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A localization framework for 𝕋m-Invariant Toeplitz Algebras: Applications to Gelfand Theory

  • Miguel Angel Rodriguez Rodriguez
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Abstract

We study 𝕋m-invariant Toeplitz operators on weighted Bergman spaces over the unit ball. A natural unitary transformation represents each such operator as a direct sum of restrictions of Toeplitz operators acting on a polyball. Under an L∞-valued angular continuity assumption, the corresponding symbols extend from β„•0m to a norm-continuous family indexed by the maximal ideal space of the C*-algebra generated by k-quasi-radial Toeplitz operators. This extension yields a localization framework for Toeplitz operator algebras. As an application, we consider commutative Banach algebras obtained by adjoining Toeplitz operators whose symbols are invariant under mixed unitary and circle actions. We identify their local quotient algebras on finite-coordinate strata and at infinity and use these identifications to describe their maximal ideal spaces and Gelfand transforms explicitly.

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Cite

BibTeXRIS
BibTeX
@misc{rodriguez2026localization,
  title = {{A localization framework for $\mathbbT^m$-Invariant Toeplitz Algebras: Applications to Gelfand Theory}},
  author = {Miguel Angel Rodriguez Rodriguez},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.12368},
  eprint = {2609.12368},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.12368},
}
RIS
TY  - UNPB
TI  - A localization framework for 𝕋^m-Invariant Toeplitz Algebras: Applications to Gelfand Theory
AU  - Miguel Angel Rodriguez Rodriguez
PY  - 2026
DA  - 2026/09/11
PB  - arXiv
DO  - 10.48550/arxiv.2609.12368
UR  - https://arxiv.org/abs/2609.12368
AB  - We study 𝕋^m-invariant Toeplitz operators on weighted Bergman spaces over the unit ball. A natural unitary transformation represents each such operator as a direct sum of restrictions of Toeplitz operators acting on a polyball. Under an L^∞-valued angular continuity assumption, the corresponding symbols extend from β„•_0^m to a norm-continuous family indexed by the maximal ideal space of the C^*-algebra generated by k-quasi-radial Toeplitz operators. This extension yields a localization framework for Toeplitz operator algebras. As an application, we consider commutative Banach algebras obtained by adjoining Toeplitz operators whose symbols are invariant under mixed unitary and circle actions. We identify their local quotient algebras on finite-coordinate strata and at infinity and use these identifications to describe their maximal ideal spaces and Gelfand transforms explicitly.
ER  -