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

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.

💡 Research Summary

The paper investigates the K‑theory and index theory of a class of comparison C*‑algebras that arise naturally from manifolds with corners. The authors begin by recalling the construction of the comparison algebra (\mathcal{A}(M)) associated to a compact manifold (M) whose boundary may have stratified corner faces. This algebra, introduced in the framework of abstract pseudodifferential algebras by Connes and Moscovici, encodes the full symbolic calculus of pseudodifferential operators that respect the corner structure, and it is inherently non‑commutative.

A central technical achievement of the work is the identification of (\mathcal{A}(M)) as a homomorphic image of a groupoid C*‑algebra. The authors construct a Lie groupoid (\mathcal{G}{M}) whose space of units is (M) and whose arrows model the admissible changes of variables near each stratum of the corner. The source and target maps are carefully defined so that the groupoid captures both interior diffeomorphisms and the dilations that occur along boundary faces. By proving that (\mathcal{G}{M}) admits a smooth Haar system and that its convolution algebra is dense in (\mathcal{A}(M)), they obtain a surjective *‑homomorphism \