In the context of noncommutative geometry, according to Serre-Swan theorem, a non commutative vector bundle is defined as a finitely generated projective module over a noncommutative C* algebra.
Now assume that E is a vector bundle over a compact topological space X. Assume that E is quiped with an inner product. then the completion of A=Hom(E,E) is a C* algebra and $\Gamma(E)$, the space of continuous sections of E, is a A- module, in a natural manner.
Is this module, a finitely generated projective A-module? If yes, is there any relation between invariants of this NC bundle and invariants(characteristic classes) of the initial (commutative) bundle?
Can one introduce me some references?
Dear Prof. Naschie
Thank you very much for your answer.
Can I ask you to let me know what part of the book "Non Commutative Geometry" by Alain Connes contains materials on such NC vector bundles, its finite projectivity and some computations of its invariants in term of characteristic classes of E.
Best regards
Ali Taghavi
The sections of a vector bundle are even a Morita equivalence bimodule. Thus they are also fgp for the algebra of endomorphisms.
Dear Stefan Waldman
Thank you for the answer.
Assume that E is a n-dimensional vector bundle over X. Put M=The spaces of sections and A=Hom(E,E), the endomorphism algebra.
As you said, M is a fgp A-module. So it gives us an idempotent e\in M_{k}(A), for some k. We can define a formal trace Tr(e)\in A. So Tr(e) is a morphism on E. what can be said about the fibrewise trace of Tr(e). Is it constant?(independent of the base point)? Is it equal to n=the fibre dimension of the initial bundle?
The motivation is that for the classical case the fibre dimension is equal to the trace of corresponding idempotent.
Many years ago, I asked an speaker(about NCG):"what is NC analogy of fibre dimension" I do not remember what was his answer. But if I am not mistaken, he said some thing about "LOCALIZATION". Do you have any idea about NC analogy of dimension of fibres of a vector bundle?
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