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.

中文翻译：

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逼近性失败的 BANACH 空间上紧逼算子的商代数

我们开始研究商代数的结构性质${\mathcal{K}}(X)/{\mathcal{A}}(X)$Banach 空间上的紧逼逼近算子$X$不符合近似属性。我们的主要结果和示例包括以下内容：（i）有一个线性同构嵌入来自$c_{0}$进入${\mathcal{K}}(Z)/{\mathcal{A}}(Z)$， 在哪里$Z$属于由 Willis 构造的 Banach 空间类，具有度量紧逼近属性但不具备逼近属性，(ii) 存在来自不可分空间的线性同构嵌入$c_{0}(\unicode[STIX]{x1D6E4})$进入${\mathcal{K}}(Z_{FJ})/{\mathcal{A}}(Z_{FJ})$， 在哪里$Z_{FJ}$是由 Johnson 和 Figiel 的工作产生的通用紧凑因子分解空间。