# Exactness of a \$C^*\$-algebra

Let $$A$$ be a $$C^*$$-algebra. Then $$A$$ is called exact if there exists a faithful nuclear representation $$pi: A hookrightarrow B(H)$$.

Question: If $$A$$ is an exact $$C^*$$-algebra, is every faithful representation $$pi’: A hookrightarrow B(H’)$$ automatically nuclear?

When $$A$$ is unital, we can view use Arveson’s extension theorem to find a contractive completely positive map $$Phi: B(H) to B(H’)$$ such that
$$Phi pi = pi’$$. Since $$pi$$ is nuclear, so is $$pi’$$.

How can we deal with the non-unital case?