Toeplitz algebra and Symbol map via Berezin transform on H²(Dⁿ)

Let T(L∞(T)) be the Toeplitz algebra, that is, the C^*-algebra generated by the set φ : φ∈ L∞(T)\. Douglas's theorem on symbol map states that there exists a C^*-algebra homomorphism from T(L∞(T)) onto L∞(T) such that Tφ↦ φ and the kernel of the homomorphism coincides with commutator ideal in T(L∞(T)). In this paper, we use the Berezin transform to study results akin to Douglas's theorem for operators on the Hardy space H²(Dⁿ) over the open unit polydisc Dⁿ for n≥ 1. We further obtain a class of bigger C^*-algebras than the Toeplitz algebra T(L∞(Tⁿ)) for which the analog of symbol map still holds true.