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 TecnologĂa (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 - http://www.scopus.com/inward/record.url?scp=85133143089&partnerID=8YFLogxK
U2 - 10.1007/s11785-022-01248-1
DO - 10.1007/s11785-022-01248-1
M3 - Article
AN - SCOPUS:85133143089
VL - 16
JO - Complex Analysis and Operator Theory
JF - Complex Analysis and Operator Theory
SN - 1661-8254
IS - 6
M1 - 77
ER -