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arXiv · 2607.15075

A Unital Banach Algebra Which Is Not a Calkin Algebra

Antonio Acuaviva·Pablo Acuaviva
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Abstract

Let $\mathscr{A}(X)$ and $\mathscr{K}(X)$ denote the ideals of approximable and compact operators on a Banach space $X$, respectively. We construct a unital Banach algebra $A$ of density character $\mathfrak c=2^{\aleph_0}$, with exactly one non-zero proper closed two-sided ideal, such that, for every Banach space $X$, the algebra $A$ is isomorphic to neither $\mathscr{B}(X)/\mathscr{A}(X)$ nor $\mathscr{B}(X)/\mathscr{K}(X)$. Thus $A$ is not a Calkin algebra under either of the two customary conventions.

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Antonio Acuaviva, Pablo Acuaviva. 2026-07-16. A Unital Banach Algebra Which Is Not a Calkin Algebra. https://arxiv.org/abs/2607.15075

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