  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.
