# On commutative C^{∗}-algebras generated by Toeplitz operators with \( \mathbb{T}^{m}\)-invariant symbols

## Abstract

It is known that Toeplitz operators, whose symbols are invariant under the action a maximal Abelian subgroups of biholomorphisms of the unit ball \( \mathbb{B}^{n}\), generate the *C*^{∗}-algebra being commutative in each standardly weighted Bergman space. In case of the unit disk (*n* = 1) this condition on generating symbols is also necessary in order that the corresponding *C*^{∗}-algebra be commutative. In this paper, for *n* > 1, we describe a wide class of symbols that are not invariant under the action of any maximal Abelian subgroup of biholomorphisms of the unit ball, and which, nevertheless, generate via corresponding Toeplitz operators *C*^{∗}-algebras being commutative in each standardly weighted Bergman space. These classes of symbols are certain proper subsets of functions that are invariant under the action of the group \( \mathbb{T}^{m}\), with *m* ≤ *n*, being a subgroup of the maximal Abelian subgroup \( \mathbb{T}^{n}\) of biholomorphisms of \( \mathbb{B}^{n}\).

## Keywords

Toeplitz operators Bergman space commutative C^{∗}-algebras

## Mathematics Subject Classification (2010)

Primary 47B35 Secondary 47L80 32A36## Preview

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