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

Abstract:

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.