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Media type:
E-Book;
Electronic Thesis;
Doctoral Thesis
Title:
Bivariant K -theory of groupoids and the noncommutative geometry of limit sets
Contributor:
Mesland, Bram
[Author]
imprint:
Universitäts- und Landesbibliothek Bonn, 2009-07-21
Language:
English
DOI:
https://doi.org/20.500.11811/4109
Origination:
Footnote:
Diese Datenquelle enthält auch Bestandsnachweise, die nicht zu einem Volltext führen.
Description:
We present a categorical setting for noncommutative geometry in the sense of Connes. This is done by introducing a notion of morphism for spectral triples. Spectral triples are the unbounded cycles for $K$-homology (\cite{Con}), and their bivariant generalization are the cycles for Kasparov's $KK$-theory (\cite{Kas}). The central feature of $KK$-theory is the Kasparov product \[KK_{i}(A,B)\otimes KK_{j}(B,C)\rightarrow KK_{i+j}(A,C).\] Here $A,B$ and $C$ are $C^{*}$-algebras, and the product allows one to view $KK$ as a category. The unbounded picture of this theory was introduced by Baaj and Julg (\cite{BJ}). In this picture the external product \[KK_{i}(A,B)\otimes KK_{j}(A',B')\rightarrow KK_{i+j}(A\minotimes B,A'\minotimes B'),\] is given by an algebraic formula, as opposed to Kasparov's original approach, which is more analytic in nature, and highly technical. In order to describe the internal Kasparov product of unbounded $KK$-cycles, we introduce a notion of connection for unbounded cycles $(\mathpzc{E},D)$. This is a universal connection \[\nabla:\mathpzc{E}\rightarrow\mathpzc{E}\tildeotimes_{B}\Omega^{1}(B),\] in the sense of Cuntz and Quillen (\cite{CQ}), such that $[\nabla,D]$ extends to a completely bounded operator. The topological tensor product used here is the Haagerup tensor product for operator spaces. Blecher (\cite{Blech}) showed this tensor product coincides with the $C^{*}$-module tensor product, in case both operator spaces are $C^{*}$-modules. His work plays a crucial role in our construction. The product of two cycles with connection is given by an algebraic formula and the product of connections can also be defined. Thus, cycles with connection form a category, and the bounded transform \[(\mathpzc{E},D,\nabla)\mapsto(\mathpzc{E},D(1+D^{2})^{-\frac{1}{2}}),\] defines a functor from this category to the category $KK$. We also describe a general construction for obtaining $KK$-cycles from real-valued groupoid cocycles. If $\mathcal{G}$ is a locally compact Hausdorff groupoid with Haar ...