Let $A$ and $B$ be two C$^{*}$-algebras (assume $B=\mathcal{O}_{2}\otimes\mathcal{K}$ if it helps) and suppose we have a non-zero projection $p\in A\otimes B$. (We can assume $A$ is nuclear, so that there is only one possible tensor product.)
Does there exist a choice of elements $a_{1},\ldots,a_{n}\in A$ and $b_{1},\ldots,b_{n}\in B$ such that:
- $\left\|\left(\sum_{i=1}^{n}a_{i}\otimes b_{i}\right)-p\right\|<\frac{1}{2}$;
- If $\pi$ is a non-zero irreducible representation of $A$ and $(\pi\otimes\operatorname{id})(p)=0$, then $\pi\left(\sum_{i=1}^{n}|a_{i}|\right)=0$?
All I can deduce is that for a given choice of $a_{i}$'s and $b_{i}$'s, and and irreducible representation $\pi$ of $A$, we have $$ \left\|\sum_{i=1}^{n}\pi(a_{i})\otimes b_{i}\right\|=\left\|(\pi\otimes \operatorname{id})\left(\left(\sum_{i=1}^{n}a_{i}\otimes b_{i}\right)-p\right)\right\|\leq \left\|\left(\sum_{i=1}^{n}a_{i}\otimes b_{i}\right)-p\right\|<\frac{1}{2}, $$ provided that $(\pi\otimes\operatorname{id})(p)=0$.