On cyclic representations of a C*-algebras

Question

I feel the following assertion is true but have no evidence to prove:

There exists an infinite dimensional C*-algebra such any cyclic representation $\pi$ of $A$ is finite dimensional! Probably $\bigoplus_{1}^{\infty}M_2(\mathbb{C})$ works.

Answer 1

I think that one doesn't work.

In fact, if $A$ is any separable infinite-dimensional C$^*$-algebra, then it has a faithful state $\varphi$. If we do GNS for $\varphi$, then $\pi_\varphi$ is a cyclic representation. And it is infinite-dimensional, because $A\subset H_\varphi$ (due to the fact that $\varphi $ is faithful).