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

$L^{\infty}$ .

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