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} $ apurestate 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.