Takesaki in his operator theory says
A C*-algebra $M$ of operators on Hilbert space $H$ means a nondegenerate ( $\text {cl} (MH) = H$) $*-$ subalgebra of $B(H)$ which is closed under the uniform topology.
Suppose $M$ is a C*-algebra. Clearly $M$ is a $*-$ Banach algebra and $\text{cl}(MH)\subset H$. Just show that $H\subset \text{cl}(MH)$ . Let $\{u_i\}$ be a approximate identity of $M$. Thus $u_i\to1$(sot) and $\xi = \lim u_i\xi \in \text{cl} (MH)$ for $\xi\in H$.
For converse direction I know that $\|x\|^2=\|xx^*\|$ in general. but I would like to know the relation between nondegenerate and C*-property. Please give me a hint. Thanks.