Commutative C* algebras and Gelfand theory through phase space methods

TL;DR

This paper introduces a phase space approach to analyze the Gelfand spectrum of commutative operator algebras, extending quantum spectral synthesis techniques to broader abelian phase spaces and operator-valued cases.

Contribution

It presents a novel method combining phase space techniques with Gelfand theory, applicable to a wider class of commutative algebras and their operator-valued extensions.

Findings

Characterization of Gelfand spectra for certain commutative operator algebras

Extension of spectral characterization to operator-valued algebras

Application of phase space methods to abelian phase spaces

Abstract

We show how the Gelfand spectrum of certain commutative operator algebras can be studied based on the theorem of Stone and von Neumann. The method presented is a natural addition to the tools of quantum spectral synthesis, which were recently used to characterize certain commutative Toeplitz algebras on the Fock space. Our method applies to this setting and also to more general abelian phase spaces. Besides characterizing Gelfand spectra of such commutative operator algebras, we also prove an extension of this result to the operator-valued case.