Infinite tensor product and partial isometries Consider a countably infinite tensor product of the algebra of complex $2\times 2$ matrices:
$\otimes_{k=1}^\infty M^{(k)}_2 (\mathbb{C})$, and two different states: 
1) $\phi$ that is the tensor product of the density matrix $\frac{1}{2}\left(\begin{matrix} 
1 & 0 \\
0 & 1 
\end{matrix}\right)$ at each site.
2) $\psi$ that is the tensor product of the density matrix $\frac{1}{1+\lambda}\left(\begin{matrix} 
1 & 0 \\
0 & \lambda 
\end{matrix}\right)$ with
$\lambda\neq 0$ at each site.
The von Neumann algebra one obtains using $\psi$ is the hyperfinite type II$_1$ factor, and is type III$_\lambda$ if we use $\psi$. 
Now, consider the projector operator $p=\left(\begin{matrix} 
1 & 0 \\
0 & 0
\end{matrix}\right)$ on the first qubit. In the case of type III factor there exists an isometry $W$ in the algebra such that $W^\dagger W=1$ and $WW^\dagger=p$, what is this operator $W$? and why is it not in the algebra of the II$_1$ factor?
 A: I don't think it makes sense to expect any explicit description of $W$. We know that $W$ is a sot limit (in the GNS representation of $\psi$), $W=\lim_j W_j$ with $W_j\in\bigotimes_kM_2(\mathbb C)$; and the $W_j$ are not unitaries, because if they were we would have $p=WW^*=\lim_jW_jW_j^*=1$ (multiplication is sot continuous on bounded nets). But then the $W_j$ cannot even be isometries, because if they were they would be unitaries (as $\bigotimes_kM_2(\mathbb C)$ has a trace). 
The reason $W\in \overline{\bigotimes_kM_2(\mathbb C)}^{\rm sot -\psi}$ while $W\not\in \overline{\bigotimes_kM_2(\mathbb C)}^{\rm sot -\phi}$ is that these are different completions (i.e., completions under different topologies). It's no different (in "weirdness") than taking a locally compact space, say $(0,1)$ and taking two different completions, like the one-point compactification $\overline{(0,1)}^{o}$and the Stone-Cech compactification $\overline{(0,1)}^{sc}$: why are points in the corona $\overline{(0,1)}^{sc}\setminus (0,1)$ not in $\overline{(0,1)}^{o}$?
Also, it's easy to forget that taking sot/wot closure gives you scores of new elements. Concretely, your C$^*$-algebra $\bigotimes_kM_2(\mathbb C)$ (commonly denoted by UHF$(2^\infty)$ ) is norm-separable, while its sot closure is not. A dramatic example of this appearance of new elements occurs when you create a II$_1$-factor as a group algebra: namely, take an icc group $G$, consider its left-regular representation $\lambda:G\to B(\ell^2(G))$, and get the II$_1$-factor $L(G)=\lambda(G)''$. As you now, II$_1$-factors have lots and lots of projections; yet, the dense C$^*$-algebra $\overline{\operatorname{span}\{\lambda(G)\}}^{\|\cdot\|}$ often has no nontrivial projections (this is true for instance if $G=\mathbb F_2$). 
