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Weak pure infiniteness and O-infinity absorption
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Abstract
We prove that every complex C∗-algebra in which each positive element is properly infinite is strongly purely infinite. This answers the ordinary-to-strong part of Kirchberg–Rørdam's comparison question. For exact algebras, we also prove that proper infiniteness of one fixed finite amplification of every positive element implies proper infiniteness of each positive element. Consequently, every separable nuclear algebra with this fixed-amplification property absorbs $\mathcal O_\infty$, without assumptions of unitality or simplicity.
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work in progress
Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Formalization
- OpenAI math release, Lean scope notes for family 303 (lean/docs/303.md, release 2026-10-06). The scope notes say the Lean development covers Theorem 1.1 (property (P) implies strong pure infiniteness, with no exactness, nuclearity, unitality or simplicity assumptions) and Theorem 1.2 (fixed amplification implies (P) for exact algebras), with comparator files
ExactInfiniteness.leanandIndividualStrongInfiniteness.lean; the O∞-absorption corollary is not among the listed results. [family-level] - OpenAI, lean/OAI/Analysis/WeakInfiniteness/Main.lean (github.com/openai/math, commit adc7f12). Top-level Lean file in the release repository that imports the pure-infiniteness modules and is the location given for the formal exact-algebra statement.
Related prior literature mentioned by others
- Christopher Schafhauser, Aaron Tikuisis and Stuart White, Nuclear C-algebras: 99 problems* (arXiv:2506.10902v2, 2026-05-08). The AI Proved What? exposition identifies Problem LXXII in this survey as the weak/ordinary/strong pure-infiniteness comparison question addressed by the manuscript.
Possibly relevant literature (found by the agent, not checked)
- Eduard Vilalta, An introduction to the Global Glimm Problem (arXiv:2512.13334, December 2025). Survey treating the weakly purely infinite problem as an open question tied to the Global Glimm Problem; relevant to the paper's remark that the nonexact weak-to-ordinary implication is still open and to its use of Global Glimm results [TV2023, NTV2026].
- David Pask, Nuclear dimension, pure infiniteness and real rank for higher rank graph C*-algebras (arXiv:2607.27691, July 2026). Proves that purely infinite higher-rank graph C*-algebras of topological dimension zero are strongly purely infinite and O∞-stable, a special case of the conclusions of Theorem 1.1 and the O∞-absorption corollary; it was posted before the release.
Digests and expositions
- Latent Seminar, Pure infiniteness: compatible corners and amplification removal (2026-10-07). Technical exposition of hereditary transport, uniform separation, coefficient bounds and amplification removal, limited to selected proof mechanisms and disclaiming independent verification of the complete proofs. [family-level]
- AI Proved What?, Infinite enough? Kirchberg–Rørdam’s pure-infiniteness question (episode 303, October 2026). Animated exposition and transcript introducing the three infiniteness notions, outlining the manuscript's two main arguments, and distinguishing the exact-algebra conclusion from the nonexact weak-to-ordinary case left outside its scope. [AI-generated per source] [family-level]
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 303
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:Weak-pure-infiniteness-and-O-infinity-absorption-September-25-2026,
author = {{OpenAI}},
title = {{Weak pure infiniteness and $\mathcal{O}_{\infty}$ absorption}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Weak-pure-infiniteness-and-O-infinity-absorption-September-25-2026/paper.pdf}{OAI:Weak-pure-infiniteness-and-O-infinity-absorption-September-25-2026}},
year = {2026}
}Lean formalization
OpenAI provides Lean files for this family: lean/docs/303.md. The Lean statements have not been checked by people against the paper's claims.
Family 303: Weak pure infiniteness and Cuntz-algebra absorption
Resolves the ordinary-to-strong pure-infiniteness question of Kirchberg and Rørdam for complex C∗-algebras. For exact algebras, proper infiniteness of one fixed finite amplification of every positive element also suffices. Consequently, every separable nuclear algebra with this property absorbs $\mathcal O_\infty$, without unitality or simplicity assumptions.
This is the only paper in this family.