In the book of Rordam, Larsen, Lautsen (exercise 5.7) they give three equivalent definitions for a simple, unital, purely infinite $C^*$-algebra. Assume $A$ is unital, simple and not equal to $\mathbb C$. Then the following are equivalent
- For every non-zero positive $a \in A$ there is some $x \in A$ with $1_A = x^*ax$.
- Every non-zero hereditary sub-$C^*$-algebra of $A$ contains a projection equivalent to $1_A$
- Every non-zero projection in $A$ is properly infinite and every hereditary sub-$C^*$-algebra of $A$ contains a non-zero projection.
I was wondering where the fact is needed that $A$ is simple. For example, I think that the first condition already implies that $A$ is simple.
It is needed for showing $(c) \Rightarrow (a)$ :
Following the Hint the book suggests you, with the same notations, you find a non-zero, properly infinite projection $p$ in $\overline{aAa}$. A is simple, so $p$ must be a full projection. By 4.9(i) we know that $1_{A} \preceq p$, i.e. there exists a projection $q$ in $A$ s.t. $1 \sim q \leq p$.
So, if you take $v\in A$ that implements the equivalence: $v^*v=1_{A}$ and $vv^*=q$ then you get:
$v^*pv= v^*vv^*pv= v^*qpv=v^*qv=1_{A}$, exactly as the book want you to find.
I hope this helps.