# Concrete examples of von Neumann algebras

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

• Lovely, thank you for the reference :) – Munk Dec 26 '17 at 21:53

The current list, is missing one very important example, which has hardly any rigidity results. The ultrapower construction, see Popa's book on $II_1$ factors chapter 5 for its construction.

http://www.math.ucla.edu/~popa/Books/IIun-v13.pdf

• Thank you for that reference :) – Munk Jan 28 '18 at 22:15