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

In: Multivariable Operator Theory

Author

Listed:
  • Wolfram Bauer

    (Institut für Analysis, Leibniz Universität)

  • Miguel Angel Rodriguez Rodriguez

    (Institut für Analysis, Leibniz Universität)

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.

Suggested Citation

  • Wolfram Bauer & Miguel Angel Rodriguez Rodriguez, 2023. "Commutative Toeplitz Algebras and Their Gelfand Theory: Old and New Results," Springer Books, in: Ernst Albrecht & Raúl Curto & Michael Hartz & Mihai Putinar (ed.), Multivariable Operator Theory, pages 77-113, Springer.
    • Handle: RePEc:spr:sprchp:978-3-031-50535-5_4
    DOI: 10.1007/978-3-031-50535-5_4
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