# On Direct integral decomposition of von Neumann algebras

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!

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.
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).