KK-theory for Banach algebras and proper groupoids [Elektronische Ressource] / vorgelegt von Walther Dietrich Paravicini

MathematikDissertationsthemaKK Theory for Banach Algebrasand Proper GroupoidsInaugural Dissertationzur Erlangung des Doktorgradesder Naturwissenschaften im FachbereichMathematik und Informatikder Mathematisch Naturwissenschaftlichen Fakultätder Westfälischen Wilhelms Universität Münster vorgelegt vonWalther Dietrich Paraviciniaus Boulogne Billancourt 2006 Dekan: Prof. Dr. Joachim CuntzErster Gutachter: Prof. Dr. Siegfried EchterhoffZweiter Gutachter: Prof. Dr. Joachim CuntzTag der mündlichen Prüfung: 25. Januar 2007Tag der Promotion: 25. Januar 2007AbstractIn analogy to the definition of the assembly map of Baum Connes one can construct a homomorphismtopB ∗μ fromK (G,B) toK (A(G,B)), whereG is a locally compact group,B is aG C algebra and0AA(G) is an unconditional completion ofC (G), that is, a completion with respect to a normk·k suchc ABthatkfk only depends on the functiong7→|f(g)|. Isμ an isomorphism? This question was raisedA Aby Vincent Lafforgue, who has also given affirmative answers in many important cases. Moreover, heconsidered the more general situation where the groupG is replaced by a locally compact HausdorffgroupoidG.In the present thesis the setting is generalised further, takingB to be a non degenerateG Banach∗ Balgebra instead of a G C algebra. The main result asserts that the mapμ is split surjective ifAthe G Banach algebraB is proper (and A(G) satisfies some mild condition).