# Index theory and groupoids for filtered manifolds

 dc.contributor.advisor Meyer, Ralf Prof. Dr. dc.contributor.author Ewert, Eske Ellen dc.date.accessioned 2020-12-21T13:35:33Z dc.date.available 2020-12-21T13:35:33Z dc.date.issued 2020-12-21 dc.identifier.uri http://hdl.handle.net/21.11130/00-1735-0000-0005-152D-2 dc.identifier.uri http://dx.doi.org/10.53846/goediss-8375 dc.language.iso eng de dc.rights.uri http://creativecommons.org/licenses/by-nc-nd/4.0/ dc.subject.ddc 510 de dc.title Index theory and groupoids for filtered manifolds de dc.type doctoralThesis de dc.contributor.referee Schick, Thomas Prof. Dr. dc.date.examination 2020-10-26 dc.description.abstracteng In this thesis, we propose to use generalized fixed point algebras as an approach to the pseudodifferential calculus on filtered manifolds. de A filtered manifold is a manifold $$M$$ with a filtration of its tangent bundle which is compatible with the Lie bracket. This filtration allows to define a new notion of order for the differential operators on $$M$$. As a result, the highest order part of a differential operator is a family of right-invariant model operators acting on certain nilpotent Lie groups. These groups form the bundle of osculating groups of the filtered manifold. The new order can be encoded by a dilation action of $$\mathbb{R}_{>0}$$ on this bundle. The tangent groupoid of a filtered manifold $$M$$ describes the relation between the operators acting on $$M$$ and their model operators on the osculating groups. It is equipped with a "zoom" action of $$\mathbb{R}_{>0}$$ that extends the dilations. In this thesis, we build the generalized fixed point algebra for the zoom action on a certain ideal $$J$$ in the groupoid $$C^*$$-algebra of the tangent groupoid. This generalized fixed point algebra $$\mathrm{Fix}^{\mathbb{R}_{>0}}(J)$$ is a $$C^*$$-subalgebra of the bounded operators on $$L^2(M)$$. Moreover, there is a "principal symbol map" which takes values in another generalized fixed point algebra $$\mathrm{Fix}^{\mathbb{R}_{>0}}(J_0)$$, where $$J_0$$ is an ideal in the $$C^*$$-algebra of the bundle of osculating groups. The kernel of the symbol map consists of the compact operators. The symbol algebra is, in general, noncommutative. It is unital if $$M$$ is compact. In this case, call $$P\in\mathrm{Fix}^{\mathbb{R}_{>0}}(J)$$ elliptic if its principal symbol is invertible. We show that $$P$$ is elliptic if and only if all model operators satisfy the Rockland condition. Furthermore, it is shown that the sequence above coincides with the $$C^*$$-completion of the order zero pseudodifferential extension by van Erp and Yuncken. When viewing a graded Lie group as a filtered manifold, we show that the same holds for the calculus by Fischer, Ruzhansky and Fermanian-Kammerer. We prove that $$\mathrm{Fix}^{\mathbb{R}_{>0}}(J_0)$$ is $$KK$$-equivalent to the usual principal symbol algebra of functions on the cosphere bundle of $$M$$. Lastly, we present an index theorem, up to inverting the Connes-Thom isomorphism, for order zero pseudodifferential operators on a compact filtered manifold that are elliptic in this calculus. dc.contributor.coReferee Schrohe, Elmar Prof. Dr. dc.contributor.thirdReferee Møller, Niels Martin Assoc. Prof. dc.subject.eng Filtered manifolds de dc.subject.eng Rockland condition de dc.subject.eng Generalized fixed point algebras de dc.subject.eng K-theory de dc.subject.eng Noncommutative Geometry de dc.subject.eng Groupoids de dc.subject.eng Tangent groupoid de dc.subject.eng Index theory de dc.subject.eng Graded Lie groups de dc.subject.eng Pseudodifferential calculus de dc.subject.eng C*-algebras de dc.subject.eng Representation theory de dc.identifier.urn urn:nbn:de:gbv:7-21.11130/00-1735-0000-0005-152D-2-6 dc.affiliation.institute Fakultät für Mathematik und Informatik de dc.subject.gokfull Mathematik (PPN61756535X) de dc.identifier.ppn 1743472412
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