C*-algebra without finite-dimensional representations is simple?

Question

Suppose $A$ is an infinite dimensional simple $C^*$-algebra. Then it has no non-zero finite dimensional representations. Is the converse also true? That is to say, if a $C^*$-algebra has no finite dimensional representation, can we conclude that the $C^*$-algebra is simple?

Answer 1

Take any simple infinite-dimensional $A_0$, and form $A=A_0\oplus A_0$. Then $A$ is not simple. And $A$ has no finite-dimensional representations, because all representations restrict to representations of $A_0$.

Also, a trivial explicit example is $B(H)$ with $H$ infinite-dimensional, which is not simple and has no finite-dimensional represenations.