Toeplitz C*-algebras on radially weighted Fock spaces: commutativity and spectral representation
Abstract
We study Toeplitz operators acting on radial weighted Fock spaces. We use tools from representation theory to construct commutative families of C*-algebras that are generated by Toeplitz operators whose symbols are invariant under the action of U(n). For a partition m=(b1,...,bk) of an integer n, we realize Um:=U(b1)×...×U(bk) as a block diagonally subgroup of U(n) and describe the decomposition of the weighted Fock space into irreducible Um-modules. This allows us to study Toeplitz operators with Um-invariant, equivalently k-quasi-radial, symbols. Also, we provide an explicit integral representation of their eigenvalues. More generally, for an arbitrary compact subgroup H⊆U(n), we characterize the commutativity of the C*-algebra generated by H-invariant Toeplitz operators in terms of the multiplicity-free property of the representation π|H. Finally, for a logarithmically growing radial weight, we construct a bounded radial symbol for which the corresponding eigenvalue sequence is not uniformly continuous with respect to the square-root metric. Consequently, the uniform closure of the set of eigenvalue sequences does not coincide with the C*-algebra of bounded sequences that are uniformly continuous with respect to the square-root metric.
Related
- On the mapping properties of a radially power-weighted k-plane transform
- Absolutely summing Toeplitz operators on weighted Fock spaces
- Compact Toeplitz operators via the Berezin transform on radial weighted Bergman spaces
- IDA symbols and Schatten--Lorentz theory for Toeplitz operators on weighted Fock spaces
- Matrix algebras with orthogonal Cauchy-like transforms and Toeplitz approximation: a negative result
BibTeX
@misc{bdarneh2026toeplitz,
title = {{Toeplitz $C^*$-algebras on radially weighted Fock spaces: commutativity and spectral representation}},
author = {Khalid Bdarneh},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.20652},
eprint = {2609.20652},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.20652},
}
RIS
TY - UNPB
TI - Toeplitz C^*-algebras on radially weighted Fock spaces: commutativity and spectral representation
AU - Khalid Bdarneh
PY - 2026
DA - 2026/09/20
PB - arXiv
DO - 10.48550/arxiv.2609.20652
UR - https://arxiv.org/abs/2609.20652
AB - We study Toeplitz operators acting on radial weighted Fock spaces. We use tools from representation theory to construct commutative families of C^*-algebras that are generated by Toeplitz operators whose symbols are invariant under the action of U(n). For a partition m=(b_1,...,b_k) of an integer n, we realize U_m:=U(b_1)×...×U(b_k) as a block diagonally subgroup of U(n) and describe the decomposition of the weighted Fock space into irreducible U_m-modules. This allows us to study Toeplitz operators with U_m-invariant, equivalently k-quasi-radial, symbols. Also, we provide an explicit integral representation of their eigenvalues. More generally, for an arbitrary compact subgroup H⊆U(n), we characterize the commutativity of the C^*-algebra generated by H-invariant Toeplitz operators in terms of the multiplicity-free property of the representation π|_H. Finally, for a logarithmically growing radial weight, we construct a bounded radial symbol for which the corresponding eigenvalue sequence is not uniformly continuous with respect to the square-root metric. Consequently, the uniform closure of the set of eigenvalue sequences does not coincide with the C^*-algebra of bounded sequences that are uniformly continuous with respect to the square-root metric.
ER -