TITLE: Hochschild/cyclic homology of von Neumann algebras: useless?
QUESTION [9 upvotes]: Hochschild homology gives invariants of (unital) $k$-algebras for $k$ a unital, commutative ring. If we let our algebra $A$ be the group ring $k[G]$ for $G$ a finite group, we get group homology. There are plenty of other connections to homological algebra. If we use cyclic homology, there are connections to geometry and topology involving the Chern character.
Von Neumann algebras are complex algebras, so we can take their Hochschild and cyclic homologies. When I have asked experts in the fields of von Neumann algebras and non-commutative geometry about what you get, I usually hear some approximation of the following: "There's also analysis in von Neumann algebras, so I wouldn't expect an algebraic invariant like Hochschild or cyclic homology to tell you anything useful."
Although this answer makes some sense, I find it very displeasing and cryptic. Why shouldn't it tell you something? Is there some way to make "it doesn't tell you anything" quantitative? Is there an example of a von Neumann algebra with nontrivial Hochschild or cyclic homology (different from that of the complex numbers)?
EDIT: After reading the responses so far, I should specify that I really want to know if there is a $II_1$-factor with nontrivial Hochschild or cyclic (co)homology.

REPLY [4 votes]: There is important work by Alain Connes and Dimitri Shlyakhtenko (see $L^2$-homology for von Neumann algebras (MSN)). They come up with a definition of $\ell^2$-homology for finite von Neumann algebras and define numerical invariants 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 Kadison, Liu, and Thom - A note on commutators in algebras of unbounded operators (MSN)) 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.