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 \T_{: 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 T_ 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.