Extensions and renormalized traces |
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Authors: | Email author" target="_blank">Denis?PerrotEmail author |
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Institution: | (1) Mathematisches Institut, Universität Münster, Einsteinstr. 62, 48149 Münster, Germany |
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Abstract: | It has been shown by Nistor (Doc Math J DMV 2:263–295, 1997) that given any extension of associative algebras over
\mathbb C{\mathbb C}, the connecting morphism in periodic cyclic homology is compatible, under the Chern–Connes character, with the index morphism
in lower algebraic K-theory. The proof relies on the abstract properties of cyclic theory, essentially excision, which does not provide explicit
formulas a priori. Avoiding the use of excision, we explain in this article how to get explicit formulas in a wide range of
situations. The method is connected to the renormalization procedure introduced in our previous work on the bivariant Chern
character for quasihomomorphisms Perrot (J Geom Phys 60:1441–1473, 2010), leading to “local” index formulas in the sense of non-commutative geometry. We illustrate these principles with the example
of the classical family index theorem: we find that the characteristic numbers of the index bundle associated to a family
of elliptic pseudodifferential operators are expressed in terms of the (fiberwise) Wodzicki residue. |
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