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Journal of Operator Theory

Volume 60, Issue 1, Summer 2008  pp. 149-163.

On the commutator ideal of the Toeplitz algebra on the Bergman space of the unit ball in $\mathbb{C}^n$

Authors Trieu Le
Author institution: Department of Mathematics, University of Toronto, Toronto, ON, Canada M5S 2E4

Summary:  Let $L_{a}^2$ denote the Bergman space of the open unit ball $B_n$ in $\mathbb{C}^n,$ for $n\geqslant 1.$ The Toeplitz algebra $\mathfrak{T}$ is the C$^{*}$-algebra generated by all Toeplitz operators $T_{f}$ with $f\in L^{\infty}.$ It was proved by D. Su{\'a}rez that for $n=1,$ the closed bilateral commutator ideal generated by operators of the form $T_{f}T_{g}-T_{g}T_{f},$ where $f,g\in L^{\infty},$ coincides with $\mathfrak{T}$. With a different approach, we can show that for $n\geqslant 1,$ the closed bilateral ideal generated by operators of the above form, where $f,g$ can be required to be continuous on the open unit ball or supported in a nowhere dense set, is also all of $\mathfrak{T}.$

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