The function-operator convolution algebra over the Bergman space of the ball and its Gelfand theory

TL;DR

This paper explores the algebraic and Gelfand theoretical structure of a convolution algebra formed by radial functions and operators on the Bergman space, revealing insights into its Fourier transform and spectral properties.

Contribution

It introduces and analyzes a new convolution algebra over the Bergman space, connecting quantum harmonic analysis with operator theory and Gelfand theory.

Findings

Characterization of the Gelfand spectrum of the algebra

Development of a Fourier transform for the algebra

Insights into the spectral properties of radial operators

Abstract

We investigate the structure of the commutative Banach algebra formed as the direct sum of integrable radial functions on the disc and the radial operators on the Bergman space, endowed with the convolution from quantum harmonic analysis as the product. In particular, we study the Gelfand theory of this algebra and discuss certain properties of the appropriate Fourier transform of operators which naturally arises from the Gelfand transform.