Spectrum of a Schwartz space of cotangent bundle

Question

Let $E \to X$ be a smooth vector bundle over a $C^\infty$-manifold. There is an isomorphism between the algebra $(\mathcal{S}_c(E),*)$ and $(\mathcal{S}_c(E^*),.)$ by the Fourier transformation. My question is about the latter algebra. Can I say this algebra is commutative and by Gelfand transformation make the following conclusion : $Sp(\mathcal{S}_c(E^*)) =E^*$ ? On the other hand, I was wondering that maybe I can consider the Fourier transformation as a Gelfand transformation, and then the result would not be the same.