrafilter. Conversely, proving the required absence establishes absorption. The tensor-power result identifies a specific construction that produces this stability from character-free starting algebras.

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

The Kirchberg–Rørdam character criterion

September 25, 2026 21 pages Main result formalized in Lean

We prove the Kirchberg–Rørdam character criterion: for every free ultrafilter on ℕ, a nonzero unital separable complex C∗-algebra absorbs the Jiang–Su algebra if and only if its norm central-sequence algebra has no characters. We also prove that the infinite minimal tensor power of every nonzero unital separable complex C∗-algebra without characters is Z\mathcal Z-stable and contains a unital copy of Z\mathcal Z.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:The-Kirchberg-Rordam-character-criterion-September-25-2026,
  author = {{OpenAI}},
  title = {{The Kirchberg--R\o rdam character criterion}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/The-Kirchberg-Rordam-character-criterion-September-25-2026/paper.pdf}{OAI:The-Kirchberg-Rordam-character-criterion-September-25-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/299.md.

The Kirchberg–Rørdam character criterion and infinite tensor-power Jiang–Su stability

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

Scope

The Kirchberg–Rørdam criterion relates Jiang–Su absorption to characters of the central-sequence algebra. The formalized result proves that, for every nonzero unital separable complex C∗C^*-algebra AA and every free ultrafilter on N\mathbb N, the norm central-sequence algebra has no nonzero character exactly when A≅A⊗min⁡ZA\cong A\otimes_{\min}\mathcal Z. No nuclearity, simplicity, trace, or comparison hypothesis is imposed.

Comparator links

Result Comparator statement
Kirchberg–Rørdam character criterion CharacterCriterion.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.