Grothendieck's approach to algebraic geometry in particular tells us to treat all rings as rings of functionson some sort of space. This can also be applied outside of scheme theory (e.g., Gelfand-Neumark theoremsays that the category of measurable spaces is contravariantly equivalent to the categoryof commutative von Neumann algebras). Even though we do not have a complete geometric descriptionfor the noncommutative case, we can still use geometric intuition from the commutative case effectively.
A generalization of this idea is given by Grothendieck's relative point of view, which says that a morphism of rings f: A → Bshould be regarded geometrically as a bundle of spaces with the total space Spec B fibered over the base space Spec Aand all notions defined for individual spaces should be generalized to such bundles fiberwise.For example, for von Neumann algebras we have operator valued weights, relative L^p-spaces etc.,which generalize the usual notions of weight, noncommutative L^p-space etc.
In noncommutative geometry this point of view is further generalized to bimodules.A morphism f: A → B can be interpreted as an A-B-bimodule B, with the right action of B given by the multiplicationand the left action of A given by f.Geometrically, an A-B-bimodule is like a vector bundle over the product of Spec A and Spec B.If a bimodule comes from a morphism f, then it looks like a trivial line bundle with the support beingequal to the graph of f.In particular, the identity morphism corresponds to the trivial line bundle over the diagonal.
For the case of commutative von Neumann algebras all of the above can be made fully rigoroususing an appropriate monoidal category of von Neumann algebras.
This bimodule point of view is extremely fruitful in noncommutative geometry (think of Jones' index,Connes' correspondences etc.)
However, I have never seen bimodules in other branches of geometry (scheme theory,smooth manifolds, holomorphic manifolds, topology etc.) used to the same extent as they are usedin noncommutative geometry.
Can anybody state some interesting theorems (or theories) involving bimodules in such a setting?Or just give some references to interesting papers?Or if the above sentences refer to the empty set, provide an explanation of this fact?
