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There is important work by Alain Connes and Dimitri Shlyakhtenko (see here). They come up with a definition of $\ell^2$-homology for finite von Neumann algebras and define numerical invariants calles called $\ell^2$-Betti numbers for finite von Neumann algebras. This approach builds on the more classical theory of $\ell^2$-invariants developed by Atiyah, Cheeger-Gromov and also Lück. So far, there are no really interesting computations.

However, it seems that this homology group (or some variant of it) is more likely to be able to detect the differences among free group factors. Of course, this is only speculation. There is also a cohomological picture (see here) which boils down (in dimension one) to a study of derivations with values in the algebra of affiliated operators. Unfortunately, this more algebraic approach has not been very successful so far.

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There is important work by Alain Connes and Dimitri Shlyakhtenko (see here). They come up with a definition of $\ell^2$-homology for finite von Neumann algebras and define numerical invariants calles $\ell^2$-Betti numbers for finite von Neumann algebras. This approach builds on the more classical theory of $\ell^2$-invariants developed by Atiyah, Cheeger-Gromov and also Lück. So far, there are no really interesting computations.

However, it seems that this homology group (or some variant of it) is more likely to be able to detect the differences among free group factors. Of course, this is only speculation. There is also a cohomological picture (see here) which boils down (in dimension one) to a study of derivations with values in the algebra of affiliated operators. Unfortunately, this more algebraic approach has not been very successful so far.