AI 中文总结
本文构造了一个密度特征为2^{\aleph_0}的单位Banach代数A，其具有唯一非零闭两面理想，并且在两种常规定义下都不是Calkin代数。
AI 中文摘要
令\mathscr{A}(X)和\mathscr{K}(X)分别表示Banach空间X上的近可表算子和紧算子的理想。我们构造了一个密度特征为\mathfrak c=2^{\aleph_0}的单位Banach代数A，其恰好有一个非零的闭两面理想，使得对于每一个Banach空间X，代数A既不与\mathscr{B}(X)/\mathscr{A}(X)也不与\mathscr{B}(X)/\mathscr{K}(X)同构。因此，在两种常规约定下，A都不是Calkin代数。
英文摘要
Let $\mathscr{A}(X)$ and $\mathscr{K}(X)$ denote the ideals of approximable and compact operators on a Banach space $X$, respectively. We construct a unital Banach algebra $A$ of density character $\mathfrak c=2^{\aleph_0}$, with exactly one non-zero proper closed two-sided ideal, such that, for every Banach space $X$, the algebra $A$ is isomorphic to neither $\mathscr{B}(X)/\mathscr{A}(X)$ nor $\mathscr{B}(X)/\mathscr{K}(X)$. Thus $A$ is not a Calkin algebra under either of the two customary conventions.
