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

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