Let $H$ be a [Hilbert space](/page/Hilbert%20Space), let $M\subseteq\mathcal{L}(H)$ be a von Neumann algebra with identity $1$, and let $\varphi:M\to\mathbb C$ be a positive linear functional, meaning $\varphi(x^*x)\ge 0$ for every $x\in M$. Write $\|x\|_{\mathrm{op}}$ for the operator norm of $x\in\mathcal{L}(H)$. Let $M_+$ denote the positive cone of $M$, and let $\mathcal P(M)$ denote the set of projections in $M$. The following are equivalent.

1. $\varphi$ is normal, meaning that for every bounded increasing net $(a_i)_{i\in I}$ in $M_+$ with supremum $a\in M_+$ in the self-adjoint order, one has

\begin{align*} \varphi(a)=\sup_{i\in I}\varphi(a_i). \end{align*}

2. For every increasing net $(p_i)_{i\in I}$ of projections in $M$ with supremum $p\in\mathcal P(M)$, one has

\begin{align*} \varphi(p)=\sup_{i\in I}\varphi(p_i). \end{align*}

3. For every family $(p_j)_{j\in J}$ of pairwise orthogonal projections in $M$, if $p=\bigvee_{j\in J}p_j$, then

\begin{align*} \varphi(p)=\sum_{j\in J}\varphi(p_j), \end{align*}

where the right-hand side denotes the supremum of the finite partial sums

\begin{align*} \sup\left\{\sum_{j\in F}\varphi(p_j):F\subset J,\ F\text{ finite}\right\}. \end{align*}