The Algebraic Bivariant Connes-Chern Character

Abstract

In this thesis we present many properties of bivariant periodic cyclic homology with the purpose of then constructing two bivariant Connes-Chern characters from algebraic versions of Kasparov's KK-theory with values in bivariant periodic cyclic homology. The thesis is naturally divided into three parts.

In the first part, which spans the two first chapters, periodic cyclic theory is presented, starting with the very basic definitions in cyclic theory. The properties of differential homotopy invariance, Morita invariance, and excision, all of which are important for the construction of bivariant Connes-Chern characters, are discussed.

In the second part we discuss algebraic KK-theory based on the reformulations of Kasparov's KK-theory by Cuntz and Zekri. By using the properties of bivariant periodic cyclic theory from the first part, we construct two different bivariant Connes-Chern characters.

In the third part we discuss possible extensions of the theory to topological algebras, in particular a well-behaved class of topological algebras known as m-algebras.