Definition 43.4.4 (Normal Linear Functional).label Let $H$ be a complex Hilbert space, $M \subset B(H)$ be a von Neumann algebra, and $\omega: M \to \complex$ be a positive linear functional, then the following are equivalent:
- (1)
For any bounded, upward-directed family $A \subset M$, $\omega(\sup(A)) = \sup(\omega(A))$.
- (2)
For any mutually orthogonal projections $\mathcal{P}\subset M$, $\omega\paren{\sum_{p \in \mathcal{P}}p}= \sum_{p \in \mathcal{P}}\omega(p)$.
- (3)
$\omega$ is the norm limit of a sequence of weak-operator continuous positive linear functionals.
- (4)
$\omega$ is weak-operator continuous on bounded subsets of $M$.
- (5)
$\omega$ is strong-operator continuous on bounded subsets of $M$.
Proof, [Theorem 2.5.5, AP]. (2) $\Rightarrow$ (3): Let $\mathcal{P}\subset M$ be a maximal family of mutually orthogonal projections such that for each $p \in \mathcal{P}$, there exists $\xi_{p} \in pH$ such that $\omega(x) = \dpn{x\xi_p, \xi_p}{H}$ for all $x \in pMp$. Let $q = \sum_{p \in \mathcal{P}}p$, then by maximality of $\mathcal{P}$ and Lemma 43.4.3 applied to $(1 - q)M(1 - q)$, $\sum_{p \in \mathcal{P}}p = 1_{M}$.
For each $x \in M$ and $p \in \mathcal{P}$, by the Cauchy-Schwarz inequality,
so the mapping $y \xi_{p} \mapsto \omega(yp)$ extends into a bounded linear functional on $\ol{M\xi_p}$. By the Riesz Representation Theorem, there exists $\eta_{p} \in \ol{M\xi_p}$ such that $\omega(xp) = \dpn{x\xi_p, \eta_p}{H}$ for all $x \in M$.
Let $\mathcal{F}\subset 2^{\mathcal{P}}$ be the collection of all finite subsets of $\mathcal{P}$, directed by inclusion. For each $F \in \mathcal{F}$, let $q_{F} = \sum_{p \in F}p$ and $\omega_{F} \in M^{*}$ be defined by $\omega_{F}(x) = \omega(q_{F}xq_{F})$, then for each $x \in M$,
so $\omega_{F}$ is weak-operator continuous. Finally, for any $x \in M$,
By (2), $\lim_{F \in \mathcal{F}}\omega(1 - q_{F})^{1/2}= 0$. Therefore $\omega = u\text{-}\lim_{F \in \mathcal{F}}\omega_{F}$.$\square$
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