central cover of a representation of $C^*$ algebra $A$

Question

In G.K.Pedersen's book "$C^*$ algebras and their automorphism groups",there is a theorem

Two representations $(\pi_1,H_1)$ and $(\pi_2,H_2)$ of a $C^*$ algebra $A$are equivalent iff $c(\pi_1)=c(\pi_1)$,where $c(\pi_i)$ is the central cover of $\pi_i,i=1,2.$

I don't understand the following statement in the proof

$(\pi_u,H_u)$ is the univeral representation of $A$,for each projection $p\ne 0$ in $\pi_u(A)''\cong A^{**}$,the map $x\rightarrow xp,x\in A$ is a representation of $A$ on $pH_u$ with central cover $p$.

How to check that the central cover of $A\rightarrow pA$ is $p$?

Answer 1

The book says,

for each central projection $p\ne0$ in $A''$...

Since $p$ is central, the map $\pi:x\longmapsto px$ is a representation. The central cover of this representation is, by definition, the central projection $q$ such that $A''q = \pi(A)''=(Ap)''=A''p$. So $q=p$.