sequence of projections tending to a projection in semifinite von Neumann algebra

Question

Let $M$ be a semifinite von Neumann algebra with a faithful normal semifinite trace $\tau$. For any projection $P\in M$ with $\tau(P)=\infty$, we can always find an increasing net $P_\alpha$ with $\tau(P_\alpha)<\infty$ such that $P_\alpha \uparrow P$. Since $\tau(P)=\sup_\alpha\tau(P_\alpha)$, we can find a set $\{P_n\}$ of projections from $\{P_\alpha\}$ with $\tau(P_n) \ge n$.

Then, for every $P_\alpha$, we always can find an $n$ such that $\tau(P_n)\ge n > \tau(P_\alpha)$ (i.e. $P_n \ge P_\alpha$). So, $\{P_n\}$ is a subnet of $\{P_\alpha\}$. Therefore $P_n\uparrow_n P$.

Well, so, can I say that for every projection in a semifinite von Neumann algebra, we can always find an increasing sequence of $\tau$-finite projections tending to this projection? I have never seen something like that. I think probably I am wrong. Can anyone explain a little bit for me?

Now, I have realized that the problem is the net is not totally ordered！

Answer 1

As you mention, the issue is that a net is not necessarily totally ordered. Let us consider an example:

Let $H=\ell^2(\mathbb N)$, with $\{e_n\}$ the canonical basis. Take $M=B(H)$, with the canonical trace. Let $\mathcal A$ be the family of finite subsets of $\mathbb N$, ordered by inclusion. For each $\alpha\subset\mathcal A$, define $$ P_\alpha=\sum_{n\in\alpha} E_n, $$ where $E_n$ is the rank-one projection onto the span of $e_n$. It is standard that $P_\alpha\nearrow I$.

Now you want to choose your sequence. Since you only choose by the size of the trace, say we choose $$ P_n=P_{\{2,4,6,\ldots,2n\}}=\sum_{k=1}^n E_{2k}. $$ Then $\tau(P_n)=n$, and the sequence satisfies your condition. But it is not a subnet: for example $E_1=P_{\{1\}}$ is not below any $P_n$. And $$P_n\nearrow\sum_{k=1}^\infty E_{2k},$$ which is of course not the identity.