# A Question About Pure States, Support Projections and Central Covers

I am trying to study the paper Consistency of a Counterexample to Naimark’s Problem by Charles Akemann and Nik Weaver, and there is a claim in Lemma 1 of the paper that I am stuck at, which is as follows.

Claim: Let $\mathscr{A}$ be a $C^{*}$-algebra and $g: \mathscr{A} \to \mathbb{C}$ a pure state on $\mathscr{A}$. Let $q \in \mathscr{A}^{**}$ be the support projection of $g$. Also, let $y$ be the central cover of $q$ in $\mathscr{A}^{**}$; in other words, $y$ is the smallest projection $p \in \mathcal{Z}(\mathscr{A}^{**})$ such that $q \leq p$. Then $y \mathscr{A}^{**}$ is isomorphic to $B(\mathcal{H})$ for some Hilbert space $\mathcal{H}$.

The claim may seem obvious to the many operator theorists more capable than myself, but I am just not getting it. I believe that the GNS construction is needed here (as the claim is about a pure state, which corresponds to an irreducible $*$-representation of $\mathscr{A}$ on some Hilbert space $\mathcal{H}$), but this is the furthest that I have gotten.

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For each state $\phi$ on $A$ let $\pi_\phi: A \to B(H_\phi)$ be the corresponding GNS representation. Let $\pi: A \to B(H)$ be the direct sum of all of these representations. The von Neumann algebra $\pi(A)'' \subseteq B(H)$ is called the enveloping von Neumann algebra of $A$ and it is naturally isomorphic to $A^{**}$. (See C${}^*$-algebras and Their Automorphism Groups by Pedersen.)
For each $\phi$, since $H_\phi \subseteq H$ we have a natural projection from $\pi(A)'' \subseteq B(H)$ onto $\pi_\phi(A)'' \subseteq B(H_\phi)$. The kernel of this projection is a weak* closed ideal of $\pi(A)'' \cong A^{**}$ and hence it has the form $zA^{**}$ for some central projection $z$ in $A^{**}$. Then $y = 1-z$ is the central cover of $\phi$ and $A^{**} = yA^{**} \oplus zA^{**}$, so that $\pi_\phi(A)'' \cong A^{**}/zA^{**} \cong yA^{**}$.
So the general fact is that $\pi_\phi(A)'' \cong yA^{**}$. If $\phi$ is pure then $\pi_\phi$ is irreducible and $\pi_\phi(A)'' = B(H_\phi)$.