Let $A$ be a C$^*$-algebra. Let $Irr(A)=\{[\pi]: \pi$ is an irreducible representation of A} and $\rho\in [\pi]$ if there is an unitary operator $V:H_{\pi}\to H_{\rho}$ such that $V\pi(a)=\rho(a)V$ for all $a\in A$. We had in lecture:
If $A\subseteq K(H)$ is a C$^*$-subalgebra, $Irr(A)$ is endowed with the discrete topology ($K(H)$ are the compact operators on a complex Hilbert space), then $$A\cong \bigoplus_{[\pi]\in Irr(A)}K(H_{\pi}).$$
As a consequence of this, we had: If $A\subseteq K(H)$ is a $C^*$-subalgebra, then the ker-hull-topology on $Irr(A)$ is the discrete topology.
We didn't prove the consequence and I have no idea, why the ker-hull-topology on $Irr(A)$ is the discrete topology. Can you explain me this?
Edit: I answered my question on mathoverflow https://mathoverflow.net/questions/221276/why-is-the-ker-hull-topology-on-irra-is-the-discrete-topology
The answer is: It is $$\hat{A}\cong\widehat{ \bigoplus_{[\pi]\in Irr(A)}K(H_{\pi})}\cong \coprod_{[\pi]}\widehat{K(H_{\pi})} ,$$ everything is endowed with the ker-hull-topology. Now, the $K(H_{\pi})$ are simple, it follows $\widehat{K(H_{\pi})}=\{[id_\pi]\}$ and $\hat{A}$ is a disjoint union of onepoint-sets. It follows that the ker-hull-topology on $\hat{A}$ is the discrete topology.