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On the essential commutant of $ {\mathcal T}($QC$ )$


Author: Jingbo Xia
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
Published electronically: July 23, 2007
MathSciNet review: 2346484
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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).


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Additional Information

Jingbo Xia
Affiliation: Department of Mathematics, State University of New York at Buffalo, Buffalo, New York 14260
Email: jxia@acsu.buffalo.edu

DOI: https://doi.org/10.1090/S0002-9947-07-04345-0
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
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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