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Infinite Dimensional Lie Theory from the Point of View of Functional

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Infinite Dimensional Lie Theory from the Point of View of Functional Infinite dimensional Lie Theory from the point of view of Functional Analysis Josef Teichmann F¨urmeine Eltern, Michael, Michaela und Hannah Zsofia Hazel. Abstract. Convenient analysis is enlarged by a powerful theory of Hille-Yosida type. More precisely asymptotic spectral properties of bounded operators on a convenient vector space are related to the existence of smooth semigroups in a necessary and sufficient way. An approximation theorem of Trotter-type is proved, too. This approximation theorem is in fact an existence theorem for smooth right evolutions of non-autonomous differential equations on convenient locally convex spaces and crucial for the following applications. To enlighten the generically "unsolved" (even though H. Omori et al. gave interesting and concise conditions for regularity) question of the existence of product integrals on convenient Lie groups, we provide by the given approximation formula some simple criteria. On the one hand linearization is used, on the other hand remarkable families of right invariant distance functions, which exist on all up to now known Lie groups, are the ingredients: Assuming some natural global conditions regularity can be proved on convenient Lie groups. The existence of product integrals is an essential basis for Lie theory in the convenient setting, since generically differential equations cannot be solved on non-normable locally convex spaces. The relationship between infinite dimensional Lie algebras and Lie groups, which is well under- stood in the regular case, is also reviewed from the point of view of local Lie groups: Namely the question under which conditions the existence of a local Lie group for a given convenient Lie algebra implies the existence of a global Lie group is treated by cohomological methods. It is shown that the considerations do not depend on the convergence of the Campbell-Baker-Hausdorff-Formula as in the original paper of W.T. van Est and Th.J. Korthagen. Contents Danksagung v Introduction vii Chapter 1. Convenient Analysis 1 1. Smooth Calculus 1 2. Analytic Calculus 9 3. Convenient Manifolds 11 Chapter 2. Convenient Hille-Yosida-Theory 17 1. Classical concepts and modern points of view 17 2. Smooth semigroups 25 3. ACP and Zoology of locally convex Spaces 30 4. The Trotter approximation theorem for product integrals 38 5. Analytic Semigroups 42 6. Refinements, Applications and Examples 44 Chapter 3. Product Integrals on infinite dimensional groups 51 1. Convenient Lie groups 51 2. Tempered Lie groups 55 3. Lipschitz-metrizable smooth groups 61 4. Product integration via Metrization 66 5. Product integration via Linearization 71 Chapter 4. The inverse of S. Lie's third theorem 73 1. Local Lie groups and the CBH-Formula 73 2. Local Lie groups without the CBH-Formula 79 3. Cohomological conditions for enlargibility 79 4. Enlargibility of Lie Algebras 90 Appendix A. Smooth bump functions on convenient vector spaces 95 Appendix B. Homological algebra and spectral sequences 97 Bibliography 99 iii iv CONTENTS Danksagung Ein Mensch setzt sich zur Aufgabe, die Welt abzuzeichnen. Im Laufe der Jahre bev¨olkerter einen Raum mit Bildern von Provinzen, K¨onigreichen, Gebirgen, Buchten, Schiffen, Inseln, Fischen, Behausungen, Werkzeugen, Gestirnen, Pferden und Personen. Kurz bevor er stirbt, entdeckt er, daß dieses geduldige Labyrinth das Bild seines Gesichts wiedergibt. (Jorge Luis Borges) Im wesentlichen glaube ich ja, daß ich niemals sterben werde, aber trotzdem, daß in all dem, was man so tut und schafft mehr von einem steckt als auch auf dem zweiten Blick erkennbar ist. Damit schimmern auch die Personen, die durch Unterst¨utzungjedweder Art und durch Freundschaft in meinem Leben wichtig gewesen sind und sein werden zwischen der hochabstrakten Mathematik durch. Meinen Eltern Erna und Josef Teichmann geb¨uhrtder erste und gr¨oßteDank, unter vielen Dingen ist f¨urmich besonders zu erw¨ahnen,daß Diskussion ¨uber alles m¨ogliche, um den Sachen auf den Grund zu gehen, bei uns zu Hause eine große Rolle spielten. Im speziellen bin ich meinen Eltern daf¨urdankbar, diesen Stil mitbekommen zu haben f¨urmein Leben. Meinem Bruder, der die ersten Worte dieses Textes vor Jahren zu nachtschlafener Stunde in mein Hermes-Baby tippte, m¨ochte ich danken, nat¨urlich f¨urdie vielen interessanten Diskussionen, f¨urdie neuen Zug¨angezum Leben, f¨urseine fortw¨ahrendeUnterst¨utzungund die Liebe zur K-Theorie. Meiner Tante Hilde Darani m¨ochte ich f¨urdie sch¨onenGespr¨ache ¨uber Religion und das Leben im allgemeinen danken, f¨urdie fortw¨ahrendeUnterst¨utzungund ihren Charme. Michaela und ich sind zu Beginn unserer Zeit in Wien in das jeweilig andere Leben getreten, Monate sp¨aterbegannen wir dann uns auf Hannah Zsofia Hazel zu freuen. Michaela m¨ochte ich f¨urdie ungarischen T¨anzeim Santo Spirito danken, nat¨urlich die vielen Diskussionen, den Widerstandsgeist und die Liebe. H.Z. Hazel, obwohl noch ein sehr junger Mensch, hat den gr¨oßtenEinfluß auf uns alle. Bei ihr sind Eigenschaften wie Lust, Freude, Widerstandskraft und Wille wunderbar ausgepr¨agt. An dieser Stelle m¨ochte ich auch Sabine und Ingo Franz, Herbert Oberhuber und Sigrid Winkler f¨urdie Unterst¨utzungunserer kleinen Familie danken. Mit Stefan Haller hatte ich die Freude den Raum zu teilen, den wir regelm¨aßigin eine Spelunke mit tiefh¨angendenRauchschwaden verwandelten, vielen Dank f¨urdie sch¨onenDiskussionen. Dank geb¨uhrtauch den TeilnehmerInnen an den Wissenschaftlichen Arbeiten, insbesondere Andreas Cap, f¨urdie anregende und profunde Kritik und das Zuh¨oren. Mein mathematischer Weg wurde von zwei Pers¨onlichkeiten an der Universit¨atGraz gepr¨agt: Peter Sch¨opfhabe ich gleich zu Beginn meines Studiums bei einer Erl¨auterunglogischer Probleme der Analysis 1 kennengelernt, Franz Halter-Koch etwas sp¨aterbei seinen Algebra-Vorlesungen und auch als sein Sekret¨ar. Beiden verdanke ich mathematische Bildung, viele Diskussionen und Un- terst¨utzungen.Besonders erw¨ahnenm¨ochte ich dabei, daß Franz Halter-Koch mich auch pers¨onlich finanziell unterst¨utzte,um mir die M¨oglichkeit zu bieten, eine andere Universit¨atkennenzulernen und dort weiter zu studieren. Danken m¨ochte ich außerdem Alfred Geroldinger, Ludwig Pittner, Ernst Seidel und Bernd Thaller. Wolgang Arendt habe ich in Besan¸conim Studienjahr 1994/95 kennengelernt: Ihm m¨ochte ich f¨urdie beiden Einladungen nach Ulm, wo ich meine Arbeiten pr¨asentieren konnte, danken, auch f¨ur die sch¨onenDiskussionen ¨uber Osterreich¨ donauaufw¨arts. Mein Weg nach Wien war m¨oglicherweise mathematisch etwas unerwartet, da meine Diplomarbeit im funktionalanalytischen Bereich angesiedelt gewesen war, aber die Offenheit Peter Michors ließ alle Zweifel schwinden. Die liberale und intellektuelle Atmosph¨areum Peter Michor war f¨urmeine pers¨onliche Entwicklung und f¨urdas Gedeihen der Dissertation wesentlich. Außerdem hat er dadurch, v vi DANKSAGUNG daß er an meine Arbeit glaubte und sie aus Mitteln des FWF (Projekt P-10037-PHY) unterst¨utzte, nicht nur mein Leben als reiner Mathematiker erm¨oglicht, sondern auch mein Leben als Hausmann. F¨urdiese liberale und seltene Haltung, n¨amlich bez¨uglich der Zeiteinteilung mir freie Hand zu lassen, m¨ochte ich besonders danken. Andreas Kriegl m¨ochte ich f¨urdie vielen detaillierten Hinweise und die Redlichkeit beim Disku- tieren danken, auch f¨urdie Signale der Bereitschaft dazu. Anregungen zur Dissertation und zu meinem mathematischen Denken kamen auch von nicht- mathematischer Seite, einerseits durch die Aufforderung zu erkl¨aren,was ich da so mache, andererseits durch kluge und pr¨aziseStatements nach meinen Erl¨auterungen.So war Martin Feldbacher vor zwei Jahren fest davon ¨uberzeugt, daß das Konzept mit den Lipschitz-Metriken aufgehen wird. Sebastian Markt legte mir nahe, einige Aussagen zur mathematischen Logik noch einmal zu ¨uberlegen. Roman Urbaner sah, daß die Erstfassung meiner dilletierenden Ausf¨uhrungenzu Robert Musil doch etwas im Dunkeln liegen, wo sie vielleicht noch immer sind. Niko Wahl und Gregor Weiss m¨ochte ich f¨urdie Tatsache danken, mir die Bibel und das Kapital n¨ahergebracht zu haben. Josef Teichmann, Wien im Juli 1999 Introduction Er war weniger wissenschaftlich als menschlich verliebt in die Wissenschaft. Er sah, daß sie in allen Fragen, wo sie sich f¨urzust¨andigh¨alt,anders denkt als gew¨ohnlicheMenschen. Wenn man statt wissenschaftlicher Anschauungen Lebensanschauung setzen w¨urde, statt Hypothese Versuch und statt Wahrheit Tat, so g¨abe es kein Lebenswerk eines ansehnlichen Naturforschers oder Mathematikers, das an Mut und Umsturzkraft nicht die gr¨oßtenTaten der Geschichte weit ¨ubertreffen w¨urde. Der Mann war noch nicht auf der Welt, der zu seinen Gl¨aubigenh¨attesagen k¨onnen: Stehlt, mordet, treibt Unzucht - unsere Lehre ist so stark, daß sie aus der Jauche eurer S¨undensch¨aumendhelle Bergw¨assermacht; aber in der Wissenschaft kommt es alle paar Jahre vor, daß etwas was bis dahin als Fehler galt, pl¨otzlichalle Anschauungen umkehrt oder daß ein unscheinbarer und verachteter Gedanke zum Herrscher ¨uber ein neues Gedankenreich wird, und solche Vorkommnisse sind dort nicht bloß Umst¨urze,sondern f¨uhren wie eine Himmelsleiter in die H¨ohe. (Robert Musil, Der Mann ohne Eigenschaften) Mathematics is not politics, but the universe of mathematics has been influencing in a subtle and silent way, but revolutionary in result, the discourses of human beings in this century. Mathematics is the most dangerous science. Here I do not think of people exploiting mathematical knowledge for their own purposes, I have
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