Localizations and Essential Commutant of Toeplitz Algebra on Polydisk

Usually, the norm closure of a family of operators is not equal to the $C^*$-algebra generated by this family of operators. But, similar with the Bergman space $L^2_a(\textbf{B}, dv)$ of the unit ball in $\mathbb{C}^n$, we show that the norm closure of $\{T_f : f\in L^{\infty}(\mathbb{D}, dv)\}$ on Bergman space $L^2_a(\mathbb{D}, dv)$ of the ploydisk $\mathbb{D}$ in $\mathbb{C}^n$ actually coincides with the Toeplitz algebra $\mathcal{T}(\mathbb{D})$. A key ingredient in the proof is the class of operators $\mathcal{D}$ recently introduced by Yi Wang and Jingbo Xia. In fact, as a by-product, we simultaneously proved that $\mathcal{T}(\mathbb{D})$ also coincides with $\mathcal{D}$. Based on these results, we further proved that the essential commutant of Toeplitz algebra $\mathcal{T}(\mathbb{D})$ equals to $\{T_g: g\in VO_{bdd}\} + \mathcal{K}$ where $VO_{bdd}$ is the collection of functions of vanishing oscillation on polydisk $\mathbb{D}$ and $\mathcal{K}$ denotes the collection of compact operators on $L^2_a(\mathbb{D}, dv)$. On the other hand, we also prove that the essential commutant of $\{T_g: g\in VO_{bdd}\}$ is $\mathcal{T}(\mathbb{D})$, which implies that image of $\mathcal{T}(\mathbb{D})$ in the Calkin algebra satisfies the double commutant relation: $\pi(\mathcal{T}(\mathbb{D}))=\pi(\mathcal{T}(\mathbb{D}))''$.