The cyclic category makes an appearance in the study of planar algebras (see arXiv:0912.1320). In particular, every planar algebra is a cyclic module in two ways. To motivate some work on planar structure for infinite index subfactors, I want to know if there is a $II_1$-factor with nontrivial Hochschild or cyclic homology.

I understand that there are other types of homology theories that make more sense for von Neumann algebras, but the algebraic theory is the one that surfaces in my work.

NOTE: I have edited this post to make it a focused question with a specific goal, and it no longer resembles the original post.

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 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.

The cyclic category makes an appearance in the study of planar algebras (see arXiv:0912.1320). In particular, every planar algebra is a cyclic module in two ways. To motivate some work on planar structure for infinite index subfactors, I want to know if there is a $II_1$-factor with nontrivial Hochschild or cyclic homology.

I understand that there are other types of homology theories that make more sense for von Neumann algebras, but the algebraic theory is the one that surfaces in my work.

NOTE: I have edited this post to make it a focused question with a specific goal, and it no longer resembles the original post.