TITLE: What is a noncommutative fiber bundle?
QUESTION [16 upvotes]: Given a spectral triple (A,H,D) in the sense of Connes, what would be the right notion of a fiber bundle or a principal fiber bundle on it? An example of this type is the Connes' cosphere algebra S*A, which is the noncommutative analogue of the cosphere bundle of a 
Riemannian manifold. It is defined as the image in the Calkin algebra Q(H) of the C*-algebra generated by the compact opertators in H and all dialations of A by the continuous 1-parameter family of unitaries generated by the the derivation [|D|,.] on B(H). Connes showed that when 
the spectral triple is the canonical Dirac triple of a compact spin^c Riemannian manifold, the cosphere algebra is the C*-algebra of continuous functions on the cosphere bundle of the manifold. 
The question is: can we put this into a more general framework by finding the right notion of a fiber bundle on a spectral triple?

REPLY [2 votes]: It might not be the context you want (although if you think about this, it might be very close to what you want) but for some people a non-commutative bundle is a Banach bundle with non-commutative fibres. unlike a fibre bundle, the fibres are not necessarily isomorphic. Also bundles related to categories of Morita equivalence bimodules might be related at to the context I'm referring to because Connes' differential calculus is built into Morita equiv bimodules. The latter is a way to generalise the "tangent bundle" over the space described by A. I'm not an expert. Anyway, this approach seems to be working.