I am trying to study the paper [*Consistency of a Counterexample to Naimark’s Problem*](http://www.pnas.org/content/101/20/7522.full.pdf+html) 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.



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> **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} $.
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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.



A Question About Pure States, Support Projections and Central Covers
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 $\pi\_\phi(A)'' = B(H\_\phi)$.


