$K$-theory of smooth manifolds: continuous vs. smooth vector bundles

Suppose I have a smooth manifold $M$, and want to consider the $K$-theory $K^0(M)$. I could define this in the usual way (by taking the Grothendieck group of the monoid of equivalence classes of vector bundles) or in a 'smooth' way (considering only smooth vector bundles, and taking the Grothendieck group as usual).

I haven't seen this discussed anywhere. Is there any difference between the two approaches? And are there any references in which this question is discussed?

(Even better: if $M$ is a $G$-space for a compact Lie group $G$, is the equivariant $K$-theory affected by taking only smooth bundles?)

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2022-07-25 20:46:51
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