On a class of ideals of the Toeplitz algebra on the Bergman space

Let $\mathfrak{T}$ denote the full Toeplitz algebra on the Bergman space of the unit ball $\mathbb{B}_n$. For each subset $G$ of $L^{\infty}$, let $\mathfrak{CI}(G)$ denote the closed two-sided ideal of $\mathfrak{T}$ generated by all $T_fT_g-T_gT_f$ with $f,g\in G$. It is known that $\mathfrak{CI}(C(\overline{\mathbb{B}}_n))=\mathcal{K}$, the ideal of compact operators, and $\mathfrak{CI}(C(\mathbb{B}_n)\cap L^{\infty})=\mathfrak{T}$. Despite these ``extreme cases'', there are subsets $G$ of $L^{\infty}$ so that $\mathcal{K}\subsetneq\mathfrak{CI}(G)\subsetneq\mathfrak{T}$. This paper gives a construction of a class of such subsets.