Approximation and symbolic calculus for Toeplitz algebras on the Bergman space

If f ∈ L∞ (double-struck D sign) let Tf be the Toeplitz operator on the Bergman space La2 of the unit disk double-struck D sign. For a C*-algebra A ⊂ L∞(double-struck D sign) let script I sign(A) denote the closed operator algebra generated by {Tf : f ∈ A}. We characterize its commutator ideal c(A) and the quotient script I sign(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 5, and prove that if f ∈ L∞(double-struck D sign) and fn = BnTf then Tfn→Tf.