The Gelfand transform gives an equivalence of categories from the category of unital, commutative $C^\*$-algebras with unital $\*$-homomorphisms to the category of compact Hausdorff spaces with continuous maps. Hence the study of $C^\*$-algebras is sometimes referred to as non-commutative topology.


All diffuse commutative von Neumann algebras acting on separable Hilbert space are isomorphic to $L^\infty[0,1]$. Hence the study of von Neumann algebras is sometimes referred to as non-commutative measure theory.


Connes proposed that the definition of a non-commutative manifold is a spectral triple $(A,H,D)$. From a $C^\*$-algebra, we can recover the "differentiable elements" as those elements of the $C^\*$-algebra $A$ that have bounded commutator with the Dirac operator $D$. 


What happens if we start with a von Neumann algebra? Does the same definition give a "differentiable" structure? Is there a way of recovering a $C^\*$-algebra from a von Neumann algebra that contains the "differentiable" structure on our non commutative measure space? This would be akin to our von Neumann algebra being $L^\infty(M)$ for $M$ a compact, orientable manifold (so we have a volume form). Or are von Neumann algebras just "too big" for this?


One of the reasons I am asking this question is Connes' spectral characterization of manifolds ([arXiv:0810.2088v1](http://arxiv.org/abs/0810.2088v1)) which shows we get a "Gelfand theory" for Riemannian manifolds if the spectral triples satisfy certain axioms. Connes starts with the von Neumann algebra $L^\infty(M)$ instead of the $C^\*$-algebra $C(M)$.



Non-commutative geometry from von Neumann algebras?
You definitely need some extra structure on your von Neumann algebra, but I'm not quite sure what you're asking for.
Intuitively I would think that just as different topological spaces share the same measure space structure, trying to extract NC-topological information out of a von Neumann algebra is going to need extra structure. (For instance, no one does topological K-theory of von Neumann algebras as far as I know.)


I see that on page 7 of that Connes paper, he shows that the WOT-closure A'' does remember the original algebra A *if* extra data are given (the Dirac operator and its interaction with A). 


Although it's probably not what you want: if you're looking at group von Neumann algebras and looking at the "geometry of the dual group", then the original group can be recovered from a suitable coproduct on the original (Hopf) von Neumann algebra. This is vaguely on the lines of Weil's theorem that "essentially" recovers a locally compact group and its Haar measure from a measurable group.


