On equivalence of two approaches in index theory

Abstract

The algebra Ψ(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 ⊂ T*M by the algebra К of compact operators. In his proof of the index theorem, Higson defined and used an asymptotic homomorphism T from C₀(T*M) to К, which plays the role of a deformation for the commutative algebra C₀(T*M). Similar constructions exist also for operators and symbols with coefficients in a C*-algebra. Recently we have shown that the image of the above extension under the Connes–Higson construction is T and that this extension can be reconstructed out of T. That is why the classical approach to the index theory coincides with the one based on asymptotic homomorphisms. But the image of the above extension is defined only outside the zero section of T*(M), so it may seem that the information encoded in the extension is not the same as that in the asymptotic homomorphism. We show that this is not the case.