Toeplitz algebra and Symbol map via Berezin transform on $H^2({\mathbb{D}}^n)$

Let $\mathcal{T}(L^{\mathrm{\infty }}(\mathbb{T}))$ be the Toeplitz algebra, that is, the $C^{\ast }$-algebra generated by the set $\{T_{\phi }:\phi \in L^{\mathrm{\infty }}(\mathbb{T})\}$. Douglas’s theorem on symbol map states that there exists a $C^{\ast }$-algebra homomorphism from $\mathcal{T}(L^{\mathrm{\infty }}(\mathbb{T}))$ onto $L^{\mathrm{\infty }}(\mathbb{T})$ such that $T_{\phi }\mapsto \phi$ and the kernel of the homomorphism coincides with commutator ideal in $\mathcal{T}(L^{\mathrm{\infty }}(\mathbb{T}))$. 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 $n\ge 1$. We further obtain a class of bigger $C^{\ast }$-algebras than the Toeplitz algebra $\mathcal{T}(L^{\mathrm{\infty }}({\mathbb{T}}^n))$ for which the analog of symbol map still holds true.