Closed ideals of operators on the Tsirelson and Schreier spaces

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 $\mathscr{L}\in\Gamma_L$, $\mathscr{M}\in\Gamma_M$.