What is the definition of commutant for an abstract von Neumannn algebra?

Question

Let's suppose I have an abstract (not concrete, i.e. not acting on a Hilbert space) von Neumman algebra $\mathcal{M}$. Is there a notion of commutant for it? Because up to my knowledge (I'm reading "An invitation to von Neumann algebras" of Sunder) the commutant is defined for subsets of operators acting on aHilbert space.

Any comment will be helpful.

Answer 1

There isn't, as far as I know; the commutant is strongly dependent on the representation. There is a canonical way of representing a von Neumann algebra (the standard form), but there are many others.

Here is an example of how extreme the situation can be. Let $R\subset B(H)$ be the hyperfinite II$_1$ factor, represented in its standard form. Then

• $R'$ is a II$_1$-factor.

Now let $R_1=R\otimes I\subset B(H\otimes H)$. Then $R_1$ is still the hyperfinite II$_1$-factor, but now

• $R_1'=R'\otimes B(H)$, a II$_\infty$-factor.

To make things more extreme, let $\pi:R\to B(K)$ be an irreducible representation. Such $\pi$ exists, for instance because we can take $\phi$ a pure state on $R$, and let $\pi$ be its GNS representation. Because $R$ is simple, $\pi$ is faithful. So $R_2=\pi(R)$ is the hyperfinite II$_1$-factor. And because $\pi$ is irreducible,

• $R_2'=\mathbb C$.

Note that $R_2\subset B(K)$ is not a von Neumann algebra. Also $K$ is known to be non-separable.