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

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.