Approximate unit for a certain C*-algebra

Question

Let $A$ be a C*-algebra and $p$ a projection in $A^{**}$. To prove $p$ is the smallest unit for $B: = \{a\in A; pap=a\}$, suppose $\{u_i\}$ is an approximate unit for $B$. It's easy to see $q: = w^*-\lim u_i$ is a projection. Also $q\leq p$. How can prove $p=q$?

Answer 1

Indeed this assertion holds if and only if $p$ is an open projection.

Def. A projection $p$ in $A^{**}$ is called open if there is an increasing net $\{a_i\}$ of positive elements of the unit ball of $A$ with $p=w^*-\lim a_i$.

Example. Let us consider the C*-algebra $A=C[0,1]$. If $p$ is a minimal projection in $A^{**}=C[0,1]^{**}$ then $a=pap$ if and only if $a=0$, since (always) minimal projections in $A^{**}$ are just support of pure states on $A$ and so in this particular case they are just Dirac measures.