[proofplan] We prove the forward implication directly from normality, because projections are positive elements and their supremum is also their supremum in the positive cone. For the converse, we test normality after composing with arbitrary positive normal functionals on $N$. The assumed preservation of suprema of increasing projection nets makes each composite positive functional $\omega\circ\rho$ normal by the projection criterion for positive functionals; then normal positive functionals on $N$ separate order, so $\rho$ preserves suprema of bounded increasing positive nets. This is exactly normality for a positive map between von Neumann algebras. [/proofplan] [step:Deduce the projection condition from normality] Assume first that $\rho$ is normal. Let $I$ be a directed set, let $(p_i)_{i\in I}$ be an increasing net in $\mathcal P(M)$, and let $p=\sup_{i\in I}p_i$ in $\mathcal P(M)$. Let $M_+=\{a\in M:a\ge 0\}$ and $N_+=\{b\in N:b\ge 0\}$ denote the positive cones. Since each $p_i$ is positive and $p_i\le p$, the element $p$ is an upper bound of $(p_i)_{i\in I}$ in $M_+$. It is also the least such upper bound: if $a\in M_+$ and $p_i\le a$ for every $i\in I$, then in any faithful normal representation of $M$ the increasing projection net $(p_i)_{i\in I}$ converges strongly to its projection supremum $p$, and passing to vector-state limits in $p_i\le a$ gives $p\le a$. Hence $p=\sup_{i\in I}p_i$ in $M_+$. Normality of $\rho$ means that $\rho$ preserves suprema of bounded increasing nets in $M_+$, so \begin{align*} \rho(p)=\sup_{i\in I}\rho(p_i) \end{align*} in $N_+$. Because $\rho$ is a $*$-homomorphism, each $\rho(p_i)$ is a projection. Indeed, \begin{align*} \rho(p_i)^2=\rho(p_i^2)=\rho(p_i). \end{align*} Also, \begin{align*} \rho(p_i)^*=\rho(p_i^*)=\rho(p_i). \end{align*} Thus the supremum is a projection supremum in $\mathcal P(N)$, and the projection condition follows. [/step] [step:Turn the projection condition into normality of composed functionals] Assume now that the projection-supremum condition holds. Let $\omega:N\to\mathbb C$ be a positive normal functional. Define \begin{align*} \psi:M&\to\mathbb C \end{align*} \begin{align*} x&\mapsto \omega(\rho(x)). \end{align*} Then $\psi$ is a positive functional, because if $x\in M_+$ then $\rho(x)\in N_+$ and hence $\omega(\rho(x))\ge 0$. We prove that $\psi$ is normal. Let $I$ be a directed set, let $(p_i)_{i\in I}$ be an increasing net of projections in $M$, and let $p=\sup_{i\in I}p_i$. By the projection-supremum hypothesis, \begin{align*} \sup_{i\in I}\rho(p_i)=\rho(p) \end{align*} in $\mathcal P(N)$. Since $\omega$ is normal and positive, [citetheorem:9272] applied in $N$ gives \begin{align*} \omega(\rho(p))=\sup_{i\in I}\omega(\rho(p_i)). \end{align*} By the definition of $\psi$, this is \begin{align*} \psi(p)=\sup_{i\in I}\psi(p_i). \end{align*} Again using [citetheorem:9272], now applied in $M$, this projection-supremum preservation implies that $\psi$ is normal. [guided] The point of introducing $\psi=\omega\circ\rho$ is that positive normal functionals are scalar-valued, and scalar-valued normality has a projection criterion already available. We must check every hypothesis of that criterion. Let $\omega:N\to\mathbb C$ be a positive normal functional and define the composite map \begin{align*} \psi:M&\to\mathbb C \end{align*} \begin{align*} x&\mapsto \omega(\rho(x)). \end{align*} The map $\psi$ is linear because both $\rho$ and $\omega$ are linear. It is positive because for $x\in M_+$, positivity of the $*$-homomorphism $\rho$ gives $\rho(x)\in N_+$, and positivity of $\omega$ gives $\omega(\rho(x))\ge 0$. To prove that $\psi$ is normal, we use the projection criterion for positive functionals, [citetheorem:9272]. That criterion requires us to test increasing nets of projections. Let $I$ be a directed set, let $(p_i)_{i\in I}$ be an increasing net in $\mathcal P(M)$, and let $p=\sup_{i\in I}p_i$. The hypothesis of the theorem says exactly that the image projections have supremum \begin{align*} \sup_{i\in I}\rho(p_i)=\rho(p) \end{align*} in $\mathcal P(N)$. Now apply [citetheorem:9272] to the positive normal functional $\omega$ on $N$. Its hypotheses are satisfied because $\omega$ is positive and normal by choice, and $(\rho(p_i))_{i\in I}$ is an increasing net of projections with supremum $\rho(p)$. Therefore \begin{align*} \omega(\rho(p))=\sup_{i\in I}\omega(\rho(p_i)). \end{align*} Using the definition of $\psi$, this becomes \begin{align*} \psi(p)=\sup_{i\in I}\psi(p_i). \end{align*} Thus $\psi$ satisfies the projection-supremum condition for every increasing net of projections in $M$. Applying [citetheorem:9272] to $\psi$ on $M$, we conclude that $\psi$ is normal. [/guided] [/step] [step:Use normal functionals to detect suprema in the target algebra] Let $I$ be a directed set, let $(x_i)_{i\in I}$ be a bounded increasing net in $M_+$, and let \begin{align*} x=\sup_{i\in I}x_i \end{align*} in $M_+$. Since $\rho$ is positive, $(\rho(x_i))_{i\in I}$ is an increasing net in $N_+$ and $\rho(x_i)\le \rho(x)$ for every $i\in I$. Hence \begin{align*} y:=\sup_{i\in I}\rho(x_i) \end{align*} exists in $N_+$ and satisfies $y\le \rho(x)$. We show that equality holds. Let $\omega:N\to\mathbb C$ be any positive normal functional. By the previous step, $\omega\circ\rho$ is a positive normal functional on $M$. Therefore normality of $\omega\circ\rho$ gives \begin{align*} \omega(\rho(x))=(\omega\circ\rho)(x)=\sup_{i\in I}(\omega\circ\rho)(x_i)=\sup_{i\in I}\omega(\rho(x_i)). \end{align*} Since $\omega$ is normal and $y=\sup_i\rho(x_i)$, we also have \begin{align*} \omega(y)=\sup_{i\in I}\omega(\rho(x_i)). \end{align*} Thus \begin{align*} \omega(\rho(x)-y)=0 \end{align*} for every positive normal functional $\omega$ on $N$. The element $\rho(x)-y$ belongs to $N_+$ because $y\le \rho(x)$. Let $N_*$ denote the predual of $N$, equivalently the [Banach space](/page/Banach%20Space) of normal linear functionals on $N$ from [citetheorem:9273]. Let $N_*^+=\{\omega\in N_*:\omega(a)\ge 0\text{ for every }a\in N_+\}$ denote the cone of positive normal functionals on $N$. Positive normal functionals separate the positive cone of a von Neumann algebra: if $a\in N_+$ and $\omega(a)=0$ for every $\omega\in N_*^+$, then $a=0$. This follows from [citetheorem:9273], since the canonical embedding $N\to (N_*)^*$ is isometric, and a nonzero positive element has nonzero norm detected by a normal functional whose real and imaginary parts decompose into positive normal functionals. Applying this to $a=\rho(x)-y$ gives \begin{align*} \rho(x)-y=0. \end{align*} Therefore \begin{align*} \rho(x)=\sup_{i\in I}\rho(x_i). \end{align*} [/step] [step:Conclude normality of the homomorphism] We have shown that for every bounded increasing net $(x_i)_{i\in I}$ in $M_+$ with supremum $x\in M_+$, \begin{align*} \rho(x)=\sup_{i\in I}\rho(x_i). \end{align*} This is exactly the definition of normality for $\rho$ stated in the theorem, namely preservation of suprema of bounded increasing nets in the positive cone. Hence $\rho$ is normal. Combining this converse with the first step proves the equivalence. [/step]