Commutative algebras of Toeplitz operators on the unit disk

The C*-algebras generated by Toeplitz operators which are commutative on each weighted Bergman space over the unit disk were completely classified recently. Under some technical assumption the result was as follows. A C*-algebra generated by Toeplitz operators is commutative on each weighted Bergman space if and only if the corresponding symbols of Toeplitz operators are constant on cycles of a pencil of hyperbolic geodesics on the unit disk, or if and only if the corresponding symbols of Toeplitz operators are invariant under the action a maximal commutative subgroup of the Moebius transformations of the unit disk.

Generalizing this result to Toeplitz operators on the unit ball, it was proved that, given a maximal commutative subgroup of biholomorphisms of the unit ball, the C*-algebra generated by Toeplitz operators, whose symbols are invariant under the action of this subgroup, is commutative on each weighted Bergman space. It was firmly expected that the above algebras exhaust all possible algebras of Toeplitz operators on the unit ball which are commutative on each weighted Bergman space.

Unexpectedly it turns out that for the unit ball of dimension grater then one there exist many other Banach algebras generated by Toeplitz operators which are commutative on each weighted Bergman space. These algebras are non conjugated via biholomorphisms of the unit ball, non of them is a C*-algebra, and for the one-dimensional case all of them collapse to already known commutative C*-algebras generated by Toeplitz operators on the unit disk.