If $f\in L^\infty(\mathbb{D})$ let $T_f$ be the Toeplitz operator on
the Bergman space $L^2_a$ of the unit disk $\mathbb{D}$. For a
$C^\ast$-algebra $A\subset L^\infty(\mathbb{D})$ let $\mathfrak{T}(A)$ denote
the closed operator algebra generated by $\{ T_f : f\in A \}$. We
characterize its commutator ideal $\comm(A)$ and the quotient
$\mathfrak{T}(A)/ \mathfrak{C}(A)$ for a wide class of algebras $A$. Also, for
$n\geq 0$ integer, we define the $n$-Berezin transform $B_nS$ of a
bounded operator $S$, and prove that if $f\in L^\infty(\mathbb{D})$ and
$f_n = B_n T_f$ then $T_{f_n} \rightarrow T_f$.