About GCR C*algebra

Question

I cannot understand that a simple GCR C*algebra is *-isomorphic to the set of all compact operators on some Hilbert space.

Please tell me how to show this.

Answer 1

Let $A$ be a simple GCR C*-algebra and $\pi:A\to B(H)$ be an irreducible C*-algebra. Since $A$ is simple then $\pi$ is one to one and so it is an isometry. Since $A$ is GCR then $K(H)\subseteq \pi(A)$. Clearly $K(H)$ is a closed ideal in $\pi(A)$ , so if it were proper in $\pi(A)$ then $\pi^{-1}(K(H))$ forms a non-trivial closed ideal in $A$ (since $\pi$ is 1-1) which is impossible. This proves that $\pi(A)=K(H)$.