Bivariant K-theory of groupoids and the noncommutative geometry of limit sets [Elektronische Ressource] / vorgelegt von Bram Mesland

BivariantK-theory of groupoids and thenoncommutative geometry of limit setsDissertationzurErlangung des DoktorgradesderMathematisch-Naturwissenschaftlichen Fakult atderRheinischen Friedrich-Wilhelms-Universit at BonnVorgelegt vonBram MeslandausAmstelveen, NiederlandeBonn 2009Diese Dissertation ist auf dem Hochschulschriftenserver der ULB Bonn unterhttp :==hss:ulb:uni-bonn:de=diss onlineelektronisch publiziert.Angefertigt mit Genehmigung derMathematisch-Naturwissenschaftlichen Fakult at derRheinischen Friedrich-Wilhelms-Universit at Bonn.1.Gutachter: Prof. Dr. Matilde Marcolli2.Gutachter: Prof.Dr. Matthias LeschTag der Promotion: Freitag, 17.Juli 2009iiiSummaryWe 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 arethe unbounded cycles for K-homology ([11]), and their bivariant generalization are thecycles for Kasparov’sKK-theory ([32]). The central feature ofKK-theory is the KasparovproductKK (A;B)KK (B;C)!KK (A;C):i j i+jHere 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 ([4]). In thispicture the external product0 0 0 0KK (A;B)KK (A;B )!KK (AB;AB );i j i+jis given by an algebraic formula, as opposed to Kasparov’s original approach, which ismore analytic in nature, and highly technical.