Commutative $G$-invariant Toeplitz C$^\ast$ algebras on the Fock space and their Gelfand theory through Quantum Harmonic Analysis

TL;DR

This paper explores spectral synthesis in Quantum Harmonic Analysis to identify new commutative Toeplitz C*-algebras on the Fock space and examines their Gelfand theory.

Contribution

It introduces a new class of commutative Toeplitz C*-algebras on the Fock space and analyzes their Gelfand theory using Quantum Harmonic Analysis techniques.

Findings

Identified a new class of commutative Toeplitz C*-algebras.

Analyzed the Gelfand theory of these algebras.

Connected spectral synthesis with the structure of Toeplitz algebras.

Abstract

We discuss the notion of spectral synthesis for the setting of Quantum Harmonic Analysis. Using these concepts, we study subalgebras of the full Toeplitz algebra with certain invariant symbols and their commutators. In particular, we find a new class of commutative Toeplitz C$^\ast$ algebras on the Fock space. In the end, we investigate the Gelfand theory of those commutative C$^\ast$ algebras.