A Unital Banach Algebra That Is Not a Calkin Algebra
The realisation problem asks which unital Banach algebras arise as the Calkin algebra of some Banach space. Recorded in Tarbard's thesis and studied by Horváth and Kania. The paper exhibits a unital Banach algebra that cannot be one.
- Result
- Proved
- Status
- Resolved
- AI contribution
- AI co-developed
- Method
- Construction
- Field
- Banach Algebras, Operator Theory
- Posed by
- Tomasz Kania
- Year posed
- —
- Years open
- —
- Solved
- 2026-07-19
- Model
- ChatGPT 5.5 Pro
- Vendor
- OpenAI
- Collaborators
- Antonio Acuaviva, Pablo Acuaviva
- Verification
- Unreviewed
- Publication
- Preprint
- Significance
- 15 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What the AI did
ChatGPT 5.5 Pro, driven directly and through Codex agents over a filesystem, generated the key ideas and the proof; the authors verified and refined it. The paper reports that the model sometimes misattributed a theorem, cited a result imprecisely, or presented steps as immediate when they still needed checking, so the human verification was load-bearing. The authors state none of these results came from the newer generation of models.
Verification
No independent check. The proof stands on verification by the paper's authors, who describe correcting misattributions and filling in steps the model presented as immediate. Preprint, not refereed.
Source
Submitted by Curator34