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

We compute the $K$-theory of comparison $C^{\ast }$-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 $C^{\ast }$-algebra. We then prove an index theorem with values in the $K$-theory groups of the comparison algebra.