On a class of ideals of the Toeplitz algebra on the Bergman space
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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.References
- L. A. Coburn, Singular integral operators and Toeplitz operators on odd spheres, Indiana Univ. Math. J. 23 (1973/74), 433–439. MR 322595, DOI 10.1512/iumj.1973.23.23036
- Trieu Le, On the commutator ideal of the Toeplitz algebra on the Bergman space of the unit ball in $\mathbb {C}^n$, J. Operator Theory, to appear.
- Young Joo Lee, Pluriharmonic symbols of commuting Toeplitz type operators on the weighted Bergman spaces, Canad. Math. Bull. 41 (1998), no. 2, 129–136. MR 1624149, DOI 10.4153/CMB-1998-020-7
- Walter Rudin, Function theory in the unit ball of $\textbf {C}^{n}$, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 241, Springer-Verlag, New York-Berlin, 1980. MR 601594
- Daniel Suárez, The Toeplitz algebra on the Bergman space coincides with its commutator ideal, J. Operator Theory 51 (2004), no. 1, 105–114. MR 2055807
Additional Information
- Trieu Le
- Affiliation: Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1
- Email: trieu.le@utoronto.edu
- Received by editor(s): August 16, 2007
- Published electronically: June 6, 2008
- Communicated by: Marius Junge
- © Copyright 2008
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 136 (2008), 3571-3577
- MSC (2000): Primary 47B35; Secondary 47B47
- DOI: https://doi.org/10.1090/S0002-9939-08-09569-5
- MathSciNet review: 2415041