Localizations and Essential Commutant of Toeplitz Algebra on Polydisk

Usually, the norm closure of a family of operators is not equal to the $C^{\ast }$-algebra generated by this family of operators. But, similar with the Bergman space $L_a^2(\mathbf{\text{B}},dv)$ of the unit ball in ${\mathbb{C}}^n$, we show that the norm closure of $\{T_f:f\in L^{\mathrm{\infty }}(\mathbb{D},dv)\}$ on Bergman space $L_a^2(\mathbb{D},dv)$ 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 $\{T_g:g\in V{O}_{bdd}\}+\mathcal{K}$ where $V{O}_{bdd}$ is the collection of functions of vanishing oscillation on polydisk $\mathbb{D}$ and $\mathcal{K}$ denotes the collection of compact operators on $L_a^2(\mathbb{D},dv)$. On the other hand, we also prove that the essential commutant of $\{T_g:g\in V{O}_{bdd}\}$ is $\mathcal{T}(\mathbb{D})$, which implies that image of $\mathcal{T}(\mathbb{D})$ in the Calkin algebra satisfies the double commutant relation: $\pi (\mathcal{T}(\mathbb{D}))=\pi (\mathcal{T}(\mathbb{D}){)}^{\mathrm{\prime }\mathrm{\prime }}$.