Commutative C*-Algebras of Toeplitz Operators on the Unit Ball, II. Geometry of the Level Sets of Symbols

Abstract.

In the first part [16] of this work, we described the commutative C*-algebras generated by Toeplitz operators on the unit ball ${\mathbb{B}}^{n}$ whose symbols are invariant under the action of certain Abelian groups of biholomorphisms of ${\mathbb{B}}^{n}$. 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 ${\mathbb{B}}^{n}$. 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.