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

TL;DR

This paper computes the K-theory of comparison C*-algebras related to manifolds with corners and establishes an index theorem connecting these algebras to groupoid C*-algebras.

Contribution

It provides the first computation of K-theory for comparison C*-algebras associated with manifolds with corners and links these algebras to groupoid C*-algebras for index theory.

Findings

K-theory of comparison C*-algebras is computed.

Comparison algebras are homomorphic images of groupoid C*-algebras.

An index theorem with values in K-theory groups is proved.

Abstract

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.