I'd like to prove that the identity, $I$, of a unital, simple, purely infinite $C^*$-algebra is always an infinite projection. What I'm hoping is that the following is true:
If $p$ in $\mathfrak{A}$ is an infinite projection and $p\leq q$, then $q$ is also infinite.
The original result would of course follow from this, as every projection is bounded above by $I$. Any suggestions would be much appreciated!
Since $p$ is infinite, there exists $r\leq p$ with $r\ne p$ and $r$ equivalent to $p$. That is, there exists $v$ with $v^*v=r$, $vv^*=p$.
Note that $v=pvr$, so in particular $(q-p)v=0$, and $v(q-p)=0$.
Now let $s=r+q-p$. Then $s$ is a proper subprojection of $q$. Let $w=v+q-p$. Then $$ w^*w=(v^*+q-p)(v+q-p)=v^*v+q-p=r+q-p=s, $$ $$ ww^*=vv^*+q-p=p+q-p=q. $$ So $q$ is infinite.