A question on simple and unital $C^\star$-algebra

Question

There is a quotation of a book "$C^\star$-algebras Finite-Dimensional Approximations"

Definition 1.7.4. A representation $\pi: A \rightarrow B(H)$ is called essential if $\pi(A)$ contains no nonzero compact operators. ($A$ is a $C^\star$-algebra.)

My question is If $A$ is a simple and unital $C^\star$-algebra, then all the representations on $A$ will be faithful and essential?

Answer 1

If $A$ is simple, then indeed all representations will be faithful.

As for essential, the answer is trivially no if you consider $A=M_n(\mathbb C)$. In the infinite-dimensional case, the answer is yes: the question is reduced to whether a simple unital infinite-dimensional C$^*$-algebra can contain compact operators.

Let $J=\{x\in \pi(A):\ x \text{ is compact }\}$. Then one can show that $J$ is an ideal of $\pi(A)$, so either $J=\pi(A)$ (this is the finite-dimensional case) or $J=0$.