On a Class of Ideals of the Toeplitz Algebra on the Bergman Space of the Unit Ball

摘要

Let $\mathfrak{T}$ denote the full Toeplitz algebra on the Bergman space ofthe 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$\mathfrak{CI}(C(\bar{\mathbb{B}}_n))=\mathcal{K}$ - the ideal of compactoperators and $\mathfrak{CI}(C(\mathbb{B}_n))=\mathfrak{T}.$ Despite these``extremal cases'', $\mathfrak{T}$ does contain other non-trivial ideals. Thispaper gives a construction of a class of subsets $G$ of $L^{\infty}$ so that$\mathcal{K}\subsetneq\mathfrak{CI}(G)\subsetneq\mathfrak{T}.$