## Details

Original language | English |
---|---|

Pages (from-to) | 493-524 |

Number of pages | 32 |

Journal | Complex Analysis and Operator Theory |

Volume | 13 |

Issue number | 2 |

Early online date | 25 Aug 2018 |

Publication status | Published - 13 Mar 2019 |

## Abstract

We study C ^{∗} -algebras generated by Toeplitz operators acting on the standard weighted Bergman space Aλ2(Bn) over the unit ball B ^{n} in 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 S _{a} and S _{c} over lower dimensional unit balls B ^{ℓ} and B ^{n} ^{-} ^{ℓ} , respectively, and by assuming the invariance of a∈ S _{a} under some torus action we obtain C ^{∗} -algebras T _{λ} (S _{a} , S _{c} ) of whose structural properties can be described. In the case of k-quasi-radial functions S _{a} and bounded uniformly continuous or vanishing oscillation symbols S _{c} we describe the structure of elements from the algebra T _{λ} (S _{a} , S _{c} ) , derive a list of irreducible representations of T _{λ} (S _{a} , 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 Aλ2(Bn) 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

- Irreducible representations, Operator C -algebra, Weighted Bergman spaces

## ASJC Scopus subject areas

- Mathematics(all)
**Computational Mathematics**- Computer Science(all)
**Computational Theory and Mathematics**- Mathematics(all)
**Applied Mathematics**

## Cite this

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**Algebras of Toeplitz Operators on the n-Dimensional Unit Ball.**/ Bauer, Wolfram; Hagger, Raffael; Vasilevski, Nikolai.

In: Complex Analysis and Operator Theory, Vol. 13, No. 2, 13.03.2019, p. 493-524.

Research output: Contribution to journal › Article › Research › peer review

*Complex Analysis and Operator Theory*, vol. 13, no. 2, pp. 493-524. https://doi.org/10.48550/arXiv.1808.10372, https://doi.org/10.1007/s11785-018-0837-y

*Complex Analysis and Operator Theory*,

*13*(2), 493-524. https://doi.org/10.48550/arXiv.1808.10372, https://doi.org/10.1007/s11785-018-0837-y

}

TY - JOUR

T1 - Algebras of Toeplitz Operators on the n-Dimensional Unit Ball

AU - Bauer, Wolfram

AU - Hagger, Raffael

AU - Vasilevski, Nikolai

N1 - Funding Information: This work was partially supported by CONACYT Project 238630, México and by DFG (Deutsche Forschungsgemeinschaft), Project BA 3793/4-1.

PY - 2019/3/13

Y1 - 2019/3/13

N2 - We study C ∗ -algebras generated by Toeplitz operators acting on the standard weighted Bergman space Aλ2(Bn) over the unit ball B n in 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 S a and S c over lower dimensional unit balls B ℓ and B n - ℓ , respectively, and by assuming the invariance of a∈ S a under some torus action we obtain C ∗ -algebras T λ (S a , S c ) of whose structural properties can be described. In the case of k-quasi-radial functions S a and bounded uniformly continuous or vanishing oscillation symbols S c we describe the structure of elements from the algebra T λ (S a , S c ) , derive a list of irreducible representations of T λ (S a , 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 Aλ2(Bn) 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.

AB - We study C ∗ -algebras generated by Toeplitz operators acting on the standard weighted Bergman space Aλ2(Bn) over the unit ball B n in 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 S a and S c over lower dimensional unit balls B ℓ and B n - ℓ , respectively, and by assuming the invariance of a∈ S a under some torus action we obtain C ∗ -algebras T λ (S a , S c ) of whose structural properties can be described. In the case of k-quasi-radial functions S a and bounded uniformly continuous or vanishing oscillation symbols S c we describe the structure of elements from the algebra T λ (S a , S c ) , derive a list of irreducible representations of T λ (S a , 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 Aλ2(Bn) 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.

KW - Irreducible representations

KW - Operator C -algebra

KW - Weighted Bergman spaces

UR - http://www.scopus.com/inward/record.url?scp=85052920157&partnerID=8YFLogxK

U2 - 10.48550/arXiv.1808.10372

DO - 10.48550/arXiv.1808.10372

M3 - Article

AN - SCOPUS:85052920157

VL - 13

SP - 493

EP - 524

JO - Complex Analysis and Operator Theory

JF - Complex Analysis and Operator Theory

SN - 1661-8254

IS - 2

ER -