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

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!

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.