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

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. <br /> The dissertation consists of two main parts, discussed in chapters 1 and 2 respectively. <br /> 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). <br /> 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.