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

The algebra $\mathrm{\Psi }(M)$ of order zero pseudodifferential operators on a compact manifold $M$ defines a well-known $C^{\ast }$-extension of the algebra $C(S^{\ast }M)$ of continuous functions on the cospherical bundle $S^{\ast }M\subset T^{\ast }M$ by the algebra $\text{K}$ of compact operators. In his proof of the index theorem, Higson defined and used an asymptotic homomorphism $T$ from $C_0(T^{\ast }M)$ to $\text{K}$, which plays the role of a deformation for the commutative algebra $C_0(T^{\ast }M)$. Similar constructions exist also for operators and symbols with coefficients in a $C^{\ast }$-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.