[proofplan] We prove both directions using the predual characterization of the ultraweak topology. In one direction, bounded-set ultraweak continuity makes every composition $\omega\circ\Phi$ with a positive normal functional $\omega\in N_*^+$ into a normal positive functional on $M$; testing bounded increasing nets against all such $\omega$ then forces $\Phi$ to preserve suprema. Conversely, if $\Phi$ is normal, then every $\omega\circ\Phi$ is a normal functional on $M$, so ultraweak convergence against all functionals in $M_*$ transfers to ultraweak convergence of the images against all functionals in $N_*$. The boundedness hypothesis is used because the ultraweak-continuity condition is imposed only on bounded subsets and because bounded increasing nets are the nets appearing in the normality criterion. [/proofplan]

[step:Fix the topology and normality conventions]We use the following standard conventions. The ultraweak topology on a von Neumann algebra $M$ is the [weak topology](/page/Weak%20Topology) induced by its predual $M_*$, whose existence is given by [citetheorem:9273]. We write $M_{\mathrm{sa}}$ and $N_{\mathrm{sa}}$ for the self-adjoint parts of $M$ and $N$, $M_+$ and $N_+$ for their positive cones, and $N_*^+$ for the positive cone of the predual $N_*$. Thus a net $(x_i)_{i\in I}$ in $M$ converges ultraweakly to $x\in M$ exactly when \begin{align*} \eta(x_i)\to \eta(x) \end{align*} for every $\eta\in M_*$. For a positive [linear map](/page/Linear%20Map) $\Phi:M\to N$, normality means preservation of suprema of bounded increasing nets in the positive cone: whenever $(x_i)_{i\in I}$ is an increasing bounded net in $M_+$ with supremum $x\in M_+$, the net $(\Phi(x_i))_{i\in I}$ has supremum $\Phi(x)$ in $N_+$. We also use the monotone convergence characterization of normal positive functionals from [citetheorem:9272]: a positive functional $\eta:M\to\mathbb C$ is normal iff, for every increasing net of projections with supremum $p$, equivalently for bounded increasing positive nets, it preserves the corresponding supremum by monotone convergence. Finally, normal positive functionals separate the positive cone of a von Neumann algebra: if $a,b\in N_{\mathrm{sa}}$ and $\omega(a)\le \omega(b)$ for every $\omega\in N_*^+$, then $a\le b$.[/step]

[guided]We first pin down exactly what is being tested. The ultraweak topology is not defined by all bounded linear functionals on $M$, but by the normal ones, namely the elements of the predual $M_*$. By [citetheorem:9273], $M_*$ is a [Banach space](/page/Banach%20Space) whose dual is canonically $M$, so the ultraweak topology is precisely the weak-star topology coming from the pairing between $M$ and $M_*$. Therefore a net $(x_i)_{i\in I}$ converges ultraweakly to $x$ exactly when every normal functional $\eta\in M_*$ satisfies \begin{align*} \eta(x_i)\to \eta(x). \end{align*} For positive maps, the normality condition is order-theoretic: $\Phi$ must preserve suprema of bounded increasing nets in the positive cone. Thus, if $(x_i)_{i\in I}$ is increasing in $M_+$, bounded above, and has supremum $x\in M_+$, then normality says that $\Phi(x)$ is the least upper bound of the increasing net $(\Phi(x_i))_{i\in I}$ in $N_+$. We will verify this equality by testing against normal positive functionals on $N$. This is legitimate because normal positive functionals separate the positive cone: order inequalities between self-adjoint elements are detected by all $\omega\in N_*^+$.[/guided]

[step:Show bounded ultraweak continuity makes scalar compositions normal] Assume that $\Phi$ is ultraweakly continuous on bounded subsets of $M$. Let $\omega\in N_*^+$ be a positive normal functional, and define the positive linear functional \begin{align*} \eta: M \to \mathbb C,\qquad x \mapsto \omega(\Phi(x)) \end{align*} . Let $(x_i)_{i\in I}$ be a bounded increasing net in $M_+$ with supremum $x\in M_+$. For bounded increasing nets in a von Neumann algebra, $x_i\to x$ ultraweakly. Since the set $\{x_i:i\in I\}\cup\{x\}$ is bounded, the bounded-set ultraweak continuity of $\Phi$ gives $\Phi(x_i)\to\Phi(x)$ ultraweakly in $N$. Applying $\omega\in N_*$ yields \begin{align*} \eta(x_i)=\omega(\Phi(x_i))\to\omega(\Phi(x))=\eta(x). \end{align*} Because $(\eta(x_i))_{i\in I}$ is increasing and converges to $\eta(x)$, the functional $\eta=\omega\circ\Phi$ preserves bounded increasing suprema in $M_+$. By [citetheorem:9272], $\eta$ is normal. [/step]

[step:Deduce normality of $\Phi$ from scalar normality] Let $(x_i)_{i\in I}$ be a bounded increasing net in $M_+$ with supremum $x\in M_+$. Since $\Phi$ is positive, $(\Phi(x_i))_{i\in I}$ is increasing in $N_+$ and $\Phi(x_i)\le \Phi(x)$ for every $i\in I$. Let $y\in N_+$ denote the supremum of $(\Phi(x_i))_{i\in I}$ in $N_+$. For every $\omega\in N_*^+$, the previous step shows that $\omega\circ\Phi$ is normal. Hence \begin{align*} \omega(y)=\sup_{i\in I}\omega(\Phi(x_i))=\omega(\Phi(x)). \end{align*} The equality for all $\omega\in N_*^+$ and the separation of the positive cone by normal positive functionals imply $y=\Phi(x)$. Therefore $\Phi$ preserves suprema of bounded increasing positive nets, so $\Phi$ is normal. [/step]

[step:Show normality sends normal functionals to normal functionals] Conversely, assume that $\Phi$ is normal. Let $\omega\in N_*^+$, and define \begin{align*} \eta: M \to \mathbb C,\qquad x \mapsto \omega(\Phi(x)) \end{align*} . The functional $\eta$ is positive because both $\Phi$ and $\omega$ are positive. Let $(x_i)_{i\in I}$ be a bounded increasing net in $M_+$ with supremum $x\in M_+$. By normality of $\Phi$, the net $(\Phi(x_i))_{i\in I}$ has supremum $\Phi(x)$ in $N_+$. Since $\omega$ is normal, [citetheorem:9272] gives \begin{align*} \eta(x_i)=\omega(\Phi(x_i))\uparrow\omega(\Phi(x))=\eta(x). \end{align*} Thus $\eta$ is normal by the monotone convergence criterion [citetheorem:9272], so $\omega\circ\Phi\in M_*$. Every normal functional in $N_*$ is a complex linear combination of positive normal functionals, by the real and imaginary decompositions followed by the Jordan decomposition for normal self-adjoint functionals. Hence, for every $\omega\in N_*$, the composition $\omega\circ\Phi$ belongs to $M_*$. [/step]

[step:Transfer bounded ultraweak convergence through $\Phi$] Let $(x_i)_{i\in I}$ be a bounded net in $M$ and let $x\in M$ satisfy $x_i\to x$ ultraweakly in $M$. To prove $\Phi(x_i)\to\Phi(x)$ ultraweakly in $N$, it is enough to test against an arbitrary $\omega\in N_*$. By the previous step, the functional $\omega\circ\Phi$ belongs to $M_*$. Therefore ultraweak convergence in $M$ gives \begin{align*} \omega(\Phi(x_i))=(\omega\circ\Phi)(x_i)\to(\omega\circ\Phi)(x)=\omega(\Phi(x)). \end{align*} Since this holds for every $\omega\in N_*$, the net $(\Phi(x_i))_{i\in I}$ converges ultraweakly to $\Phi(x)$ in $N$. Thus $\Phi$ is ultraweakly continuous on bounded subsets of $M$. [/step]