Algebra generated by Toeplitz operators with \({\mathbb {T}}\)-invariant symbols

Abstract

We study the structure of the \(C^*\)-algebras generated by Toeplitz operators acting on the weighted Bergman space \({\mathcal {A}}^2_{\lambda }({\mathbb {B}}^2)\) on the two-dimensional unit ball, whose symbols are invariant under the action of the group \({\mathbb {T}}\). We consider three principally different basic cases of its action \(t:\,(z_1,z_2) \mapsto (tz_1,t^{k_2}z_2)\), with \(k_2=1,0,-1\). The properties of the corresponding Toeplitz operators as well as the structure of the \(C^*\)-algebra generated by them turn out to be drastically different for these three cases.

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Acknowledgements

This work was partially supported by CONACYT Project 238630, México.

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Correspondence to Nikolai Vasilevski.

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Vasilevski, N. Algebra generated by Toeplitz operators with \({\mathbb {T}}\)-invariant symbols. Bol. Soc. Mat. Mex. 26, 1217–1242 (2020). https://doi.org/10.1007/s40590-020-00306-y

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Keywords

  • Toeplitz operators
  • \({\mathbb {T}}\)-invariant symbol
  • Bergman space

Mathematics Subject Classification

  • Primary 47B35
  • Secondary 32A36