Translation invariant asymptotic homomorphisms: equivalence of two approaches in the index theory

Abstract

The algebra $\Psi(M)$ of order zero pseudodifferential operators on a compact manifold $M$ defines a well-known $C^*$-extension of the algebra $C(S^*M)$ of continuous functions on the cospherical bundle $S^*M\subset T^*M$ by the algebra $\K$ of compact operators. In his proof of the index theorem, Higson defined and used an asymptotic homomorphism $T$ from $C_0(T^*M)$ to $\K$, which plays the role of a deformation for the commutative algebra $C_0(T^*M)$. Similar constructions exist also for operators and symbols with coefficients in a $C^*$-algebra. We show that the image of the above extension under the Connes--Higson construction is $T$ and that this extension can be reconstructed out of $T$. This explains, why the classical approach to the index theory coincides with the one based on asymptotic homomorphisms.