Closed ideals in the algebra of compact-by-approximable operators

We construct various examples of non-trivial closed ideals of the compact-by-approximable algebra ${\mathfrak{A}}_X=:\mathcal{K}(X)\mathrm{/}\mathcal{A}(X)$ on Banach spaces $X$ failing the approximation property. The examples include the following: (i) if $X$ has cotype $2$, $Y$ has type $2$, $\mathfrak{A}_X \neq {0}$and$\mathfrak{A}Y \neq {0}$,then$\mathfrak{A}{X \oplus Y}$hasatleast$2$closedideals,(ii)thereareclosedsubspaces$X \subset \ell^p$for$4 < p < \infty$and$X \subset c_0$suchthat$\mathfrak{A}_X$containsanon-trivialclosedideal,(iii)thereisaBanachspace$Z$ such that ${\mathfrak{A}}_Z$ contains an uncountable lattice of closed ideal having the reverse order structure of the power set of the natural numbers. Some of our examples involve non-classical approximation properties associated to various Banach operator ideals. We also discuss the existence of compact non-approximable operators $X\to Y$, where $X\subset {\mathrm{\ell }}^p$ and $Y \subset \ell^q$areclosedsubspacesfor$p \neq q$.