Norm continuity of the predual of a von Neumann algebra

Question

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?

Answer 1

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 $\lim_i \|p_if\|=0$.