Abstract
We found that in the polydiskDnthere exist(n+1)(n+2)/2different classes of commutativeC⁎-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 theC⁎-algebra generated by Toeplitz operators with this kind of symbol is commutative. Thus we relate to each Abelian subgroup of biholomorphisms a commutativeC⁎-algebra of Toeplitz operators.
Highlights
Vasilevski, Quiroga, and coauthors found a connection that the commutative C∗-algebras generated by Toeplitz operators acting on the weighted Bergman space were described and classified for the case of the unit disk and the unit ball in Cn; see [1, 2] for further results and details
The classification result states that given a maximal Abelian subgroup G of biholomorphisms of the unit ball or the unit disk, the C∗-algebra generated by Toeplitz operators whose symbols are invariant under the action of G is commutative on each weighted Bergman space
We apply the above method to the polydisk Dn and we find that 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
Summary
Vasilevski, Quiroga, and coauthors found a connection that the commutative C∗-algebras generated by Toeplitz operators acting on the weighted Bergman space were described and classified for the case of the unit disk and the unit ball in Cn; see [1, 2] for further results and details. The classification result states that given a maximal Abelian subgroup G of biholomorphisms of the unit ball or the unit disk (i.e., ball of dimension one), the C∗-algebra generated by Toeplitz operators whose symbols are invariant under the action of G is commutative on each weighted Bergman space. One important advantage of the point of view of the moment map is that if we consider the symbol set invariant under the action of a nonmaximal Abelian subgroup of biholomorphisms, the C∗-algebra generated by Toeplitz operators with symbols in this set may not be commutative. In a sense, this symbol set is too big for our purposes, so this method no longer works for nonmaximal Abelian subgroups. The method to find commutative C∗-algebras of Toeplitz operators using the moment map has not been used in other manifolds and we suppose that we can apply this technique to other symplectic manifolds
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