Commutative C*-algebras of Toeplitz operators on the unit ball, II.: Geometry of the level sets of symbols

In the first part [16] of this work, we described the commutative C*- algebras generated by Toeplitz operators on the unit ball B-n whose symbols are invariant under the action of certain Abelian groups of biholomorphisms of B-n. Now we study the geometric properties of these symbols. This allows us to prove that the behavior observed in the case of the unit disk (see [3]) admits a natural generalization to the unit ball B-n. Furthermore we give a classification result for commutative Toeplitz operator C*-algebras in terms of geometric and "dynamic" properties of the level sets of generating symbols.