On the structure of commutative Banach algebras generated by Toeplitz operators on the unit ball. Quasi-elliptic case. I: Generating subalgebras

Extending recent results in [3] to the higher dimensional setting n⩾3n⩾3 we provide a further step in the structural analysis of a class of commutative Banach algebras generated by Toeplitz operators on the standard weighted Bergman space over the n  -dimensional complex unit ball. The algebras Bk(h)Bk(h) under study are subordinated to the quasi-elliptic group of automorphisms of BnBn and in terms of their generators they were described in [23]. We show that Bk(h)Bk(h) is generated in fact by an essentially smaller set of operators, i.e., the Toeplitz operators with k-quasi-radial symbols and a finite set of Toeplitz operators with “elementary” k-quasi-homogeneous symbols. Then we analyze the structure of the commutative subalgebras corresponding to these two types of generating symbols. In particular, we describe spectra, joint spectra, maximal ideal spaces and the Gelfand transform.