A Unital Banach Algebra That Is Not a Calkin Algebra
Statement
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.
Context
Answers a recorded realisation question for Calkin algebras by explicit construction; the surrounding programme includes the Argyros–Haydon theorem.
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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.
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