The {$K$}-groups and the index theory of certain comparison {$C^*$}-algebras

@article{Monthubert2011TheA,
  title={The \{\$K\$\}-groups and the index theory of certain comparison \{\$C^*\$\}-algebras},
  author={Bertrand Monthubert and Victor Nistor},
  journal={arXiv: K-Theory and Homology},
  year={2011}
}
We compute the $K$-theory of comparison $C^*$-algebra associated to a manifold with corners. These comparison algebras are an example of the abstract pseudodifferential algebras introduced by Connes and Moscovici \cite{M3}. Our calculation is obtained by showing that the comparison algebras are a homomorphic image of a groupoid $C^*$-algebra. We then prove an index theorem with values in the $K$-theory groups of the comparison algebra. 
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