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

TL;DR

This paper explores the structure of ideals within the Toeplitz algebra on the Bergman space of the unit ball, constructing new non-trivial ideals between the compact operators and the full algebra.

Contribution

It introduces a method to construct subsets of functions whose generated ideals lie strictly between the compact operators and the entire Toeplitz algebra.

Findings

Constructed new classes of ideals between compact operators and the full algebra.

Showed that these ideals are strictly larger than the compact operators but smaller than the entire algebra.

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 $\mathfrak{CI}(C(\bar{\mathbb{B}}_n))=\mathcal{K}$ - 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 $\mathcal{K}\subsetneq\mathfrak{CI}(G)\subsetneq\mathfrak{T}.$