Commutative C⁎-Algebras of Toeplitz Operators via the Moment Map on the Polydisk

Abstract

We found that in the polydisk Dn there exist (n+1)(n+2)/2 different classes of commutative C⁎-algebras generated by Toeplitz operators whose symbols are invariant under the action of maximal Abelian subgroups of biholomorphisms. On the other hand, using the moment map associated with each (not necessary maximal) Abelian subgroup of biholomorphism we introduced a family of symbols given by the moment map such that the C⁎-algebra generated by Toeplitz operators with this kind of symbol is commutative. Thus we relate to each Abelian subgroup of biholomorphisms a commutative C⁎-algebra of Toeplitz operators.