A Radon-Nikodym-type theorem due to S. Sakai

Question

I got stuck with the following problem while going through the proof of Lemma $2^{\circ}$, Chapter 10 from the book 'Lectures on von Neumann Algebras' by Strătilă and Zsidó.

Problem: Let $\mathscr{M}$ be a von Neumann algebra and $\mathscr{M}_*\subseteq\mathscr{M}^*$ be the predual of $\mathscr{M}$. Let $\phi\in\mathscr{M}_*^+$ be a normal positive form and let $\lambda\in\mathbb{C}$ with $\lambda +\overline{\lambda}=1$. Then prove that the set ${\Large\chi}:=\{\phi (\lambda a\bullet +\overline{\lambda}\bullet a):a=a^*,\|a\|\leq 1\}$ is $\sigma (\mathscr{M}_*,\mathscr{M})$-compact.

Note that the $\sigma (\mathscr{M}_*,\mathscr{M})$-topology (also called weak-topology) on $\mathscr{M}_*$ is generated by the functionals $\Psi_m:\mathscr{M}_*\rightarrow\mathbb{C},\,m\in\mathscr{M}$ defined by $\Psi_m(m_*):=m_*(m)$ for $m_*\in\mathscr{M}_*$. The above problem almost looks like the conclusion of Banach-Alaoglu theorem only if one could show that $\large\chi$ is a $\sigma (\mathscr{M}_*,\mathscr{M})$-closed subset of the closed unit ball $(\mathscr{M}^*)_1$ of $\mathscr{M}^*$. Of course, $\Large\chi$ is a subset of $(\mathscr{M}^*)_1$, but I don't know how to prove the closedness. Thanks in advance for any help.

Answer 1

Let $\Psi_\alpha$ be a net in $\large\chi$. This means there is some net $a_\alpha \in\mathscr M$ with $\|a_\alpha\|≤1$ and $\Psi_\alpha(b)= \phi(\lambda a_\alpha b+ \overline\lambda ba_\alpha)$ for all $b\in\mathscr M$. By passing to a sub-net we may assume that $a_\alpha$ converges in the weak* topology on $\mathscr M$, let $a$ denote its limit.

Now we must remember some facts about von Neumann algebras:

1. The normal functionals are weak* continuous on $\mathscr M$.
2. For any fixed $b$ the maps $\mathscr M\to\mathscr M$, $x\mapsto bx$ and $x\mapsto xb$ are weak* continuous.

Point 1. should be covered in any book containing von Neumann algebras (sometimes normal functionals are even defined as the weak* continuous functionals). Point 2. follows for example from noting that the weak* topology agrees with the $\sigma$-weak operator topology (also called the ultra-weak operator topology), and checking that the maps are Lipschitz wrt the generating semi-norms of that topology.

Point 2. then means that $\lambda a_\alpha b+ \overline\lambda ba_\alpha\to \lambda ab + \overline{\lambda }ba$ in the weak* topology, so: $$\Psi_\alpha(b) = \phi(\lambda a_\alpha b+\overline\lambda ba_\alpha)\to\phi(\lambda ab+\overline \lambda ba)$$ holds for any $b$ by point 1. But this just means that $\Psi_\alpha\to [b\mapsto \phi(\lambda ab +\overline \lambda ba)]$ in the weak topology on $\mathscr M_*$. This limit lies in $\large\chi$, so any net in $\large\chi$ has a weakly convergent sub-net, implying that the set is weakly compact.