## Unbounded Bivariant K-theory and an Approach to Noncommutative Fréchet Spaces

 dc.contributor.advisor Manin, Yuri I. dc.contributor.author Ivankov, Nikolay dc.date.accessioned 2020-04-17T06:58:49Z dc.date.available 2020-04-17T06:58:49Z dc.date.issued 19.09.2011 dc.identifier.uri http://hdl.handle.net/20.500.11811/5028 dc.description.abstract In 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 structure of an operator algebra. We prove that the notion of smoothness introduced in the paper 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 and its relation to the classical $KK$-theory, as well as an approach to smoothness in Banach algebras, introduced by Cuntz and Quillen. The rest of the chapter is devoted to operator spaces, operator algebras and operator modules. We introduce the notion of stuffed modules, that will be used for the construction of smooth modules, and study their properties. This part also contains an original research, devoted to characterization of operator algebras with a completely bounded anti-isomorphism (an analogue of involution). In Chapter 2 we introduce the notion of smooth system over a not necessarily commutative $C^*$-algebra and establish the relation of this definition of smoothness to the Baaj-Julg picture of $KK$-theory. For that we define the notion of fr{\'e}chetization as a way of construction of a smooth system form a given unbounded $KK$-cycle. For a given smooth system $\mathscr{A}$ on a $C^*$-algebra $A$ we define the set $\Psi^{(n)}_{\mu}(\mathscr{A},B)$, $n\in\mathbb{N}\cup\{\infty\}$ of the unbounded $(A,B)$-$KK$-cycles that are $n$ smooth with respect to the smooth system $\mathscr{A}$ on $A$ and fr{\'e}chetization $\mu$. Then we subsequently prove two main results of the dissertation. The first one shows that for a certain class of fr{\'e}chetizations it holds that for any set of $C^*$-algebras $\Lambda$ there exists a smooth system $\mathscr{A}$ on $A$ such that there is a natural surjective map $\Psi^{(\infty)}_{\mu}(\mathscr{A},B) \to KK(A,B)$ for all $B\in\Lambda$. The other main result is a generalization of the theorem by Bram Mesland on the product of unbounded $KK$-cycles. We also present the prospects for the further development of the theory. dc.language.iso eng dc.rights In Copyright dc.rights.uri http://rightsstatements.org/vocab/InC/1.0/ dc.subject KK-Theory dc.subject Operator Algebras dc.subject Noncommutative Geometry dc.subject.ddc 510 Mathematik dc.title Unbounded Bivariant K-theory and an Approach to Noncommutative Fréchet Spaces dc.type Dissertation oder Habilitation dc.publisher.name Universitäts- und Landesbibliothek Bonn dc.publisher.location Bonn dc.rights.accessRights openAccess dc.identifier.urn https://nbn-resolving.org/urn:nbn:de:hbz:5N-26240 ulbbn.pubtype Erstveröffentlichung ulbbnediss.affiliation.name Rheinische Friedrich-Wilhelms-Universität Bonn ulbbnediss.affiliation.location Bonn ulbbnediss.thesis.level Dissertation ulbbnediss.dissID 2624 ulbbnediss.date.accepted 2011-08-15 ulbbnediss.fakultaet Mathematisch-Naturwissenschaftliche Fakultät dc.contributor.coReferee Ballmann, Werner
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