Diffeotopy Functors of ind-Algebras and Local Cyclic Cohomology

We introduce a new bivariant cyclic theory for topological algebras, called local cyclic cohomology. It is obtained from bivariant periodic cyclic cohomology by an appropriate modification, which turns it into a deformation invariant bifunctor on the stable diffeotopy category of topological ind-algebras. We set up homological tools which allow the explicit calculation of local cyclic cohomology. The theory turns out to be well behaved for Banach- and $C^*$-algebras and possesses many similarities with Kasparov's bivariant operator K-theory. In particular, there exists a multiplicative bivariant Chern-Connes character from bivariant K-theory to bivariant local cyclic cohomology.

2000 Mathematics Subject Classification: Primary 46L80; Secondary 46L85, 18G60, 18E35, 46M99.

Keywords and Phrases: topological ind-algebra, infinitesimal deformation, almost multiplicative map, stable diffeotopy category, Fréchet algebra, Banach algebra, bivariant cyclic cohomology, local cyclic cohomology, bivariant Chern-Connes character.

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