Consider the following lemma from Takesaki's book "Theory of operator algebra II":
Questions:
(1) What is a $\sigma$-finite projection? I don't see a definition anywhere in either volume I or volume II.
I suspect it is a strong limit of a bounded increasing sequence of finite-rank operators in the von Neumann algebra?
(2) If $x \in M_0$, then why is $s_l(x) \in \Sigma$?
We know that $x= pyp$ where $p$ is $\sigma$-finite and $y \in M$. Write $p = \lim_n p_n$ where the $p_n's$ have finite rank. Then $x= \lim_n p_n y p_n$ strongly. Can I deduce from this that $s_l(x)$ is $\sigma$-finite? Or maybe a polar decomposition argument works here?
A $\sigma$-finite projection in $M$ is a projection where each family of pairwise orthogonal subprojections in $M$ is countable. When $M=B(H)$, this is equivalent with being a sot limit of an increasing sequence of finite-rank projections.
A von Neumann algebra is $\sigma$-finite if the identity is. Takesaki defined this in II.3.18, where he also shows that for a von Neumann algebra being $\sigma$-finite is equivalent to having a faithful normal state, and also equivalent to having a countable separating set in $H$.
As for the second question, if $x=pyp$, then the range of $x$ is contained in $pH$, and thus $s_\ell(x)\leq p$. A subprojection of a $\sigma$-finite projection is $\sigma$-finite.
Suppose that $p,q$ are $\sigma$-finite. We want to show that $p\lor q$ is $\sigma$-finite. We think about it as above, in terms of $(p\lor q)M(p\lor q)$ being $\sigma$-finite. By Takesaki II.3.19 there exist countable separating sets $\{\xi_n\}$ for $pMp$ and $\{\eta_n\}$ for $qMq$. We want to show that $\{\xi_n\}\cup\{\eta_n\}$ is separating for $(p\lor q)M(p\lor q)$. Let $x\in (p\lor q)M(p\lor q)$ with $x\xi_n=x\eta_n=0$ for all $n$. We may assume without loss of generality that $p\xi_n=\xi_n$, $q\eta_n=\eta_n$, as these would still be separating. Note that $x^*x\xi_n=x^*x\eta_n=0$. So $$ px^*xp\xi_n=px^{*}x\xi_n=0,\qquad qx^{*}xq\eta_n=qx^{*}x\eta_n=0 $$ for all $n$, which gives $px^*xp=qx^*xq=0$. If $\xi\in pH$, then $px^*xp\xi=0$, and thus $$0=\langle px^*xp\xi,\xi\rangle=\|x^{}p\xi\|^2=\|x^{}\xi\|^2,$$ so $x\xi=0$. Similarly, $x\eta=0$ if $\eta\in qH$. So $x=0$ on $pH\cup qH$, which means that $x=x(p\lor q)=0$. That is, the countable set $\{\xi_n\}\cup\{\eta_n\}$ is separating for $(p\lor q)M(p\lor q)$.