for every finite dimensional $C^*$-algebra there is a faithful, non-degenerate representation-> is $\dim H_1<\infty$?

Question

Let $A$ be a $C^*$-algebra, $A$ finite dimensional. Then there is a faithful, non-degenerate representation of $A$.

How to prove it?. Take an irreducible representation $\pi_1:A\to L(H_1)$ of $A$ with $H_1=\overline{\pi(A)x}$ for a fixed $x\in H_1\setminus\{0\}$. It is $\dim\pi(A)<\infty$, but is $\dim H_1<\infty$?

The rest is clear now, I still have to construct a representation which is faithful.

Answer 1

$H_1$ is obviously finite-dimensional: if $a_1,\ldots,a_n$ is a basis for $A$, then $$ \pi(a_1)x,\ldots,\pi(a_n)x $$ span $H_1$.