Let $M$ be a von Neumann algebra and let $(p\_i)$ be a net of projections in $M$ decreasing to $0$. Let $f\in M\_{\ast} $, the predual of $M$.
It is well known that
$\| p\_i f \|\_{M\_\ast}\to\_{i} 0$
for any semifinite von Neumann algebra $M$.
I am wondering whether this result holds true for type III von Neumann algebras. If so, any references?



Norm continuity of the predual of a von Neumann algebra
Let $M$ be a von Neumann algebra and $\pi:M\to B(H)$ be a normal faithful representation of $M$ on a Hilbert space, so that we can conveniently identify $M$ with $\pi(M)\subseteq B(H)$. Since $f\in M\_\*$ is $\sigma$-weakly continuous, $$\forall x\in M \hspace{8mm} f(x) = \sum\_{k=1}^{\infty} \langle \pi(x)\xi\_k^1,\xi\_k^2 \rangle $$ for some $\xi\_k^l\in H$ with $\sum\_{k=1}^{\infty} \|\xi\_k^l\|^2 <\infty$ for $l=1,2$.


If $(p\_i)\_{i\in I}$ is a net of projections decreasing to $0$, then $\pi(p\_i)\to 0$ in SOT.
$$
\|p\_if\| = \sup\_{\|x\|\leq 1}|f(xp\_i)|
\leq \sum\_{k=1}^{\infty} \sup\_{\|x\|\leq 1}|\langle \pi(xp\_i)\xi\_k^1,\xi\_k^2 \rangle|
\leq \sum\_{k=1}^{\infty}\|\pi(p\_i)\xi\_k^1 \|\|\xi\_k^2\| \\
\leq \sum\_{k=1}^{n-1}\|\pi(p\_i)\xi\_k^1 \|\|\xi\_k^2\| + \sum\_{k=n}^{\infty}\|\xi\_k^1\|\|\xi\_k^2\|
$$
for each integer $n\geq 1$ and $i\in I$. Thus, for each $n$
$$
\limsup\_i \|p\_if\| \leq \sum\_{k=n}^{\infty}\|\xi\_k^1\|\|\xi\_k^2\|.
$$
As $n\to\infty$, we get $\lim\_i \|p\_if\|=0$.


