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The C*-Algebras of Certain Lie Groups Janne-Kathrin Günther

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The C*-Algebras of Certain Lie Groups Janne-Kathrin Günther The C*-algebras of certain Lie groups Janne-Kathrin Günther To cite this version: Janne-Kathrin Günther. The C*-algebras of certain Lie groups. Algebraic Topology [math.AT]. Université de Lorraine, 2016. English. NNT : 2016LORR0118. tel-01446979 HAL Id: tel-01446979 https://tel.archives-ouvertes.fr/tel-01446979 Submitted on 26 Jan 2017 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. AVERTISSEMENT Ce document est le fruit d'un long travail approuvé par le jury de soutenance et mis à disposition de l'ensemble de la communauté universitaire élargie. Il est soumis à la propriété intellectuelle de l'auteur. Ceci implique une obligation de citation et de référencement lors de l’utilisation de ce document. D'autre part, toute contrefaçon, plagiat, reproduction illicite encourt une poursuite pénale. Contact : [email protected] LIENS Code de la Propriété Intellectuelle. articles L 122. 4 Code de la Propriété Intellectuelle. articles L 335.2- L 335.10 http://www.cfcopies.com/V2/leg/leg_droi.php http://www.culture.gouv.fr/culture/infos-pratiques/droits/protection.htm PhD-FSTC-2016-41 Facult´edes Sciences et Technologies The Faculty of Sciences, Technology and Institut Elie Cartan de Lorraine Communication Ecole Doctorale IAEM Lorraine DISSERTATION Defense held on 22/09/2016 in Luxembourg to obtain the degree of DOCTEUR DE L’UNIVERSITE´ DU LUXEMBOURG EN MATHEMATIQUES´ AND DOCTEUR DE L’UNIVERSITE´ DE LORRAINE EN MATHEMATIQUES´ by Janne-Kathrin GUNTHER¨ Born on 4 March 1985 in Preetz (Germany) THE C∗-ALGEBRAS OF CERTAIN LIE GROUPS Dissertation defense committee Dr Martin OLBRICH, dissertation supervisor Professor, Universit´edu Luxembourg Dr Jean LUDWIG, dissertation supervisor Professor, Universit´ede Lorraine Dr Ingrid BELTITA, reporter Professor, Romanian Academy Dr Joachim HILGERT, reporter Professor, Universit¨at Paderborn Dr Anton THALMAIER, Chairman Professor, Universit´edu Luxembourg Dr Angela PASQUALE, Vice Chairman Professor, Universit´ede Lorraine Contents 1 Introduction 1 2 General definitions and results 6 2.1 Nilpotent Lie groups . 6 2.1.1 Preliminaries ................................ 6 2.1.2 Orbitmethod................................ 7 2.1.3 The C∗-algebra C∗(G/U, χℓ) ....................... 8 2.2 Generalresults ................................... 9 3 The C∗-algebras of connected real two-step nilpotent Lie groups 11 3.1 Preliminaries .................................... 11 3.1.1 Two-step nilpotent Lie groups . 11 3.1.2 Induced representations . 11 3.1.3 Orbitmethod................................ 12 3.2 Conditions 1, 2 and 3(a) . 13 3.3 Condition 3(b) . 17 3.3.1 Introductiontothesetting ........................ 17 3.3.2 Changing the Jordan-H¨older basis . 17 3.3.3 Definitions . 19 3.3.4 Formula for πk ............................... 20 3.4 Condition 3(b) – First case . 22 3.5 Condition 3(b) – Second case . 23 3.5.1 Convergence of (πk)k N in G ....................... 23 ∈ 3.5.2 Definition of νk ............................... 25 3.5.3 Theorem – Second Case . .b . 29 V 3.5.4 Transition to πk k N ........................... 37 3.6 Condition 3(b) – Third case∈ . 38 3.6.1 Convergence of (πk)k N in G ....................... 38 ∈ 3.6.2 Definition of νk ............................... 41 3.6.3 Theorem – Third Case . .b . 46 V 3.6.4 Transition to πk k N ........................... 50 3.7 Result for the connected real∈ two-step nilpotent Lie groups . 51 3.8 Example: The free two-step nilpotent Lie groups of 3 and 4 generators . 51 3.8.1 The case n =3............................... 52 3.8.2 The case n =4............................... 56 4 The C∗-algebra of SL(2, R) 64 4.1 Preliminaries .................................... 64 4.2 The spectrum of SL(2, R)............................. 67 4.2.1 Introduction of the operator Ku ..................... 67 4.2.2 Description of the irreducible unitary representations . 76 4.2.3 The SL(2, R)-representations applied to the Casimir operator . 79 4.2.4 The topology on SL(2, R)......................... 81 4.2.5 Definition of subsets Γi ofthespectrum ................. 86 4.3 Condition 3(a) . .d . 87 4.4 Condition 3(b) . 90 4.5 Result for the Lie group SL(2, R)......................... 95 5 On the dual topology of the groups U(n) ⋉Hn 100 5.1 Preliminaries .................................... 100 5.1.1 The coadjoint orbits of Gn . 101 5.1.2 The spectrum of U(n)........................... 103 5.1.3 The spectrum and the admissible coadjoint orbits of Gn . 104 5.2 Convergence in the quotient space gn‡ /Gn . 107 5.3 The topology of the spectrum of Gn . 121 5.3.1 The representation π(µ,r) . 121 5.3.2 The representation π(λ,α) . 123 5.3.3 The representation τλ . 143 5.4 Results........................................ 143 6 R´esum´e´etendu de cette th`ese en fran¸cais 146 6.1 Introduction..................................... 146 6.2 D´efinitionsetr´esultatsg´en´eraux . 150 6.2.1 Les groupes de Lie nilpotents – Pr´eliminaires . 150 6.2.2 Les groupes de Lie nilpotents – La m´ethode des orbites . 151 6.2.3 Les groupes de Lie nilpotents – La C∗-alg`ebre C∗(G/U, χℓ) . 152 6.2.4 R´esultats g´en´eraux . 153 6.3 Les C∗-alg`ebres des groupes de Lie connexes r´eels nilpotents de pas deux . 154 6.3.1 Pr´eliminaires – Les groupes de Lie nilpotents de pas deux . 154 6.3.2 Pr´eliminaires – La m´ethode des orbites . 154 6.3.3 Les Conditions 1, 2 et 3(a) . 156 6.3.4 Condition 3(b) – Introduction de la situation . 156 6.3.5 Condition 3(b) – Changement de la base de Jordan-H¨older . 157 6.3.6 Condition 3(b) – Premier cas . 159 6.3.7 Condition 3(b) – Deuxi`eme cas . 160 6.3.8 Condition 3(b) – Troisi`eme cas . 162 6.3.9 R´esultat pour les groupes de Lie connexes r´eels nilpotents de pas deux 162 6.4 La C∗-alg`ebre du groupe de Lie SL(2, R) . 163 6.4.1 Pr´eliminaires . 163 6.4.2 Le spectre de SL(2, R) – Introduction de l’op´erateur Ku . 165 6.4.3 Le spectre de SL(2, R) – Description des repr´esentations irr´eductibles unitaires................................... 168 6.4.4 Le spectre de SL(2, R) – La topologie sur SL(2, R) . 170 6.4.5 Les conditions limites duales sous contrˆole normique . 171 6.4.6 R´esultat pour le groupe de Lie SL(2, R) . .d . 172 6.5 Sur la topologie duale des groupes U(n) ⋉Hn . 173 6.5.1 Pr´eliminaires . 173 6.5.2 Le spectre et les orbites coadjointes admissibles de Gn . 174 6.5.3 Convergence dans l’espace quotient gn‡ /Gn et la topologie du spectre de Gn ...................................... 175 6.5.4 R´esultats . 177 7 Appendix 178 8 Acknowledgements 186 9 References 187 Abstract In this doctoral thesis, the C∗-algebras of the connected real two-step nilpotent Lie groups and the Lie group SL(2, R) are characterized. Furthermore, as a preparation for an analysis of its C∗-algebra, the topology of the spectrum of the semidirect product U(n)⋉Hn is described, where Hn denotes the Heisenberg Lie group and U(n) the unitary ∗ group acting by automorphisms on Hn. For the determination of the group C -algebras, the operator valued Fourier transform is used in order to map the respective C∗-algebra into the algebra of all bounded operator fields over its spectrum. One has to find the conditions that are satisfied by the image of this C∗-algebra under the Fourier transform and the aim is to characterize it through these conditions. In the present thesis, it is proved that both the C∗-algebras of the connected real two-step nilpotent Lie groups and the C∗-algebra of SL(2, R) fulfill the same conditions, namely the “norm controlled dual limit” conditions. Thereby, these C∗-algebras are described in this work and the “norm controlled dual limit” conditions are explicitly computed in both cases. The methods used for the two-step nilpotent Lie groups and the group SL(2, R) are completely different from each other. For the two-step nilpotent Lie groups, one regards their coadjoint orbits and uses the Kirillov theory, while for the group SL(2, R) one can accomplish the calculations more directly. R´esum´e Dans la pr´esente th`ese de doctorat, les C∗-alg`ebres des groupes de Lie connexes r´eels nilpotents de pas deux et du groupe de Lie SL(2, R) sont caract´eris´ees. En outre, comme pr´eparation `aune analyse de sa C∗-alg`ebre, la topologie du spectre du produit semi-direct U(n) ⋉Hn est d´ecrite, o`u Hn d´enote le groupe de Lie de Heisenberg et U(n) le groupe ∗ unitaire qui agit sur Hn par automorphismes. Pour la d´etermination des C -alg`ebres de groupes, la transformation de Fourier `avaleurs op´erationnelles est utilis´ee pour appliquer chaque C∗-alg`ebre dans l’alg`ebre de tous les champs d’op´erateurs born´es sur son spectre. On doit trouver les conditions que satisfait l’image de cette C∗-alg`ebre sous la transfor- mation de Fourier et l’objectif est de la caract´eriser par ces conditions.
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