Realization of Banach algebras as Calkin algebras
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Is every unital complex Banach algebra isomorphic to a Calkin algebra B(X)/K(X) for some Banach space X, where K(X) denotes the compact operators? The question allows X to be nonseparable.
Why it matters and what counts as progress
Why it matters: Allowing nonseparable Banach spaces removes the restriction in earlier negative Calkin-algebra realization results.
What counts as progress: Construct an algebra with no such realization at any density, or prove realization for a broader class of algebras.
How the problem developed
Matthew Tarbard Recorded the general realization problem: which unital Banach algebras can occur as B(X)/K(X) for some Banach space X?
Bence Horváth and Tomasz Kania Constructed unital Banach algebras that are not Calkin algebras of separable Banach spaces; this left nonseparable realizations unexcluded.
- 19 Jul 2026 · Reported claim [3]
Antonio and Pablo Acuaviva; AI-guided proof systems The paper claims examples of unital complex Banach algebras that are not Calkin algebras of any Banach space, extending the nonrealization result beyond separable spaces.
AI contribution
Mathematical Discovery in the Wild: AI-Guided Proofs in Banach Space Theory
Antonio and Pablo Acuaviva construct a unital complex Banach algebra not isomorphic to B(X)/K(X) for any Banach space X, using large simple algebras and separable-range operator obstructions.
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