Asymptotic Morphisms and Elliptic Operators over C*‐Algebras

Abstract

This paper provides an E-theoretic proof of an exact form, due to E. Troitsky, of the Mischenko-Fomenko Index Theorem for elliptic pseudodifferential operators over a unital C * -algebra. The main ingredients in the proof are the use of asymptotic morphisms of Connes and Higson, vector bundle modification, a Baum-Douglas-type group, and a KK-argument of Kasparov.

On graded $K$-theory, elliptic operators and the functional calculus

“…Now extend the automorphism u to the A-bundle A by defining it to be equal to the identity on the complement of π * E ⊕ π * F in A. This is the symbol class of the elliptic A-operator D as constructed in [7,14,17]. By Corollary 4.8 and stability, it follows that…”

“…Let C ∞ (E) denote the vector space of smooth sections of E, which is a module over A, similarly for C ∞ (F ). Let D : C ∞ (E) → C ∞ (F ) be an elliptic differential A-operator of order n on M [13,17]. (If A = C then D is an ordinary differential operator.)…”

“…The following result implies the exact form of the Mishchenko-Fomenko index theorem [13], hence the Atiyah-Singer index theorem [1] when A = Theorem 5.4. (Lemma 4.6 [17]) If D is an elliptic differential A-operator of order one on M then Ψ * ([σ(D)]) = Index A (D) ∈ K 0 (A).…”

“…One then proves [17] via Fourier analysis and a compactness argument that for any f ∈ C 0 (R), lim t→∞ Ψ t (f (σ)) − f (t −1 D) = 0 and so the composition {Ψ t • Φ σ } is asymptotically equivalent to {Φ t D }. Therefore, by stability and homotopy invariance of the induced map [5,6],…”

Deformations of nilpotent groups and homotopy symmetric $$C^*$$-algebras

“…These morphisms are at the heart of E-theory, the universal bifunctor from the category of separable C * -algebras to the category abelian groups which is homotopy invariant, C * -stable and half-exact. Asymptotic morphisms have become important tools in other areas, such as deformation quantization [22], index theory [4], [27], the Baum-Connes conjecture [14], shape theory [8], and classification theory of nuclear C*-algebras [24].…”