On the essential commutant of ${\mathcal T}(\text {QC})$
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Abstract:
Let ${\mathcal T}$(QC) (resp. ${\mathcal T}$) be the $C^\ast$-algebra generated by the Toeplitz operators $\{T_\varphi : \varphi \in$ QC$\}$ (resp. $\{T_\varphi : \varphi \in L^\infty \}$) on the Hardy space $H^2$ of the unit circle. A well-known theorem of Davidson asserts that ${\mathcal T}$(QC) is the essential commutant of ${\mathcal T}$. We show that the essential commutant of ${\mathcal T}$(QC) is strictly larger than ${\mathcal T}$. Thus the image of ${\mathcal T}$ in the Calkin algebra does not satisfy the double commutant relation. We also give a criterion for membership in the essential commutant of ${\mathcal T}$(QC).References
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Additional Information
- Jingbo Xia
- Affiliation: Department of Mathematics, State University of New York at Buffalo, Buffalo, New York 14260
- MR Author ID: 215486
- Email: jxia@acsu.buffalo.edu
- Received by editor(s): January 1, 2005
- Received by editor(s) in revised form: May 8, 2006
- Published electronically: July 23, 2007
- Additional Notes: This work was supported in part by National Science Foundation grant DMS-0100249
- © Copyright 2007
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 360 (2008), 1089-1102
- MSC (2000): Primary 42A38, 46L05, 47L80
- DOI: https://doi.org/10.1090/S0002-9947-07-04345-0
- MathSciNet review: 2346484