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We exhibit a Banach space Z failing the approximation property, for which there is an uncountable family \mathscr F of closed subideals contained in the Banach algebra \mathcal K(Z) of the compact operators on Z, such that the subideals in \mathscr F are mutually isomorphic as Banach algebras. This contrasts with the behaviour of closed ideals of the algebras \mathcal L(X) of bounded operators on X, where closed ideals \mathcal I \neq \mathcal J are never isomorphic as Banach algebras. We also construct families of non-trivial closed subideals contained in the strictly singular operators \mathcal S(X) for classical spaces such as X = L^p with p \neq 2, where pairwise isomorphic as well as pairwise non-isomorphic subideals occur. [ABSTRACT FROM AUTHOR]

We establish new results on the I$\mathcal {I}$‐approximation property for the Banach operator ideal I=Kup$\mathcal {I}=\mathcal {K}_{up}$ of the unconditionally p‐compact operators in the case of 1≤p<2$1\le p<2$. As a consequence of our results, we provide a negative answer for the case p=1$p=1$ of a problem posed by Kim. Namely, the Ku1$\mathcal {K}_{u1}$‐approximation property implies neither the SK1$\mathcal {SK}_1$‐approximation property nor the (classical) approximation property; and the SK1$\mathcal {SK}_1$‐approximation property implies neither the Ku1$\mathcal {K}_{u1}$‐approximation property nor the approximation property. Here, SKp$\mathcal {SK}_p$ denotes the p‐compact operators of Sinha and Karn for p≥1$p\ge 1$. We also show for all 2

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 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. [ABSTRACT FROM AUTHOR]