Algebra generated by Toeplitz operators with $${\mathbb {T}}$$-invariant symbols

Abstract

We study the structure of the C Ã-algebras generated by Toeplitz operators acting on the weighted Bergman space A 2 k ðB 2 Þ on the two-dimensional unit ball, whose symbols are invariant under the action of the group T. We consider three principally different basic cases of its action t : ðz 1 ; z 2 Þ7 !ðtz 1 ; t k 2 z 2 Þ, with k 2 ¼ 1; 0; À1. The properties of the corresponding Toeplitz operators as well as the structure of the C Ãalgebra generated by them turn out to be drastically different for these three cases.