On tracial factors

Question

Take $\mathbb{G}$ as I.C.C group for example. Now we know group von Neumann algebra is of type $II_1$ factor. My question is what are the finite projections look like in the algebra as we know all projections are finite here and in $B(l^{2}(\mathbb{G}))$ projections are at least we can see what they are but the same visualization is possible here?

Answer 1

You cannot really expect to see the projections. For a dramatic example, take $G=\mathbb F_2$. Then, even though $L(\mathbb F_2)= \overline{C_r^*(\mathbb F_2)}$, the C$^*$-algebra $C_r^*(\mathbb F_2)$ is projectionless. That is, even norm limits of linear combinations of elements of the group do not render projections; only after you take the sot/wot closure you get them.

At the other end of the spectrum, when you construct the hyperfinite II$_1$-factor from matrices (and not from an amenable group), via $R=\overline{\bigoplus_n M_{2^n}(\mathbb C)}$ you do get to see the projections. But if you write $R=L(\mathbb S_\infty)$, I wouldn't know how to write a single non-trivial projection, even when there are crazy many.