Abstract
We will discuss a quite unexpected phenomenon in the theory of Toeplitz operators on the Bergman space: the existence of a reach family of commutative C*‐algebras generated by Toeplitz operators with non‐trivial symbols. As it tuns out the smoothness properties of symbols do not play any role in the commutativity, the symbols can be merely measurable. Everything is governed here by the geometry of the underlying manifold, the hyperbolic geometry of the unit disk. We mention as well that the complete characterization of these commutative C*‐algebras of Toeplitz operators requires the Berezin quantization procedure. These commutative algebras come with a powerful research tool, the spectral type representation for the operators under study, which permit us to answer to many important questions in the area.
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