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?