  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: $\pi(\mathcal{T}(\mathbb{D}))=\pi(\mathcal{T}(\mathbb{D}))''$.
