  We 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 non-separable space $c_0(\Gamma)$ 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.
