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Lie Algebras of Linear Operators on Locally Convex Spaces Rodrigo

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Lie Algebras of Linear Operators on Locally Convex Spaces Rodrigo Lie algebras of linear operators on locally convex spaces Rodrigo Augusto Higo Mafra Cabral Tese de Doutorado Orientador: Frank Michael Forger Programa de P´os-Gradua¸c~aoem Matem´atica Aplicada Instituto de Matem´atica e Estat´ıstica da Universidade de S~aoPaulo (IME - USP) Trabalho produzido com apoio financeiro da ag^encia CNPq Outubro de 2019 Lie algebras of linear operators on locally convex spaces Esta vers~aoda tese cont´emas corre¸c~oes e altera¸c~oessuge- ridas pela Comiss~aoJulgadora no dia da defesa da ver- s~aooriginal do trabalho, realizada em 15/03/2019. Uma c´opia da vers~aooriginal est´adispon´ıvel no Instituto de Matem´atica e Estat´ıstica da Universidade de S~aoPaulo. Comiss~aoJulgadora: • Prof. Dr. Frank Michael Forger (orientador) - IME - USP • Prof. Dr. Christian Dieter J¨akel - IME - USP • Prof. Dr. Severino Toscano do R^egoMelo - IME - USP • Prof. Dr. Pedro Lauridsen Ribeiro - CMCC - UFABC • Prof. Dr. Luiz Roberto Hartmann Junior - DM - UFSCar Dedicated to my parents, Ione and Juarez Agradecimentos Primeiramente, gostaria de agradecer aos meus pais, Ione e Juarez, pelo apoio constante e incondicional, e sem os quais este trabalho n~aoseria poss´ıvel; ao meu irm~ao,Gabriel, por ser um grande amigo com quem sempre posso contar; e `aGabs, por ter sido uma grande companhia nas (muitas) madrugadas em que esta tese foi escrita. Ao meu orientador, Frank Michael Forger, por sempre compartilhar seus conhecimentos cient´ıficos de maneira t~aogenerosa, e por expor sua vis~aosobre a Ci^encia sempre de maneira franca, entusiasmada e, certamente, inspiradora. Agrade¸coimensamente por ter me a- presentado a uma ´area t~aorica e fascinante dentro da An´alise Funcional, e por ter tido a liberdade de estudar quaisquer t´opicos que me interessassem para, somente ent~ao,decidir o tema da tese. Esta abordagem heterodoxa constituiu-se numa experi^encia muito enrique- cedora, para mim. Ao Severino Toscano do R. Melo, por seu grande altru´ısmo, pelas in´umeras conversas, sobre Matem´atica ou n~ao,e pelos incont´aveis ensinamentos que tanto contribu´ıram para a minha forma¸c~aode Matem´atico, como aluno, orientando e, agora, colaborador. E´ realmente im- poss´ıvel quantificar o aprendizado que obtive durante todos esses anos. Ao professor Paulo Domingos Cordaro, por poder ter assistido como ouvinte o excelente curso \Espa¸cos Localmente Convexos e Aplica¸c~oes", ministrado em 2017, e pelos esclareci- mentos acerca de operadores fortemente el´ıpticos. Ao Eric Ossami Endo, por sempre me incentivar a expor o meu trabalho, pelos convites a tantos semin´arios e eventos e pelas v´arias aulas de Mec^anica Estat´ıstica Cl´assica. Ao Lucas Affonso, por sempre me mostrar alguma aplica¸c~ao(ou poss´ıvel aplica¸c~ao!)interes- sante de An´alise Funcional `aMec^anica Estat´ıstica Qu^antica. Resumo Palavras-chave: ´algebras de Lie, grupos de Lie, representa¸c~oesfortemente cont´ınuas, exponencia¸c~ao,espa¸coslocalmente convexos, limites projetivos, limites inversos, vetores anal´ıticos projetivos, ´algebras localmente convexas, ∗-´algebras localmente convexas, ´alge- bras localmente C∗, ´algebras de Arens-Michael, ∗-´algebras de Arens-Michael, ´algebras de von Neumann, ´algebras GB∗, automorfismos, ∗-automorfismos, deriva¸c~oes, ∗-deriva¸c~oes, operadores pseudo-diferenciais. Condi¸c~oesnecess´arias e suficientes para a exponencia¸c~aode ´algebras de Lie reais de dimens~aofinita de operadores lineares sobre espa¸coslocalmente convexos completos Haus- dorff s~aoobtidas, com foco no caso equicont´ınuo - em particular, condi¸c~oes necess´arias para a exponencia¸c~aocom respeito a grupos de Lie compactos s~aoestabelecidas. Aplica¸c~oespara ´algebras localmente convexas completas s~aodadas, com uma aten¸c~aoespecial para ´algebras localmente C∗. A defini¸c~aode vetor anal´ıtico projetivo ´eintroduzida, possuindo um papel importante em alguns dos teoremas de exponencia¸c~aoe na caracteriza¸c~aodos geradores de uma certa classe de grupos a um par^ametro fortemente cont´ınuos. i Abstract Keywords: Lie algebras, Lie groups, strongly continuous representations, exponenti- ation, locally convex spaces, projective limits, inverse limits, projective analytic vectors, locally convex algebras, locally convex ∗-algebras, locally C∗-algebras, pro-C∗-algebras, LMC∗-algebras, Arens-Michael algebras, Arens-Michael ∗-algebras, von Neumann algebras, GB∗-algebras, automorphisms, ∗-automorphisms, derivations, ∗-derivations, pseudodiffer- ential operators. Necessary and sufficient conditions for the exponentiation of finite-dimensional real Lie algebras of linear operators on complete Hausdorff locally convex spaces are obtained, focused on the equicontinuous case - in particular, necessary conditions for exponentiation to compact Lie groups are established. Applications to complete locally convex algebras, with special attention to locally C∗-algebras, are given. The definition of a projective analytic vector is given, playing an important role in some of the exponentiation theorems and in the characterization of the generators of a certain class of strongly continuous one- parameter groups. iii Contents Introduction vii 1 Preliminaries 1 1.1 One-Parameter Semigroups and Groups . 3 1.2 Lie Group Representations and Infinitesimal Generators . 8 1.2.1 The G˚arding Subspace . 12 1.2.2 The Space of Smooth Vectors is Left Invariant by the Generators . 16 1.2.3 Lie Algebra Representations Induced by Group Representations . 18 1.2.4 Group Invariance and Cores . 19 1.3 Dissipative and Conservative Operators . 27 1.4 The Kernel Invariance Property (KIP), Projective Analytic Vectors . 30 1.5 Some Estimates Involving Lie Algebras . 45 1.6 Extending Continuous Linear Maps . 49 1.7 Projective Limits . 50 2 Group Invariance and Exponentiation 57 Constructing a Group Invariant Domain . 57 Exponentiation Theorems . 86 The First Exponentiation Theorems . 86 Strongly Elliptic Operators - Sufficient Conditions for Exponentiation . 97 Strongly Elliptic Operators - Necessary Conditions for Exponentiation . 116 Exponentiation in Locally Convex Spaces - Characterization . 120 3 Some Applications to Locally Convex Algebras 123 Definitions, Examples and a Few Structure Theorems . 123 Exponentiation of Complete Locally Convex Algebras . 133 v vi Contents Exponentiation of Locally C∗-Algebras . 136 Bibliography 139 Introduction A Physics Point of View: Some Motivations1 With the advent of Quantum Theory, the theory of Lie groups and their representa- tions has assumed an important role in Physics, mainly to mathematically incorporate the notion of symmetry. Some highlights that could be mentioned are, already in the 1930's, the development of the theory of compact Lie groups and their representations, by Hermann Weyl, a typical application being the consequences of rotational symmetry in atomic spectroscopy, and the classification of relativistic elementary particles in terms of irreducible unitary representations of the Poincar´egroup, by Eugene Wigner. Another historical landmark was the \eightfold way", by Gell'Mann and Ne'eman, in the 1960's, to classify hadrons (strongly interacting particles) in terms of weight diagrams of the group SU(3). In the early stages of Lie group representation theory, studies dealt almost exclusively with unitary representations on Hilbert spaces as the state spaces of quantum systems. However, Quantum Mechanics had already exposed with clarity a phenomenon which is present even in Classical Mechanics, but had rarely been treated there in an explicit man- ner: the duality between the state space and the algebra of observables of a system, which leads, in the treatment of temporal evolution, to the distinction between the \Schr¨odinger picture" (with time-dependent states and static observables) and the \Heisenberg picture" (with static states and time-dependent observables). The transition from Quantum Me- chanics to Quantum Field Theory strongly suggests that it is, at the very least, more convenient to perform this temporal evolution - or, more generally, the dynamics - on the observables, and not on the states, since the observables, and not the states, are the ones that allow localization in regions of space-time. Hence, besides unitary representations on 1The author would like to thank his advisor, Frank Michael Forger, for the big help in the writing process of this section of the Introduction. His great knowledge of Physics contributed in an essential way to the final form of the text. vii viii Introduction Hilbert spaces one should also investigate representations by automorphisms on algebras of observables. The question that naturally arises is then: which kind of algebras must one use? A first answer to this question may be found in Algebraic Quantum Field Theory, initiated in 1964 by Haag and Kastler, which revolves around a central concept that is absent in Quantum Mechanics: locality. This is incorporated by demanding the existence of a net of local C∗-algebras A(U) associated to an adequate (suficiently large) family of bounded open sets U in space-time satisfying the property that, if U ⊂ V , then A(U) ⊂ A(V ). The total C∗-algebra of this net is defined by [ A := A(U); U where the closure indicates the C∗-completion of the algebra. Relativistic invariance of the theory is implemented by a representation α: P −! Aut(A) of the Poincar´egroup P by ∗-automorphisms of A which is compatible with this net: α(a; Λ)[A(U)] = A(ΛU + a): Locality means that A(U) and A(V ) commute when the two regions U and V are spatially separated [62]. However, since the beginning of this
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