Let $\mathscr{B}(X)$ denote the Banach algebra of bounded operators on $X$, where~$X$ is either Tsirelson's Banach space or the Schreier space of order $n$ for some $n\in\mathbb N$. We show that the lattice of closed ideals of~$\mathscr{B}(X)$ has a very rich structure; in particular $\mathscr{B}(X)$ contains at least continuum many maximal ideals. Our approach is to study the closed ideals generated by the basis projections. Indeed, the unit vector basis is an unconditional basis for each of the above spaces, so there is a basis projection $P_N\in\mathscr{B}(X)$ corresponding to each non-empty subset $N$ of $\mathbb N$. A closed ideal of $\mathscr{B}(X)$ is spatial if it is generated by $P_N$ for some $N$. We can now state our main conclusions as follows: i) the family of spatial ideals lying strictly between the ideal of compact operators and $\mathscr{B}(X)$ is non-empty and has no minimal or maximal elements; ii) for each pair $\mathscr{I}\subsetneqq\mathscr{J}$ of spatial ideals, there is a family $\{\Gamma_L\colon L\in \Delta\}$, where the index set $\Delta$ has the cardinality of the continuum, such that $\Gamma_L$ is an uncountable chain of spatial ideals, $\bigcup\Gamma_L$ is a closed ideal that is not spatial, and $$ \mathscr{I}\subsetneqq\mathscr{L}\subsetneqq\mathscr{J}\qquad\text{and}\qquad \overline{\mathscr{L}+\mathscr{M}} = \mathscr{J}$$ whenever $L,M\in\Delta$ are distinct and $\mathscr{L}\in\Gamma_L$, $\mathscr{M}\in\Gamma_M$.

中文翻译：

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Tsirelson 和 Schreier 空间上算子的闭理想

让 $\mathscr{B}(X)$ 表示 $X$ 上的有界算子的 Banach 代数，其中~$X$ 是 Tsirelson 的 Banach 空间或 $n$ 阶的 Schreier 空间对于某些 $n\in\mathbb美元。我们证明~$\mathscr{B}(X)$的闭理想格具有非常丰富的结构；特别是 $\mathscr{B}(X)$ 包含至少连续的许多极大理想。我们的方法是研究由基础投影产生的封闭理想。确实，单位向量基是上述每一个空间的无条件基，所以有一个基投影$P_N\in\mathscr{B}(X)$对应于$\mathbb的每一个非空子集$N$美元。$\mathscr{B}(X)$ 的封闭理想是空间的，如果它是由 $P_N$ 为某个 $N$ 生成的。我们现在可以陈述我们的主要结论如下：i) 严格介于紧凑算子理想和 $\mathscr{B}(X)$ 之间的空间理想族是非空的，并且没有最小或最大元素；ii) 对于每对 $\mathscr{I}\subsetneqq\mathscr{J}$ 的空间理想，存在一个族 $\{\Gamma_L\colon L\in \Delta\}$，其中索引集 $\Delta $ 具有连续统的基数，使得 $\Gamma_L$ 是不可数的空间理想链，$\bigcup\Gamma_L$ 是一个非空间的封闭理想，而 $$ \mathscr{I}\subsetneqq\mathscr{ L}\subsetneqq\mathscr{J}\qquad\text{and}\qquad \overline{\mathscr{L}+\mathscr{M}} = \mathscr{J}$$ 每当 $L,M\in\Delta $ 和 $\mathscr{L}\in\Gamma_L$、$\mathscr{M}\in\Gamma_M$ 是不同的。