Norm continuity of the predual of a von Neumann algebra

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?

• If I understand the question correctly, this is a standard textbook result and holds more generally for nets of positive elements in place of projections. See for example Sakai, C*-algebras and W*-algebras, Lemma 1.7.4: Every uniformly bounded increasing directed set [of self-adjoint elements] converges to its least upper bound. Commented Dec 25, 2022 at 8:51
• But I may have misunderstood what you mean by "decreasing to 0". Perhaps you want to clarify in the question, is it in the directed set sense or is it convergence in some topology? (Which one?) Commented Dec 25, 2022 at 9:05
• @TobiasFritz Actually, decreasing to $0$ in the directed set sense and the wot are the same in this case. Anyway, Onur has proviced an answer. Thx! Commented Dec 25, 2022 at 9:49
• Yes, I know, and I thought that this was what you were asking :) Anyway, glad that it's answered. Commented Dec 25, 2022 at 9:59

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 $$\lim_i \|p_if\|=0$$.
• @user92646 I'm glad if you found it useful. I bet you can find a similar proof in operator algebra textbooks for the same or a similar result. For instance, one can easily deduce from what's written here that the map $T_f:M\to M_*$, $T_fx=xf$ is SOT-to-norm continuous for every $f\in M_*$. Commented Dec 26, 2022 at 13:39