Given a separable $C^*$-algebra $\mathcal{A}$ can we construct a $*$-representation $\phi: \mathcal{A} \rightarrow \mathcal{B}(l_2)$ such that for every $x \neq 0$, ideal generated by $\phi(x)$ is $\mathcal{B}(l_2)$?
I only guess that the image of $\phi$ should contain all non-compact elements of $B(\mathcal{l_2})$. Thanks in advance for any idea.
I'll give you a couple of hints:
1) You're guess isn't quite right. If $H$ is a separable Hilbert space and $T\in\mathcal B(H)$ is non-compact, then the ideal generated by $T$ is $\mathcal B(H)$ (this follows from the fact that the ideal $\mathcal K(H)$ of $\mathcal B(H)$ is unique). Thus, it suffices to find a representation $\phi:A\to\mathcal B(H)$ such that for any $x\in A$ non-zero, $\phi(x)$ is non-compact.
2) Take any faithful represenation $\varphi:A\to\mathcal B(H)$ of $A$ on a separable Hilbert space $H$, and tweak it (possibly infinitely many times) until you get the desired property.
As a side note (and possible third hint), depending on who you talk to, if the representaiton $\varphi:A\to\mathcal B(H)$ is also non-degenerate (which can also be required, without any more hypotheses), it is sometimes called an ample or a standard representation of $A$.