No CrossRef data available.
Article contents
On equivalence of two approaches in index theory
Published online by Cambridge University Press: 05 March 2008
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.
Keywords
- Type
- Research Article
- Information
- Journal of K-Theory , Volume 2 , Special Issue 3: In Memory of Yurii Petrovich Solovyev October 8, 1944–September 11, 2003 , December 2008 , pp. 445 - 452
- Copyright
- Copyright © ISOPP 2008
References
1.Connes, A., Higson, N., Déformations, morphismes asymptotiques et K-théorie bivariante, C. R. Acad. Sci. Paris Sér. I Math. 311 (1990), 101–106Google Scholar
2.Higson, N., On the K-theory proof of the index theorem, Index theory and operator algebras (Boulder, CO, 1991), 67–86, Contemp. Math. 148, Amer. Math. Soc., Providence, RI, 1993Google Scholar
3.Loring, T., Almost multiplicative maps between C*-algebras, Operator Algebras and Quantum Field Theory. Internat. Press, 1997, 111–122Google Scholar
4.Luke, G., Mishchenko, A. S., Vector Bundles and Their Applications, Kluwer, 1998CrossRefGoogle Scholar
5.Manuilov, V., Thomsen, K., E-theory is a special case of KK-theory, Proc. London Math. Soc. 88 (2004), 455–478CrossRefGoogle Scholar
6.Manuilov, V., Thomsen, K., Extensions of C*-algebras and translation invariant asymptotic homomorphisms, Math. Scand., to appearGoogle Scholar
7.Manuilov, V., Translation invariant asymptotic homomorphisms: equivalence of two approaches in index theory, J. Operator Theory, to appearGoogle Scholar