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AbstractWe initiate a study of structural properties of the quotient algebra ${\mathcal{K}}(X)/{\mathcal{A}}(X)$of the compact-by-approximable operators on Banach spaces $X$failing the approximation property. Our main results and examples include the following: (i) there is a linear isomorphic embedding from $c_{0}$into ${\mathcal{K}}(Z)/{\mathcal{A}}(Z)$, where $Z$belongs to the class of Banach spaces constructed by Willis that have the metric compact approximation property but fail the approximation property, (ii) there is a linear isomorphic embedding from a nonseparable space $c_{0}(\unicode[STIX]{x1D6E4})$into ${\mathcal{K}}(Z_{FJ})/{\mathcal{A}}(Z_{FJ})$, where $Z_{FJ}$is a universal compact factorisation space arising from the work of Johnson and Figiel.