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. © 2022, The Author(s).