A Unital Banach Algebra Which Is Not a Calkin Algebra

Abstract

Let $\mathcal{A}(X)$ and $\mathcal{K}(X)$ 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 $\mathcal{B}(X)/\mathcal{A}(X)$ nor $\mathcal{B}(X)/\mathcal{K}(X)$. Thus $A$ is not a Calkin algebra under either of the two customary conventions.