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

Authors:	Jingbo Xia

Institution:	Department of Mathematics, State University of New York at Buffalo, Buffalo, New York 14260

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 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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