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1-Cocycles of Unitary Representations of Infinite–Dimensional Unitary Groups

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1-Cocycles of Unitary Representations of Infinite–Dimensional Unitary Groups 1-Cocycles of unitary representations of infinite–dimensional unitary groups Der Naturwissenschaftlichen Fakultät der Friedrich-Alexander-Universität Erlangen-Nürnberg zur Erlangung des Doktorgrades Dr. rer. nat. vorgelegt von Manuel Herbst aus Bogotá Als Dissertation genehmigt von der Naturwissenschaftlichen Fakultät der Friedrich-Alexander-Universität Erlangen-Nürnberg Tag der mündlichen Prüfung: 18.12.2018 Vorsitzender des Promotionsorgans: Prof. Dr. Georg Kreimer Gutachter: Prof. Dr. Karl-Hermann Neeb, Prof. Dr. Alain Valette I Acknowledgements I wish to express my sincere gratitude to the following people that supported me throughout my PhD time: First and foremost, I thank Prof. Dr. Karl–Hermann Neeb for his very friendly and always available help as supervisor and initiator of my project. His knowledge and all the fruitful discussions with him provided an invaluable source of inspiration for my work. I am grateful for his numerous corrections and proof-readings that helped me to improve my thesis. I am very much obliged to him and to the Friedrich–Alexander university (FAU) for providing me an office room for my project. I owe special thanks to the German National Academic Foundation (Studienstiftung des deutschen Volkes) who granted me a PhD scholarship for three years. In particu- lar, I gratefully acknowledge the helpful advices of Prof. Dr. André Kaup concerning the timing of my PhD project. I would also like to thank my family and my colleagues for encouragement in times of frustation. In particular, I would like to thank Prof. Dr. Catherine Meusburger and Stefan Wiedenmann. II Abstract The main objective of this thesis is to establish a classification theorem on the non- vanishing of the first order cohomology space H1 associated to a unitary highest weight representations (HWR), depending on the respective highest weight λ. This is real- ized for unitary HWR’s of various types of unitary groups: For the finite-dimensional unitary groups U(n), it is well-known that every continuous irreducible unitary rep- n resentation is a HWR with some highest weight λ Z . But in this case, the space 1 H is always trivial because the group U(n) is compact.2 The situation is drastically different for the direct limit U( ) := lim U(n): Here, we obtain for every λ ZN 1 2 a unitary HWR as a direct limit of unitary−! HWR’s of the groups U(n). We determine precisely for which λ the space H1 is non-trivial and find that this is true in most cases, in particular whenever λ = 0 is finitely supported. The proof is not constructive and makes use of the classical6 Branching Laws for HWR’s of the U(n). In a second step, 2 we extend the direct limit group U( ) to its Banach completions Up(` ). Here, we 1 2 allow p to take values in [1, ) and a unitary operator g belongs to Up(` ) if and only if the difference g 1 is an1 operator of pth Schatten class. For finitely supported − 2 λ, the corresponding unitary HWR of U( ) uniquely extends to Up(` ) (and even to 2 the whole unitary group U(` )). The associated1 unitary HWR has a concrete realiza- 2 2 tion in a finite tensor product of Hilbert spaces that is built from ` and its dual (` )∗. Whenever this tensor product consists of at least two factors, there is a natural way to construct unbounded 1-cocycles as infinite sums of 1-coboundaries and we derive that the spaces H1 are non-trivial (even infinite-dimensional). In the complementary 2 2 cases, which are the identical representation of Up(` ) on ` , its dual representation 2 on (` )∗ and the one-dimensional trivial representation, the first order 1-cohomology spaces are trivial. The proofs rely on automatic continuity of group 1-cocycles under mild assumptions and on their automatic smoothness in a suitable Lie group setting. The latter provides a bridge between first order cohomology of Lie groups and that of Lie algebras. These tools are developed and discussed in this thesis. Finally, if the highest weight λ `1(N, Z) is bounded but not finitely supported, then the corre- 2 2 sponding unitary HWR of U( ) only extends to the trace-class completion U1(` ), 1 1 2 and in this case we have H = 0 for U1(` ) without exceptions. This result is ob- f g 2 tained by restricting the representation to the subgroup of diagonal operators T1(` ) (usually called a “maximal torus”). For finite-dimensional semisimple Lie groups, the non-vanishing of the spaces H1 for irreducible representations is a rare phenomenon. In contrast to that, our classifica- tion theorem shows that groups like the direct limit U( ) and its Banach completions 2 1 Up(` ) have natural irreducible unitary representations with highly non-trivial first or- der cohomology spaces. In particular, these groups do not have Property (FH). Here, 1 a topological group G is said to have Property (FH) if H = 0 holds for every con- tinuous unitary representation of G. On the other hand, infinite-dimensionalf g groups 2 like U (` ) (the difference g 1 is a compact operator) and the full unitary group 2 U(` ) do1 have Property (FH) since− they belong to the class of bounded groups (or more generally groups with Rosendal’s Property (OB)). III Zusammenfassung Das Hauptziel dieser Arbeit ist die Herleitung eines Klassifikationssatzes über das Nicht-Verschwinden des Kohomologieraumes erster Ordnung, mit H1 bezeichnet, für eine höchste Gewichts-Darstellung (HGD) in Abhängigkeit vom jeweiligen höchsten Gewicht λ. Dies wird für unitäre HGDen verschiedener Klassen von unitären Gruppen realisiert: Es ist bekannt, dass jede stetige irreduzible unitäre Darstellung der endlich- n dimensionalen unitären Gruppen U(n) eine Darstellung mit höchstem Gewicht λ Z 1 ist. In diesem Fall ist jedoch der Raum H stets trivial, weil die Gruppe U(n) kompakt2 ist. Ganz anders verhält es sich im Falle des direkten Limes U( ) := lim U(n): Hier 1 erhält man zu jedem λ ZN eine unitäre HGD als direkten Limes von unitären−! HGDen 1 der Gruppen U(n). Wir2 bestimmen exakt diejenigen Gewichte λ, für die der Raum H nicht verschwindet und sehen, dass dies in den meisten Fällen zutrifft, insbesondere immer dann, wenn λ = 0 endlichen Träger hat. Der Beweis ist nicht konstruktiver Art und benutzt die klassischen6 Verzweigungsgesetze (“Branching Laws”) für HGDen der 2 U(n). In einem zweiten Schritt betrachten wir die Banach-Vervollständigungen Up(` ) der Gruppe U( ). Hierbei lassen wir Werte aus [1, ) für p zu und ein unitärer 1 2 1 1 Operator g ist aus Up(` ) genau dann wenn die Differenz g ein p-ter Schatten- Klasse Operator ist. Für Gewichte λ mit nur endlich vielen Einträgen− lässt sich die 2 entsprechende unitäre HGD von U( ) eindeutig auf Up(` ) fortsetzen (sogar auf 2 die volle unitäre Gruppe U(` )). Die1 entsprechende unitäre HGD lässt sich in einem endlichen Hilbertraum-Tensorprodukt realisieren, welches aus `2 und seinem Dual- 2 raum (` )∗ gebildet wird. Besteht dieses Tensorprodukt aus mindestens zwei Faktoren, so lassen sich unbeschränkte 1-Kozykel als unendliche Summen von 1-Korändern auf natürliche Art konstruieren und wir leiten daraus ab, dass die Räume H1 nicht triv- ial (sogar unendlich-dimensional) sind. In den komplementären Fällen, also bei der 2 2 2 identischen Darstellung von Up(` ) auf ` , deren dualer Darstellung auf (` )∗ sowie der eindimensionalen trivialen Darstellung, verschwinden die zugehörigen Räume H1. 1 2 Die Beweise bezüglich der H -Räume für die Gruppen Up(` ) fußen auf automatischer Stetigkeit von Gruppen-1-Koyzkeln unter milden Zusatzannahmen und auf deren au- tomatischer Glattheit in einem geeigneten Lie Gruppen-Kontext. Letzteres stellt eine Beziehung zwischen der 1-Kohomologie von Lie-Gruppen und jener von Lie Algebren her. Wenn schließlich das höchste Gewicht λ `1(N, Z) beschränkt aber nicht endlich getragen ist, dann setzt sich die jeweilige HGD2 von U( ) nur noch auf die Spurklasse- 2 11 Vervollständigung U1(` ) fort und in diesem Fall gilt H = 0 ausnahmslos. Dieses Resultat erhalten wir durch Einschränken der Darstellung auff dieg Untergruppe der Di- 2 agonaloperatoren T1(` ) (üblicherweise als “Maximaler Torus” bezeichnet). Für endlich-dimensionale halbeinfache Lie-Gruppen ist das Nicht-Verschwinden der H1-Räume ein seltenes Phänomen. Im Gegensatz dazu zeigt unser Klassifikationssatz, dass Gruppen wie der direkt Limes U( ) und seine Banach-Vervollständigungen 2 1 Up(` ) natürliche irreduzible unitäre Darstellungen mit hochgradig nichttrivialen Ko- homologieräumen H1 aufweisen. Insbesondere haben diese Gruppen nicht die Eigen- schaft (FH). Wir sagen hier, dass eine topologische Gruppe G die Eigenschaft (FH) hat, 1 wenn H = 0 für jede stetige unitäre Darstellung von G gilt. Auf der anderen Seite f g IV 2 haben unendlich-dimensionale Gruppen wie U (` ) (hier ist die Differenz g 1 ein 2 kompakter Operator) und die volle unitäre Gruppe1 U(` ) durchaus die Eigenschaft− (FH), weil sie zur Klasse der beschränkten Gruppen (oder allgemeiner der Gruppen mit Rosendals Eigenschaft (OB)) zählen. Contents V Contents 1 Introduction1 2 Preliminary facts on 1-cocycles 20 2.1 Basic definitions.................................. 20 2.2 1-cocycles, affine isometric actions and negative definite functions... 26 2.3 A generalized version of the Delorme–Guichardet Theorem....... 31 3 Topological Aspects 39 3.1 Baire groups and automatic continuity.................... 39 3.2 Boundedness of topological groups and Rosendal’s Property (OB)... 45 3.3 A short remark on bounded orbits and fixed points in Banach spaces. 55 4 Lie Theoretic Aspects 61 4.1 The class of locally exponential Lie groups.................. 61 4.2 Integration and differentiation of 1-cocycles................. 73 2 5 Schatten ideals, unitary subgroups of U(` ) and unbounded 1-cocycles 84 5.1 Schatten class operators............................. 84 5.2 The Banach–Lie group Up( ) ......................... 88 H 5.3 Derivations of Banach algebras and 1-cocycles............... 94 6 Unitary Highest Weight Representations (HWR) of Lie algebras 102 6.1 General facts on unitary highest weight representations........
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