A certain *-isomorphism

Question

let $A$ be a C*-algebra and $z\in A^{**}$ the supremum of all the minimal projections in $A^{**}$. How can show $* -$ homomorphism $A\to zA\subset zA^{**}$ is injective?

Answer 1

It is enough you combine the following two facts:

1- For any positive element $a\in A$ there is pure state $\phi$ on $A$ with $\phi(a)=||a||$.

2- Minimal projections are just the support of pure states on $A$.