Translation Invariant Asymptotic Homomorphisms: Equivalence of Two Approaches in the Index Theory

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 K of compact operators. In his proof of the index theorem, Higson defined and used an asymptotic homomorphism T from C0(T ∗M) to K, 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. 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. 1. Two ways to define index The standard way to define the index of a pseudodifferential elliptic operator on a compact manifold M comes from the short exact sequence of C-algebras 0 → K → Ψ(M) → C(SM) → 0, (1) where K is the algebra of compact operators on L(M), Ψ(M) denotes the norm closure of the algebra of order zero pseudodifferential operators in the algebra of bounded operators on L(M) and SM denotes the cospherical bundle, SM = {(x, ξ) ∈ T M : |ξ| = 1}, in the cotangent bundle T M . If one deals with operators having coefficients in a C-algebra A then one has to tensor the short exact sequence (1) by A: 0 → K⊗ A→ ΨA(M) → C(S M ;A) → 0, (2) where C(X;A) denotes the C-algebra of continuous functions on X taking values in A. The (main) symbol of a pseudodifferential elliptic operator of order zero is an invertible element in C(SM ;A) and the K-theory boundary map K1(C(S M ;A)) → K0(K⊗A) maps the symbol to a class in K0(A), which is called the index of the operator. Another approach, suggested by Higson in [2], is based on the notion of an asymptotic homomorphism [1]. Here one starts with a symbol σ of a pseudodifferential operator of order one and constructs a symbol class [aσ] ∈ K0(C0(T M)) (see details in [2]). Then one constructs an asymptotic homomorphism from C0(T M) to K as follows. In the local coordinates (x, ξ) in U × R ⊂ TM take a smooth function Partially supported by RFFI grant No. 02-01-00574 and by HIII-619.2003.01. 1