On Direct integral decomposition of von Neumann algebras

Question

I have a question. We know by theory that any von Neumann algebra is direct integral of factors. Then how to get the decomposition in practical situation. Basically what is the decomposition examples for abelian vN algebras, Group vN algebras such that group is not i.c.c, and $\mathbb{B}(\mathcal{H})\otimes L^{\infty}(X,\mu)$. Thanks in advance!

Answer 1

There is no "practical situation", my opinion (with a small caveat mentioned at the end). Any von Neumann algebra that is expressed to you in a somewhat concrete way, is more amenable to manipulation than a direct integral. People (very) seldom use direct integrals to prove some general fact, not to understand their algebras.

For example, an abelian von Neumann algebra is $L^\infty(X,\mu)$ for some measure space. Suppose $X$ is $\mathbb R^n$, or $\mathbb C^n$, or one of many many other nontrivial measure spaces. Writing your algebra as a direct integral of uncountable many copies of $\mathbb C$ gives you nothing. Similarly, on $B(H)\otimes L^\infty(X,\mu)$ you can see the centre directly (it's $I\otimes L^\infty(X,\mu)$) and do stuff; writing the algebra as a direct integral of uncountably many copies of $B(H)$ gives you nothing of value.

In the case of group von Neumann algebras I don't have much to say. Mostly because I don't really remember if one can characterize the centre explicitly: the only case where the direct integral decomposition can be useful is the case where you have minimal central projections, because in that case you can write the decomposition rather explicitly. That said, I don't think it is usual to be able to write projections in a group von Neumann algebras explicitly (I might be wrong, I haven't played with group algebras in a very long time).

The only case where the direct integral decomposition is meaningful is in the case of finite-dimensional algebras. A finite-dimensional von Neumann algebra is of the form $\bigoplus_{k=1}^m M_{m_k}(\mathbb C)$. That's precisely the direct integral decomposition (over a finite measure space, that's why it's tractable).