## Dense *-algebra

We say that $x$ is of the class $C^{\infty}$ if there is a smooth mapping $g\mapsto \alpha_g(x): G\rightarrow A$ where $\alpha$ is an automorphism of the C*-algebra $A$ and $G$ is a Lie group. The tripple $(A,G,\alpha)$ forms a C*-dynamical system. Define $A^{\infty}:=\{x\in A: x~ \text{is of class }~ C^{\infty}\}$. Show that $A^{\infty}$ in dense in $A$.