Answer by Andreas Thom for Hochschild/cyclic homology of von Neumann...
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...
View ArticleAnswer by Claude Schochet for Hochschild/cyclic homology of von Neumann...
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...
View ArticleAnswer by Yemon Choi for Hochschild/cyclic homology of von Neumann algebras:...
This is in response to Dmitri's remark. The reason I didn't bring up Sinclair & Smith's book (which is where I first started trying to learn Hochschild cohomology) is that it deals with continuous...
View ArticleAnswer by Dmitri Pavlov for Hochschild/cyclic homology of von Neumann...
I am surprised that nobody mentioned the book by Sinclair and Smith “Hochschild Cohomology of von Neumann algebras” so far.If I remember it correctly, they claim that nobody knows whether there is a...
View ArticleAnswer by Yemon Choi for Hochschild/cyclic homology of von Neumann algebras:...
Some further thoughts: the most striking results I know of on "purely algebraic cyclic/Hochschild homology" are due to Wodzicki, see e.g.Wodzicki, Mariusz, Homological properties of rings of...
View ArticleAnswer by Yemon Choi for Hochschild/cyclic homology of von Neumann algebras:...
As someone who works on the continuous (bounded) cohomology of Banach algebras: I think the quote is a way of saying "we don't really know". There are certainly questions which start off with extra...
View ArticleAnswer by GMRA for Hochschild/cyclic homology of von Neumann algebras: useless?
I am certainly not an expert, but I guess that when people say There's also analysis in von Neumann algebras, so I wouldn't expect an algebraic invariant like Hochschild or cyclic homology to tell you...
View ArticleHochschild/cyclic homology of von Neumann algebras: useless?
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...
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