  Let $\mathscr{T}(L^{\infty}(\mathbb{T}))$ be the Toeplitz algebra, that is,
the $C^*$-algebra generated by the set $\{T_{\phi} : \phi\in
L^{\infty}(\mathbb{T})\}$. Douglas's theorem on symbol map states that there
exists a $C^*$-algebra homomorphism from $\mathscr{T}(L^{\infty}(\mathbb{T}))$
onto $L^{\infty}(\mathbb{T})$ such that $T_{\phi}\mapsto \phi$ and the kernel
of the homomorphism coincides with commutator ideal in
$\mathscr{T}(L^{\infty}(\mathbb{T}))$. In this paper, we use the Berezin
transform to study results akin to Douglas's theorem for operators on the Hardy
space $H^2(\mathbb{D}^n)$ over the open unit polydisc $\mathbb{D}^n$ for $n\geq
1$. We further obtain a class of bigger $C^*$-algebras than the Toeplitz
algebra $\mathscr{T}(L^{\infty}(\mathbb{T}^n))$ for which the analog of symbol
map still holds true.
