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.



Hochschild/cyclic homology of von Neumann algebras: useless?
The question is not well-posed. There are various versions of cyclic theory (for instance) which differ according to continuity conditions that are assumed. In Connes' original IHES papers he deals with both discrete (useful for arbitrary rings) and topological (useful in the $C^\infty $-setting.)


The basic problem is that cyclic theory is very very sensitive. Consider the following example (using Connes' topological cyclic theory.) Let $M$ denote a compact smooth manifold with a smooth foliation. Then there are three operator algebras that you can associate with the situatio:;


a) $C(M)$, the $C^\ast$-algebra of continuous complex functions on $M$.


b) $C^\infty (M)$, the smooth functions on $M$.


c) $C\_\tau ^\infty (M)$, the continuous functions on $M$ which are smooth in the leaf directions.


The cyclic cohomology of these three rings are all different typically (e.g. for the Kronecker flow on the torus).


a) leads to measures on $M$;


b) leads to de Rham cohomology on $M$;


c) leads to "tangential cohomology" on $M$ (cf. my book [Global analysis on foliated spaces](https://doi.org/10.1007/978-1-4613-9592-8) with Cal Moore).


