GNS representation of a nuclear $C^*$-algebra

Question

Suppose $A$ is a nuclear $C^*$-algebra with a tracial state $\psi$, $(\pi_{\psi},H_{\psi})$ is the GNS reprsentation with respect to $\psi$.

My question:

Does there exist $A$ which satisfy the above condition and $A/\ker(\pi_{\psi})=K(H)$, where $H$ is separable and infinite dimensional.

Can anyone show me an example? Thanks!

Answer 1

No, if $\psi\ne 0$. You can define a state on $A/\ker\pi_\psi$ by $$ \tilde\psi(a+\ker\pi_\psi)=\psi(a). $$ This is well-defined since $\psi=0$ on $\ker\pi_\psi$. So $\psi$ is a tracial state on $A/\ker\pi_\psi$. And $K(H)$ does not admit a tracial state.