Lemma 43.4.3.label Let $H$ be a complex Hilbert space, $M \subset B(H)$ be a von Neumann algebra, and $\varphi: M \to \complex$ be a positive linear functional. If for any mutually orthogonal projections $\mathcal{P}\subset M$, $\varphi\paren{\sum_{p \in \mathcal{P}}p}= \sum_{p \in \mathcal{P}}\varphi(p)$, then there exist a non-zero projection $p \in M$ and $\xi \in pH$ such that $\varphi(x) = \dpn{x\xi, \xi}{H}$ for all $x \in pMp$.
Proof, [Lemma 2.5.6., AP]. Let $\eta \in H$ with $\varphi(1_{M}) < \dpn{\eta, \eta}{H}$. Let $\mathcal{P}\subset M$ be a maximal family of mutually orthogonal projections with $\varphi(p) \ge \dpn{p\eta, \eta}{H}$ for all $p \in \mathcal{P}$. By assumption,
Let $p := 1_{M} - \sum_{q \in \mathcal{P}}q > 0$, then by maximality of $\mathcal{P}$, $\varphi(q) < \dpn{q\eta, \eta}{H}$ for all subprojections $q \le p$.
Let $x \in (pMp)_{+}$ with $0 \le x \le 1$. By Theorem 43.7.2, there exist projections $\seq{q_n}\subset pMp$ such that $x = \sum_{n = 1}^{\infty} \frac{q_{n}}{2^{n}}$. In which case[1],
Therefore $\dpn{x\eta, \eta}{H}\ge \varphi(x)$ for all $x \in (pMp)_{+}$. By the Radon-Nikodym theorem for positive functionals, there exists $\xi \in pH$ with $\varphi(x) = \dpn{x\xi, \xi}{H}$ for all $x \in pMp$.$\square$
- Here, only continuity of $\varphi$ is used.keyboard_return
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