Hochschild/Cyclic Homology of von Neumann Algebras: Useless?

Some further thoughts: the most striking results I know of on "purely algebraic cyclic/Hochschild homology" are due to Wodzicki, see e.g.

Homological properties of rings of functional-analytic type, Proceedings of the National Academy of Sciences USA 87 (1990), 4910-4911

which states that stable C*-algebras have trivial cyclic homology. Obviously this doesn't answer your II_1 factor question...

Also: your remark that in some cases, we can ignore the analysis and make the situation a bit simpler confuses me a little. To get anywhere with cyclic or Hochschild homology, we need to do some kind of comparison of resolutions, or construction of contracting homotopies, or something like that. My intuition - but I don't work much on operator algebras, so I could well be wrong here - is that a von Neumann algebra is such a big object we usually can only get a handle on it by looking at suitable subsets which generate its unit ball in the WOT/SOT. So for group von Neumann algebras, one tries to see what's going on for translations, and thence to deduce more general results by exploiting w*-w* continuity; or else use projections and approximation arguments. If we go to a purely algebraic category, then it is no longer sufficient to define things on dense subsets - one really needs a global definition, one really needs to verify that certain putative identities are satisfied by each element of the von Neumann algebra.

Sorry if that's a bit waffly. I think my point is that imposing continuity restrictions actually makes things easier, because - intuitively - more things are going to be projective/injective/flat relative to one's restricted class of short exact sequences. This is why, for instance, we know that $H^n_{cb}(M,M)=0$ for any von Neumann algebra M, but why the analogous claim without the 'cb' is open and back-breaking. In a similar vein, if you work in a restricted category then one does indeed get some known instances of homological non-triviality (though at the level of modules, not at the level of cyclic homology):

M. E. Polyakov, An Example of a Spatially Nonflat von Neumann Algebra

I should also say that the Hilbert module stuff you mention doesn't really connect to your original question about cyclic (co)homology. It's interesting, and I think more has been done, but it's just different - so if that's what interests you, cyclic and Hochschild homology may be something of a distraction.

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