Unbounded Bivariant K-theory and an Approach to Noncommutative Fréchet Spaces [Elektronische Ressource] / Nicolay Ivankov

Unbounded Bivariant K-theoryand an Approach toNoncommutative Fréchet SpacesDissertationzurErlangerung des Doktorgrades (Dr. rer. nat.)derMathematisch-Naturwissenschaftlichen FakultätderRhienischen Friedriech-Wilhelms-Universität Bonnvorgelegt vonNikolay IvankovausKaliningrad, Moskau Gebiet (Russland)Bonn, 2011iAngefertigt mit Genehmigung der Matematisch-Naturwissenschaftlichen Fakultät derRhienischen Friedriech-Wilhelms-Universität Bonn1. Gutachter: Prof. Dr. Yuri I. Manin2. Prof. Dr. Werner BallmannTag der Promotion: 15. August 2011Erscheinungsjahr: 2011AbstractIn the current work we thread the problems of smoothness in non-commutative C -algebras arising form the Baaj-Julg picture of the KK-theory. We introduce the notion of smoothness based on the pre-C -subalgebras of C -algebras endowed with the struc-ture of an operator algebra. We prove that the notion of smoothness introduced in thepaper may then be used for simplification of calculations in classical KK-theory.The dissertation consists of two main parts, discussed in chapters 1 and 2 respectively.In the Chapter 1 we first give a brief overview to Baaj-Julg picture of KK-theory andits relation to the classical KK-theory, as well as an approach to smoothness in Banachalgebras, introduced by Cuntz and Quillen. The rest of the chapter is devoted to opera-tor spaces, operator algebras and operator modules.