Banach Algebras of Commuting Toeplitz Operators on the Unit Ball via the Quasi-hyperbolic Group

Abstract

We continue the study of commutative algebras generated by Toeplitz operators acting on the weighted Bergman spaces over the unit ball Bn in Cn. As was observed recently, apart of the already known commutative Toeplitz C * -algebras, quite unexpectedly, there exist many others, not geometrically defined, classes of symbols which generate commutative Toeplitz operator algebras on each weighted Bergman space. These classes of symbols were in a sense subordinated to the quasi-elliptic and quasi-parabolic groups of biholomorphisms of the unit ball. The corresponding commutative operator algebras were Banach, and being extended to the C * -algebras they became non-commutative. We consider here the case of symbols subordinated to the quasi-hyperbolic group and show that such classes of symbols are as well the sources for the commutative Banach algebras generated by Toeplitz operators. That is, together with the results of [11, 12], we cover the multidimensional extensions of all three model cases on the unit disk.