Unbounded Bivariant Ktheory and an Approach to Noncommutative Fréchet Spaces
Unbounded Bivariant Ktheory and an Approach to Noncommutative Fréchet Spaces
dc.contributor.advisor  Manin, Yuri I.  
dc.contributor.author  Ivankov, Nikolay  
dc.date.accessioned  20200417T06:58:49Z  
dc.date.available  20200417T06:58:49Z  
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 noncommutative $C^*$algebras arising form the BaajJulg 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 BaajJulg 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 antiisomorphism (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 BaajJulg 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  KKTheory  
dc.subject  Operator Algebras  
dc.subject  Noncommutative Geometry  
dc.subject.ddc  510 Mathematik  
dc.title  Unbounded Bivariant Ktheory 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://nbnresolving.org/urn:nbn:de:hbz:5N26240  
ulbbn.pubtype  Erstveröffentlichung  
ulbbnediss.affiliation.name  Rheinische FriedrichWilhelmsUniversität Bonn  
ulbbnediss.affiliation.location  Bonn  
ulbbnediss.thesis.level  Dissertation  
ulbbnediss.dissID  2624  
ulbbnediss.date.accepted  20110815  
ulbbnediss.fakultaet  MathematischNaturwissenschaftliche Fakultät  
dc.contributor.coReferee  Ballmann, Werner 
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