Embeddings for ultrapowers of the universal UHF-algebra

I've come across a little obstacle and was wondering if anyone could provide a hint/suggestion/help to the following.

First of all, let $\omega$ be some fixed free ultrafilter on $\mathbb{N}$. Let $Q$ be the universal UHF-algebra and $Q_\omega$ be its ultrapower. Let $\tau_\omega$ be the unique trace on $Q_\omega$ and suppose $p$ is some projection in $Q_\omega$ such that $\tau_\omega(p)>0$. Then there exists an integer $k$ and injective $\ast$-homomorphism $\pi \colon Q_\omega \rightarrow p Q_\omega p \otimes \mathbb{M}_k$ fulfilling $\pi(pap)=pap \otimes e_{11}$ with $e_{11}$ being the $(1,1)$'th unit matrix.

So far I have managed to produce $\pi$ in the case where $\tau_\omega(p)m=1$ for some positive integer $m$. Is there any way to reduce the proof to this case?