Extending known results for the unit disk, we prove that for the unit ball B n there exist n + 2 different cases of commutative C * -algebras generated by Toeplitz operators, acting on weighted Bergman spaces. In all cases the bounded measurable symbols of Toeplitz operators are invariant under the action of certain commutative subgroups of biholomorphisms of the unit ball.
Mathematics Subject Classification (2000). Primary 47B35; Secondary 47L80, 32A36.
We present here a quite unexpected result: Apart from already known commutative C * -algebras generated by Toeplitz operators on the unit ball, there are many other Banach algebras generated by Toeplitz operators which are commutative on each weighted Bergman space. These last algebras are non conjugated via biholomorphisms of the unit ball, non of them is a C * -algebra, and for n = 1 all of them collapse to the algebra generated by Toeplitz operators with radial symbols.
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