Commutative C∗-algebras of Toeplitz operators and quantization on the unit disk

A family of recently discovered commutative C∗-algebras of Toeplitz operators on the unit disk can be classified as follows. Each pencil of hyperbolic straight lines determines a set of symbols consisting of functions which are constant on the corresponding cycles, the orthogonal trajectories to lines forming a pencil. The C∗-algebra generated by Toeplitz operators with such symbols turns out to be commutative. We show that these cases are the only possible ones which generate the commutative C∗-algebras of Toeplitz operators on each weighted Bergman space.