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Complex Analysis and Operator Theory

, Volume 13, Issue 2, pp 493–524 | Cite as

Algebras of Toeplitz Operators on the n-Dimensional Unit Ball

  • Wolfram Bauer
  • Raffael Hagger
  • Nikolai VasilevskiEmail author
Article

Abstract

We study \(C^*\)-algebras generated by Toeplitz operators acting on the standard weighted Bergman space \(\mathcal {A}_{\lambda }^2({\mathbb {B}}^n)\) over the unit ball \({\mathbb {B}}^n\) in \({\mathbb {C}}^n\). The symbols \(f_{ac}\) of generating operators are assumed to be of a certain product type, see (1.1). By choosing a and c in different function algebras \(\mathcal {S}_a\) and \(\mathcal {S}_c\) over lower dimensional unit balls \({\mathbb {B}}^{\ell }\) and \({\mathbb {B}}^{n-\ell }\), respectively, and by assuming the invariance of \(a\in \mathcal {S}_a\) under some torus action we obtain \(C^*\)-algebras \(\varvec{\mathcal {T}}_{\lambda }(\mathcal {S}_a, \mathcal {S}_c)\) of whose structural properties can be described. In the case of k-quasi-radial functions \(\mathcal {S}_a\) and bounded uniformly continuous or vanishing oscillation symbols \(\mathcal {S}_c\) we describe the structure of elements from the algebra \(\varvec{\mathcal {T}}_{\lambda }(\mathcal {S}_a, \mathcal {S}_c)\), derive a list of irreducible representations of \(\varvec{\mathcal {T}}_{\lambda }(\mathcal {S}_a, \mathcal {S}_c)\), and prove completeness of this list in some cases. Some of these representations originate from a “quantization effect”, induced by the representation of \(\mathcal {A}_{\lambda }^2({\mathbb {B}}^n)\) as the direct sum of Bergman spaces over a lower dimensional unit ball with growing weight parameter. As an application we derive the essential spectrum and index formulas for matrix-valued operators.

Keywords

Weighted Bergman spaces Operator \(C^*\)-algebra Irreducible representations 

Mathematics Subject Classification

Primary 47B35 47L80 Secondary 32A36 

References

  1. 1.
    Akkar, Z., Albrecht, E.: Spectral properties of Toeplitz operators on the unit ball and the unit sphere. In: The Varied Landscape of Operator Theory, Theta Series in Advanced Mathematics, 17, Theta, Bucharest, pp. 1–22 (2014)Google Scholar
  2. 2.
    Bauer, W., Coburn, L.A.: Heat flow, weighted Bergman spaces and real analytic Lipschitz approximation. J. Reine Angew. Math. 703, 225–246 (2015)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Bauer, W., Coburn, L.A.: Toeplitz operators with uniformly continuous symbols. Integral Equ. Oper. Theory 83, 25–34 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bauer, W., Hagger, R., Vasilevski, N.: Uniform continuity and quantization on bounded symmetric domains. J. Lond. Math. Soc. (2) 96(2), 345–366 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bauer, W., Herrera Yañez, C., Vasilevski, N.: \((m,\lambda )\)-Berezin transform and approximation of operators on weighted Bergman spaces over the unit ball. Oper. Theory Adv. Appl. 240, 45–68 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bauer, W., Vasilevski, N.: On the structure of commutative Banach algebras generated by Toeplitz operators on the unit ball. Quasi-elliptic case. I: generating subalgebras. J. Funct. Anal. 265(11), 2956–2990 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bauer, W., Vasilevski, N.: On the structure of commutative Banach algebras generated by Toeplitz operators on the unit ball. Quasi-elliptic case. II: Gelfand theory. Complex Anal. Oper. Theory 9, 593–630 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Bauer, W., Vasilevski, N.: On algebras generated by Toeplitz operators and their representations. J. Funct. Anal. 272, 705–737 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Berger, C.A., Coburn, L.A., Zhu, K.H.: Function theory on Cartan domains and the Berezin–Toeplitz symbol calculus. Am. J. Math. 110, 921–953 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Coburn, L.A.: The \(C^*\)-algebra generated by an isometry. Bull. Am. Math. Soc. 73, 722–726 (1967)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Dixmier, J.: Les \(C^*\)-algebres et leurs représensations. Gauthier-Villars, Paris (1964)Google Scholar
  12. 12.
    Grudsky, S., Karapetyants, A., Vasilevski, N.: Toeplitz operators on the unit ball in \({\mathbb{C}}^n\) with radial symbols. J. Oper. Theory 49, 325–346 (2003)zbMATHGoogle Scholar
  13. 13.
    Le, T.: Commutator ideals of subalgebras of Toeplitz algebras on weighted Bergman spaces. J. Oper. Theory 64(1), 89–101 (2010)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Mitkovski, M., Suárez, D., Wick, B.: The essential norm of operators on \(A_{{\alpha }}^{p}({\mathbb{B}}_n)\). Integral Equ. Oper. Theory 75, 197–233 (2013)CrossRefzbMATHGoogle Scholar
  15. 15.
    Suárez, D.: The essential norm of operators in the Toeplitz algebra \(A^p({\mathbb{B}}_n)\). Indiana Univ. Math. J. 56(5), 2185–2232 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Vasilevski, N.: Quasi-radial quasi-homogeneous symbols and commutative Banach algebras of Toeplitz operators. Integral Equ. Oper. Theory 66(1), 141–152 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Venugopalkrishna, U.: Fredholm operators associated with strongly pseudoconvex domains in \(C^{n}\). J. Funct. Anal. 9, 349–373 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Xia, J., Zheng, D.: Toeplitz operators and Toeplitz algebra with symbols of vanishing oscillation. J. Oper. Theory 76(1), 107–131 (2016)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Wolfram Bauer
    • 1
  • Raffael Hagger
    • 1
  • Nikolai Vasilevski
    • 2
    Email author
  1. 1.Institut für AnalysisLeibniz UniversitätHannoverGermany
  2. 2.Departamento de MatemáticasCINVESTAVMexicoMexico

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