During some revisiting of von Neumann algebras, which I needed for understanding certain proofs regarding the injective hyperfinite $II_\text{1}$-Factor $\mathcal{R}$, I realized that I don't really know many concrete examples of von Neumann algebras. By concrete I mean something differing from "look at the von Neumann algebra generated by this particular operator". I know of:
(1) $B(H)$ of course.
(2) Group von Neumann algebras.
(3) ITPFI factors, also written as $\mathcal{R}_\lambda$ for a real-value $\lambda$.
(4) I know fairly little about the $L^\infty$-constructions and ergodic actions. Any elaboration would be appreciated.
I was wondering whether anyone could provide some good examples? Maybe even jsut pin-point some important instances of the above (eg $\mathcal{R}_1 = \mathcal{R}$ or some important group such as free-groups as they give factors).
Regarding (4) there are many references for crossed products algebras associated with a group acting on a measure space. For instance Vaughan Jones (unfinished) notes include a chapter on crossed products (https://math.berkeley.edu/~vfr/math20909.html).
Your list is already quite rich. Other sets of examples may include:
(5) Double duals of $C^\ast$-algebras
(6) Free products of of other von Neumann algebras
(7) Shlyakhtenko's free Araki-Woods factors