Journal of Operator Theory

Volume 64, Issue 1, Summer 2010 pp. 89-101.

Commutator ideals of subalgebras of Toeplitz algebras on weighted Bergman spaces

Authors: Trieu Le
Author institution: Department of Mathematics, Mail Stop 942, University of Toledo, Toledo, OH 43606, U.S.A.

Summary: For $\alpha>-1$, let $A^{2}_{\alpha}$ denote the corresponding weighted Bergman space of the unit ball. For any self-adjoint subset $G\subset L^{\infty}$, let $\mathfrak{T}(G)$ denote the $C^{*}$-subalgebra of $\mathfrak{B}(A^2_{\alpha})$ generated by $\{T_{f}: f\in G\}$. Let $\mathfrak{CT}(G)$ denote the commutator ideal of $\mathfrak{T}(G)$. It was showed by D. Su{\'a}rez (in 2004 for $n=1$) and by the author (in 2006 for all $n\geqslant 1$) that $\mathfrak{CT}(L^{\infty})=\mathfrak{T}(L^{\infty})$ in the case $\alpha=0$. In this paper we show that in the setting of weighted Bergman spaces, the identity $\mathfrak{CT}(G)=\mathfrak{T}(G)$ holds true for a class of subsets $G$ including $L^{\infty}$.

Keywords: Commutator ideals, Toeplitz operators, weighted Bergman spaces.

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