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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On a class of ideals of the Toeplitz algebra on the Bergman space


Author: Trieu Le
Journal: Proc. Amer. Math. Soc. 136 (2008), 3571-3577
MSC (2000): Primary 47B35; Secondary 47B47
Published electronically: June 6, 2008
MathSciNet review: 2415041
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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(\overline{\mathbb{B}}_n))=\mathcal{K}$, the ideal of compact operators, and $ \mathfrak{CI}(C(\mathbb{B}_n)\cap L^{\infty})=\mathfrak{T}$. Despite these ``extreme cases'', there are subsets $ G$ of $ L^{\infty}$ so that $ \mathcal{K}\subsetneq\mathfrak{CI}(G)\subsetneq\mathfrak{T}$. This paper gives a construction of a class of such subsets.


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Additional Information

Trieu Le
Affiliation: Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1
Email: trieu.le@utoronto.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-08-09569-5
PII: S 0002-9939(08)09569-5
Received by editor(s): August 16, 2007
Published electronically: June 6, 2008
Communicated by: Marius Junge
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.