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

Abstract

We study the structure of the $$C^*$$ -algebras generated by Toeplitz operators acting on the weighted Bergman space $${\mathcal {A}}^2_{\lambda }({\mathbb {B}}^2)$$ on the two-dimensional unit ball, whose symbols are invariant under the action of the group $${\mathbb {T}}$$ . We consider three principally different basic cases of its action $$t:\,(z_1,z_2) \mapsto (tz_1,t^{k_2}z_2)$$ , with $$k_2=1,0,-1$$ . The properties of the corresponding Toeplitz operators as well as the structure of the $$C^*$$ -algebra generated by them turn out to be drastically different for these three cases.