Commutative Toeplitz Algebras and Their Gelfand Theory: Old and New Results

Wolfram Bauer; Miguel Angel Rodriguez Rodriguez

doi:10.1007/s11785-022-01248-1

Commutative Toeplitz Algebras and Their Gelfand Theory: Old and New Results

Wolfram Bauer^*
, Miguel Angel Rodriguez Rodriguez

^*Corresponding author for this work

Institute of Analysis

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

Abstract

We present a survey and new results on the construction and Gelfand theory of commutative Toeplitz algebras over the standard weighted Bergman and Hardy spaces over the unit ball in Cⁿ. As an application we discuss semi-simplicity and the spectral invariance of these algebras. The different function Hilbert spaces are dealt with in parallel in successive chapters so that a direct comparison of the results is possible. As a new aspect of the theory we define commutative Toeplitz algebras over spaces of functions in infinitely many variables and present some structural results. The paper concludes with a short list of open problems in this area of research.

Original language	English
Article number	77
Journal	Complex Analysis and Operator Theory (CAOT)
Volume	16
Issue number	6
E-pub ahead of print	30 Jun 2022
DOIs	https://doi.org/10.1007/s11785-022-01248-1
Publication status	Published - Sept 2022

Keywords

Bergman and Hardy space
Commutative Banach algebras
Fock space of functions in infinitely many variables
Gaussian measure in infinite dimensions

ASJC Scopus subject areas

Computational Mathematics
Computational Theory and Mathematics
Applied Mathematics

Access to Document

10.1007/s11785-022-01248-1Licence: CC BY 4.0

Cite this

@article{4d34684d78ca444b9ff5a800b70ce1cd,

title = "Commutative Toeplitz Algebras and Their Gelfand Theory: Old and New Results",

abstract = "We present a survey and new results on the construction and Gelfand theory of commutative Toeplitz algebras over the standard weighted Bergman and Hardy spaces over the unit ball in Cn. As an application we discuss semi-simplicity and the spectral invariance of these algebras. The different function Hilbert spaces are dealt with in parallel in successive chapters so that a direct comparison of the results is possible. As a new aspect of the theory we define commutative Toeplitz algebras over spaces of functions in infinitely many variables and present some structural results. The paper concludes with a short list of open problems in this area of research.",

keywords = "Bergman and Hardy space, Commutative Banach algebras, Fock space of functions in infinitely many variables, Gaussian measure in infinite dimensions",

author = "Wolfram Bauer and \{Rodriguez Rodriguez\}, \{Miguel Angel\}",

note = "Funding Information: Open Access funding enabled and organized by Projekt DEAL. The second author was partially supported by Consejo Nacional de Ciencia y Tecnolog{\'i}a (Conacyt), Mexico.",

year = "2022",

month = sep,

doi = "10.1007/s11785-022-01248-1",

language = "English",

volume = "16",

journal = "Complex Analysis and Operator Theory (CAOT)",

issn = "1661-8254",

publisher = "Springer International Publishing",

number = "6",

}

TY - JOUR

T1 - Commutative Toeplitz Algebras and Their Gelfand Theory

T2 - Old and New Results

AU - Bauer, Wolfram

AU - Rodriguez Rodriguez, Miguel Angel

N1 - Funding Information: Open Access funding enabled and organized by Projekt DEAL. The second author was partially supported by Consejo Nacional de Ciencia y Tecnología (Conacyt), Mexico.

PY - 2022/9

Y1 - 2022/9

N2 - We present a survey and new results on the construction and Gelfand theory of commutative Toeplitz algebras over the standard weighted Bergman and Hardy spaces over the unit ball in Cn. As an application we discuss semi-simplicity and the spectral invariance of these algebras. The different function Hilbert spaces are dealt with in parallel in successive chapters so that a direct comparison of the results is possible. As a new aspect of the theory we define commutative Toeplitz algebras over spaces of functions in infinitely many variables and present some structural results. The paper concludes with a short list of open problems in this area of research.

AB - We present a survey and new results on the construction and Gelfand theory of commutative Toeplitz algebras over the standard weighted Bergman and Hardy spaces over the unit ball in Cn. As an application we discuss semi-simplicity and the spectral invariance of these algebras. The different function Hilbert spaces are dealt with in parallel in successive chapters so that a direct comparison of the results is possible. As a new aspect of the theory we define commutative Toeplitz algebras over spaces of functions in infinitely many variables and present some structural results. The paper concludes with a short list of open problems in this area of research.

KW - Bergman and Hardy space

KW - Commutative Banach algebras

KW - Fock space of functions in infinitely many variables

KW - Gaussian measure in infinite dimensions

UR - https://www.scopus.com/pages/publications/85133143089

U2 - 10.1007/s11785-022-01248-1

DO - 10.1007/s11785-022-01248-1

M3 - Article

AN - SCOPUS:85133143089

SN - 1661-8254

VL - 16

JO - Complex Analysis and Operator Theory (CAOT)

JF - Complex Analysis and Operator Theory (CAOT)

IS - 6

M1 - 77

ER -

Commutative Toeplitz Algebras and Their Gelfand Theory: Old and New Results

Abstract

Keywords

ASJC Scopus subject areas

Access to Document

Other files and links

Cite this