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