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

Abstract: 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.