Details
| Originalsprache | Englisch |
|---|---|
| Seiten (von - bis) | 705-737 |
| Seitenumfang | 33 |
| Fachzeitschrift | Journal of functional analysis |
| Jahrgang | 272 |
| Ausgabenummer | 2 |
| Frühes Online-Datum | 21 Sept. 2016 |
| Publikationsstatus | Veröffentlicht - 15 Jan. 2017 |
Abstract
We study Banach and C⁎-algebras generated by Toeplitz operators acting on weighted Bergman spaces Aλ 2(B2) over the complex unit ball B2⊂C2. Our key point is an orthogonal decomposition of Aλ 2(B2) into a countable sum of infinite dimensional spaces, each one of which can be identified with a differently weighted Bergman space Aμ 2(D) over the complex unit disk D. Moreover, all elements of the above algebras leave each of the summands in the above decomposition invariant and their restriction to each level acts as a compact perturbation of a Toeplitz operator on Aμ 2(D). The symbols of the generating Toeplitz operators are chosen to be suitable extensions to B2 of families S of bounded functions on D. Symbol classes S that generate important classical commutative and non-commutative Toeplitz algebras in L(Aμ 2(D)) are of particular interest. In this paper we discuss various examples. In the case of S=C(D‾) and S=C(D‾)⊗L∞(0,1) we characterize all irreducible representations of the resulting Toeplitz operator C⁎-algebras. Their Calkin algebras are described and index formulas are provided.
ASJC Scopus Sachgebiete
- Mathematik (insg.)
- Analysis
Zitieren
- Standard
- Harvard
- Apa
- Vancouver
- BibTex
- RIS
in: Journal of functional analysis, Jahrgang 272, Nr. 2, 15.01.2017, S. 705-737.
Publikation: Beitrag in Fachzeitschrift › Artikel › Forschung › Peer-Review
}
TY - JOUR
T1 - On algebras generated by Toeplitz operators and their representations
AU - Bauer, Wolfram
AU - Vasilevski, Nikolai
N1 - Funding information: This work was partially supported by CONACYT Project 238630 , México and through the Deutsche Forschungsgemeinschaft , DFG Sachmittelbeihilfe BA 3793/4-1 .
PY - 2017/1/15
Y1 - 2017/1/15
N2 - We study Banach and C⁎-algebras generated by Toeplitz operators acting on weighted Bergman spaces Aλ 2(B2) over the complex unit ball B2⊂C2. Our key point is an orthogonal decomposition of Aλ 2(B2) into a countable sum of infinite dimensional spaces, each one of which can be identified with a differently weighted Bergman space Aμ 2(D) over the complex unit disk D. Moreover, all elements of the above algebras leave each of the summands in the above decomposition invariant and their restriction to each level acts as a compact perturbation of a Toeplitz operator on Aμ 2(D). The symbols of the generating Toeplitz operators are chosen to be suitable extensions to B2 of families S of bounded functions on D. Symbol classes S that generate important classical commutative and non-commutative Toeplitz algebras in L(Aμ 2(D)) are of particular interest. In this paper we discuss various examples. In the case of S=C(D‾) and S=C(D‾)⊗L∞(0,1) we characterize all irreducible representations of the resulting Toeplitz operator C⁎-algebras. Their Calkin algebras are described and index formulas are provided.
AB - We study Banach and C⁎-algebras generated by Toeplitz operators acting on weighted Bergman spaces Aλ 2(B2) over the complex unit ball B2⊂C2. Our key point is an orthogonal decomposition of Aλ 2(B2) into a countable sum of infinite dimensional spaces, each one of which can be identified with a differently weighted Bergman space Aμ 2(D) over the complex unit disk D. Moreover, all elements of the above algebras leave each of the summands in the above decomposition invariant and their restriction to each level acts as a compact perturbation of a Toeplitz operator on Aμ 2(D). The symbols of the generating Toeplitz operators are chosen to be suitable extensions to B2 of families S of bounded functions on D. Symbol classes S that generate important classical commutative and non-commutative Toeplitz algebras in L(Aμ 2(D)) are of particular interest. In this paper we discuss various examples. In the case of S=C(D‾) and S=C(D‾)⊗L∞(0,1) we characterize all irreducible representations of the resulting Toeplitz operator C⁎-algebras. Their Calkin algebras are described and index formulas are provided.
KW - Index
KW - Operator algebras
KW - Semi-classical limit
KW - Weighted Bergman space
UR - http://www.scopus.com/inward/record.url?scp=84998850050&partnerID=8YFLogxK
U2 - 10.1016/j.jfa.2016.09.013
DO - 10.1016/j.jfa.2016.09.013
M3 - Article
AN - SCOPUS:84998850050
VL - 272
SP - 705
EP - 737
JO - Journal of functional analysis
JF - Journal of functional analysis
SN - 0022-1236
IS - 2
ER -