  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.
