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

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).