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Do Degree Zero Pseudo-Differential Operators on a Manifold Send Smooth Functions to Smooth Functions?
@Kashayar: This is not trivial. See math.mit.edu/~rbm/18.157-F05-Chapter5.ps, or math.mit.edu/~rbm/iml90c2.ps, for example. Your manifold is compact, so the estimates are not difficult.
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Do Degree Zero Pseudo-Differential Operators on a Manifold Send Smooth Functions to Smooth Functions?
One way to see it is that smooth functions is within the intersection of Sobolev spaces. So Deane Yang's comment just follow.
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Connes-Chern pairing, compatibility with periodicity operator in the odd case
You cannot use Khalkhali's book as a reference, it contains a lot of handwaving arguments and mistakes. I suggest you stick with Connes.
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