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

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 $T_f{T}_g-T_g{T}_f$ with $f,g\in G.$ It is known that $\mathfrak{CI}(C({\bar{\mathbb{B}}}_n))=\mathcal{K}$ - the ideal of compact operators and $\mathfrak{CI}(C({\mathbb{B}}_n))=\mathfrak{T}.$ Despite these ``extremal cases'', $\mathfrak{T}$ does contain other non-trivial ideals. This paper gives a construction of a class of subsets $G$ of $L^{\infty }$ so that $\mathcal{K}⊊\mathfrak{CI}(G)⊊\mathfrak{T}.$

8 pages