Journal of Operator Theory

Volume 55, Issue 2, Spring 2006 pp. 339-347.

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

Authors: V. Manuilov
Author institution: epartment of Mechanics and Mathematics, Moscow State University, Moscow, 119992, Russia and Harbin Institute of Technology, Harbin, P.R. China

Summary: 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 index theory coincides with the one based on asymptotic homomorphisms.

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