Approximation and symbolic calculus for Toeplitz algebras on the Bergman space

If f ∈ L∞(D) let Tf be the Toeplitz operator on the Bergman space La of the unit disk D. For a C∗-algebra A ⊂ L∞(D) let T(A) denote the closed operator algebra generated by {Tf : f ∈ A}. We characterize its commutator ideal C(A) and the quotient T(A)/C(A) for a wide class of algebras A. Also, for n ≥ 0 integer, we define the n-Berezin transform BnS of a bounded operator S, and prove that if f ∈ L∞(D) and fn = BnTf then Tfn→Tf .