Lemma 43.4.1 (Jordan Decomposition).label Let $H$ be a complex Hilbert space, $M \subset B(H)$ be a von Neumann algebra, and $\omega: M \to \complex$ be a weak-operator continuous linear functional, then there exist positive weak-operator continuous functionals $\bracsn{\omega_k}_{0}^{3}$ such that $\omega = \sum_{k = 0}^{3} i^{k} \omega_{k}$.
Proof. Assume without loss of generality that there exist $\xi, \eta \in H$ such that $\dpn{x, \omega}{M}= \dpn{x\xi, \eta}{H}$ for all $x \in M$. In which case, by polarisation,
\[\dpn{x, \omega}{M}= \dpn{x\xi, \eta}{H}= \frac{1}{4}\sum_{k = 0}^{3} i^{k} \dpn{x(\xi + i^k\eta), \xi + i^k\eta}{H}\]
for all $x \in M$.$\square$
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