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

Wolfram Bauer () and Miguel Angel Rodriguez Rodriguez ()
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Wolfram Bauer: Institut für Analysis, Leibniz Universität
Miguel Angel Rodriguez Rodriguez: Institut für Analysis, Leibniz Universität

A chapter in Multivariable Operator Theory, 2023, pp 77-113 from Springer

Abstract: 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 C n $$\mathbb {C}^n$$ . 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; Gaussian measure in infinite dimensions; Fock space of functions in infinitely many variables; Commutative Banach algebras; Primary 47B35; Seconday 47L80; 32A36; Primary 47B35; Seconday 47L80; 32A36 (search for similar items in EconPapers)
Date: 2023
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-031-50535-5_4

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DOI: 10.1007/978-3-031-50535-5_4

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