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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