Universita Degli Studi Di Catania
Università degli studi di CataniaDottorato di Ricerca in Matematica ed InformaticaSettore Scientifico Disciplinare MAT/05Representable functionals andderivations on Banach quasi*-algebrasTesi di Dottorato di:Maria Stella AdamoIl tutor:Prof. Camillo TrapaniIl Coordinatore:Prof. Giovanni RussoCiclo XXXIAnno conseguimento titolo: 2019
To my (grand)mum Nina
AcknowledgmentsFirst and foremost, I wish to express my sincere and heartfelt gratitudeto my supervisor prof. Camillo Trapani, for leading me through the worldof Operator Algebras, in particular locally convex quasi *-algebras, and forbeing a strong support along these years of my career. I got in touch withthe multicoloured world of Operator Algebras through the course of Noncommutative Analysis, delivered by him at first year of Master Degree. Forme, Operator Algebras constitute a huge and fascinating field of research. Iwish to express my gratitude to him for passing me his strong enthusiasm.This enthusiasm remains nowadays.My sincere gratitude goes to prof. Maria Fragoulopoulou for hosting metwice in Athens (Greece) at the Department of Mathematics of Nationaland Kapodistrian University of Athens, in Winter 2017 and Spring/Summer2018. This thesis has been written mostly during my visit in Athens. Manythanks for being helpful, not only in the professional role, and for sharingwith me many advices about math and the wonderful Greek culture. I willalways keep with me those memories of time spent in Athens.I’m so grateful also to prof. Pere Ara for hosting me at AutonomousUniversity of Barcelona and for giving me the opportunity of discoveringmany other interesting branches of Operator Algebras. Many thanks for thepatience and the kindness shown during my visit.I spent very pleasant moments abroad in Athens and in Barcelona. Itwas a big opportunity for me, both personally and for my career, to get thechance to visit these places and the mentioned universities. I really hope tovisit again prof. Maria Fragoulopoulou at NKUA in Greece and prof. PereAra at UAB in Spain in the near future.I’d like to warmly thank everyone that contributed somehow for the sakeof me and this thesis, supporting, advising and indicating me a way to finda solution to non-trivial problems, especially in those bad moments when Iwas ”blind”. Whoever you are, thank you very much!Among them, I’m specially grateful to my uncle Piero, for letting mediscover the existence of PhD when I was just a teenager and for encouragingI
me along my career. I will always remember the lunch in September 2009,when he got his PhD.While writing this thesis and being abroad for research and conferencepurposes, I felt a strong and warm support that I would not have expectedto feel. For this reason, I’m sincerely grateful to Yoh, as a mathematician, afriend and a partner of life.Last but not least, my immense gratitude goes to my parents, my motherFrancesca and my father Matteo, for joying with me of my success and forstrongly supporting me in those difficult moments that may occur in everyone’s life. Without them, this thesis wouldn’t have been possible.A loving thought goes to my grandmother Nina, for giving me all thelove she was capable of and for being always proud of me. I know you aresomewhere watching over me and never leaving me alone.To all of you, many thanks.II
AbstractLocally convex quasi *-algebras, in particular Banach quasi *-algebras,have been deeply investigated by many mathematicians in the last decadesin order to describe quantum physical phenomena (see [7, 8, 9, 15, 21, 35,46, 47, 61, 68, 70]).Banach quasi *-algebras constitute the framework of this thesis. Theyform a special family of locally convex quasi *-algebras, whose topology isgenerated by a single norm, instead of a separating family of seminorms (see,for instance, [14, 19, 20, 22]).The first part of the work concerns the study of representable functionalsand their properties. The analysis is carried through the key notions of fullyrepresentability and *-semisimplicity, appeared in the literature in [9, 14, 20,38]. In the case of Banach quasi *-algebras, these notions are equivalent upto a certain positivity condition. This is shown in [3], by proving first thatevery sesquilinear form associated to a representable functional is everywheredefined and continuous. In particular, Hilbert quasi *-algebras are alwaysfully representable.The aforementioned result about sesquilinear forms allows one to selectwell behaved Banach quasi *-algebras where it makes sense to reconsiderin a new framework classical problems that are relevant in applications (see[13, 25, 44, 49, 58, 69, 72, 73, 74]). One of them is certainly that of derivationsand of the related automorphisms groups (for instance see [4, 6, 12, 17, 26]).Definitions of course must be adapted to the new situation and for this reasonwe introduce weak *-derivations and weak automorphisms in [4]. We studyconditions for a weak *-derivation to be the generator of such a group. Aninfinitesimal generator of a continuous one-parameter group of uniformlybounded weak *-automorphisms is shown to be closed and to have certainproperties on its spectrum, whereas, to acquire such a group starting witha certain closed * derivation, extra regularity conditions on its domain arerequired. These results are then applied to a concrete example of weak *derivations, like inner qu*-derivation occurring in physics.Another way to study representations of a Banach quasi *-algebra is toIII
construct new objects starting from a finite number of them, like tensorproducts (see [5, 36, 37, 41, 43, 52, 53, 59]). In [2] we construct the tensorproduct of two Banach quasi *-algebras in order to obtain again a Banachquasi *-algebra tensor product. We are interested in studying their capacity to preserve properties of their factors concerning representations, likethe aforementioned full representability and *-semisimplicity. It has beenshown that a fully representable (resp. *-semisimple) tensor product Banach quasi *-algebra passes its properties of representability to its factors.About the viceversa, it is true if only the pre-completion is considered, i.e.if the factors are fully representable (resp. *-semisimple), then the tensorproduct pre-completion normed quasi *-algebra is fully representable (resp.*-semisimple).Several examples are investigated from the point of view of Banach quasi*-algebras.IV
ContentsIntroduction11 Brief review on quasi *-algebras and their1.1 Partial *-algebras of operators . . . . . . .1.2 Concrete examples of quasi *-algebras . . .1.3 A special case: Banach quasi *-algebras . .1.3.1 Bounded elements . . . . . . . . . .1.4 *-Representations of quasi *-algebras . . .1.4.1 Full representability . . . . . . . .1.4.2 *-Semisimplicity . . . . . . . . . . .representations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2 The continuity of representable functionals2.1 Full representability vs *-semisimplicity . . . . . . . . .2.2 Representable functionals on a Hilbert quasi *-algebra .2.2.1 Partial multiplication and bounded elements . .2.2.2 Positive elements and representable functionals2.2.3 Integrable Hilbert quasi *-algebras . . . . . . .2.3 Intertwining operators and representable functionals . .2.3.1 Two peculiar examples . . . . . . . . . . . . . .3 Unbounded derivations and *-automorphism groups3.1 Densely defined derivations . . . . . . . . . . . . . . . .3.2 Extension of a qu*-derivation . . . . . . . . . . . . . .3.2.1 Weak multiplication and weak topologies . . . .3.2.2 Inner qu*-derivations . . . . . . . . . . . . . . .3.2.3 Derivations as infinitesimal generators . . . . .3.3 Integrability of weak *-derivations . . . . . . . . . . . .3.4 Examples and applications . . . . . . . . . . . . . . . .3.4.1 Commutators for h self-adjoint . . . . . . . . .3.4.2 A physical example: quantum lattice systems 467
4 Tensor products of Banach quasi *-algebras4.1 Algebraic construction . . . . . . . . . . . . . . . . .4.2 *-Admissible topologies . . . . . . . . . . . . . . . . .4.3 Topological tensor product . . . . . . . . . . . . . . .4.4 Representations of tensor products quasi *-algebras .4.5 Examples of tensor product Banach quasi *-algebras .Conclusions.69697276778587A Brief overview on Banach *-algebras89A.1 Banach *-algebras and C*-algebras . . . . . . . . . . . . . . . 89A.2 Spectrum of a Banach *-algebra . . . . . . . . . . . . . . . . . 91A.3 *-Representations and positive functionals . . . . . . . . . . . 91Bibliography93VI
IntroductionIn the last century, many mathematicians put their effort in describingquantum systems with rigorous mathematical models. Among them, in acelebrated paper [40] about algebraic formulation of Quantum theories, R.Haag and D. Kastler employed C*-algebras as suitable tools in order to describe physical phenomena. Despite this, there are quantum models notfitting in this formulation. For instance, in certain spin lattice system withlong range interactions, the thermodynamical limit does not converge in anyC*-topology (see [9, 21, 50, 51]).In order to give a rigourous mathematical formulation of this kind ofproblems G. Lassner introduced and studied locally convex quasi*-algebras in[50, 51]. The simplest example is given by the completion of a locally convex*-algebra with separately continuous multiplication [9, 15, 35]. Clearly, inthis case the multiplication is not necessarily everywhere defined.For what concerns representations of locally convex quasi *-algebras,bounded operators are not enough, despite they have nice properties and theycan be handled without any trouble. For this aim, the family of L† (D, H)is employed. It is made of closable operators with the same domain D, i.e.a dense subspace of a Hilbert space H, such that the domain of the Hilbertian adjoint contains D. This family of unbounded operators can be madeinto a partial *-algebra by defining a partial product between operators. Thelatter were introduced by J.-P. Antoine and K. Karwowski in [8] and thenextensively studied by many authors (see [9]).This thesis aim to present results about continuity of representable functionals, i.e. those functionals that admit a GNS-like construction, and theirapplications to derivations arising as infinitesimal generators of *-automorphisms groups and topological tensor products in the special context of Banach quasi *-algebras.Representations constitute an important tool to look at abstract structures (see [3, 9, 16, 17, 23, 24, 38, 71, 65]). In the case of C*-algebras,*-representations have a deep link with positive functionals, because thesecan be regarded as ”blocks” used in the process of building *-representations,1
namely the GNS -construction. The lack of an everywhere defined multiplication makes it impossible to deal with positive functionals. However thenotion of representable functional, introduced in [65] plays a similar role inthis context. A representable functional is positive on the core *-algebra andsome appropriate conditions guarantee the existence of a GNS-like triple, asin the classical case.In spite of this reasonable behaviour, complete results on the continuityof representable functionals are still missing and no example of a discontinuous representable functional is known so far, whereas examples of continuosfunctionals that are not representable do exist. (see, for instance, [9, 38]).Prior to investigation, Chapter 1 is devoted to background material needed for the ongoing work in the thesis. Chapter 2 concerns representable functionals on Banach quasi *-algebras and some related concepts like full representability and *-semisimplicity, devoting a special attention to the case ofHilbert quasi *-algebras, i.e. completions of Hilbert algebras under the normdefined by their inner product. The investigation of the problem concerningcontinuity starts looking at sesquilinear forms associated to representable andcontinuous functionals. These forms turn out to be everywhere defined andbounded, hence the notion of full representability reduces to the sufficiencyof the family of these functionals, in the sense that they distinguish pointsin the Banach quasi *-algebra. In the case of a *-semisimple Banach quasi*-algebra, the family of representable and continuous functionals is shown tobe always sufficient, thus *-semisimple Banach quasi *-algebras are alwaysfully representable. The converse is true under the condition of positivity,satisfied in many examples.Having a representable functional at hand, it is possible to associate toit a second sesquilinear form defined through the GNS-representation of thefunctional (see [3, 65]). This form is everywhere defined and, in the casethe functional is also continuous, it coincides with the closure of the abovesesquilinear form defined through the funcitonal. This remark suggests thatthese sesquilinear forms might be useful to characterize continuity. Indeed,it has been shown that every representable functional is continuous if, andonly if, there exists another representable and continuous functional less orequal to the given representable functional. Nonetheless, the results gives noalgorithm to construct such a functional.Our investigation continues focusing on the case of Hilbert quasi *-algebras, that turn out to be fully representable. In this situation, representableand continuous functionals are in 1-1 correspondence with bounded and weakly positive elements of the Hilbert quasi *-algebra. The definition of weaklypositive element is indeed a generalization of the notion given in [38].Positivity plays an important role in studying the continuity for repre2
sentable functionals. Indeed, the existence of a continuous module function,i.e. a sort of generalization of the absolute value in the case of Hilbert quasi*-algebras, owning certain invariance properties, guarantees the continuity ofrepresentable functionals that are positive on the set of all weakly positive elements. The latter condition is difficult to verify though, hence we examine indetails the Hilbert space of square integrable functions, that is a Hilbert quasi*-algebra over continuous functions first and then over essentially boundedfunctions. In these examples, it is shown that every representable functionalis continuous.The previous chapters show the particular role of *-semisimple and fullyrepresentable Banach quasi *-algebras. These properties motivate the studyof specific problems usually treated in the context of C*-algebras, in particular those more relevant for applications such as derivations and the generationof groups of automorphisms (see [10, 11, 18]). For this reason in Chapter 3, weexamine derivations obtained as infinitesimal generators of automorphismsgroups. We employ sesquilinear forms in order to define what a derivation ona Banach quasi *-algebra is. In this framework, the main point is to definea suitable Leibnitz rule for the derivation, since a priori its image doesn’tbelong to the universal multipliers, i.e. those elements for which the left andright multiplication operators are everywhere defined.Our first step is to look at densely defined derivations on a Banach quasi *algebra (A, A0 ), starting our investigation from inner qu*-derivations, namelythose that can be written as δh (x) i[h, x] for x A0 and fixed h A. Inthe *-semisimple case, it is shown that every inner qu*-derivation is closable,independently by the nature of h A. The closure, in general, is not again aderivation in the classical sense, because the domain is not a quasi *-algebraover A0 . Therefore, we need to weaken the Leibnitz rule through the employment of sesquilinear forms, achieving a more general kind of derivation, i.e.a weak *-derivation.As for weak *-derivations, we need a suitable notion of *-automorphisms,namely weak *-automorphism, in order to extend the well known resultof Bratteli-Robinson about the 1-1 correspondence between certain closed*-derivations and continuous *-automorphism groups in a C*-algebra (see[25, 26]). In our case, in order to get a closed weak *-derivation as infinitesimal generator, we have to ask the group to be made of uniformly boundedweak *-automorphisms, condition automatically verified for C*-algebras. Onthe other hand, for a weak *-derivation to be the generator of a weak *automorphisms group as before, stronger conditions have to be required, asfor instance the domain should consist of bounded elements Ab . Despite that,these extra conditions are verified in some classical examples, thus appear tobe reasonable for our work.3
The last step consists of computing one parameter group generated byinner qu*-derivations and give a physical examples motivating our choiceto examine derivations in a more general context when the implementingelement is unbounded.In the last chapter, Chapter 4, we construct the tensor product Banachquasi *-algebra and we explore its properties in relation with its factors.There is few literature about tensor products of unbounded operator algebras,despite the wide applications of topological tensor products (see [36, 37, 41]).This construction aims to study representations of the factors through thetensor product.We first analyse the algebraic candidate for the tensor product Banachquasi *-algebra. A quasi *-algebra A over A0 can be regarded as a bimoduleover A0 . The problem is that two quasi *-algebras are bimodules over differentrings. In order to solve this problems, one might sum the rings and constructa bimodule structure of the direct sum, but this leads to a trivial tensorproduct, if one of the factor is unital. Then, we suppose the existence of anembedding between the *-algebras involved in the tensor product. In thisway, both the quasi *-algebras are bimodules over the same ring. Moreover,if we extend the scalars and compute the tensor product, what we obtain isthe same structure obtained constructing the tensor product on the smallest*-algebra.Having at our hands a notion of tensor product quasi *-algebra, the definition of topological tensor product of normed (resp. Banach) quasi *-algebrais given. We endow the tensor product quasi *-algebra with an admissiblenorm, for instance the injective or the projective norm, in order to get atensor product normed quasi *-algebra. The completion of the latter will befor us the tensor product Banach quasi *-algebra.At this point, the existence and the relation between representations of atensor product normed (resp. Banach) quasi *-algebra and those of the tensorfactors are explored. If the tensor product Banach quasi *-algebra possesses*-representations, hence representable functionals, then also the tensor factors do. Although, the converse is true if we consider the pre-completion, i.e.the tensor product normed quasi *-algebra possesses *-representations if thefactors Banach quasi *-algebras admit them.4
Chapter 1Brief review on quasi *-algebrasand their representations1.1Partial *-algebras of operatorsPartial *-algebras of unbounded operators play a relevant role in representation theory. We recall here the basic definitions and facts; for furtherdetails, see [8, 9].Let D be a dense subspace of a Hilbert space H[h· ·i]. Denote withL† (D, H) the set of all closable linear operators X : D H for whichD(X ) D, where X indicates the adjoint of X. In symbols,L† (D, H) : {X : D H : D(X ) D}.L† (D, H) is a complex vector space with respect to sum and scalar productdefined in the canonical way.In L† (D, H), it is possible to identify the following subspaceL† (D) {X : D D : D(X ) D, X D D}. If we define an involution as X 7 X † X Dand a partial multiplicationX Y : X † Y whenever Y D D(X † ) and X † D D(Y ) on L†
Locally convex quasi *-algebras, in particular Banach quasi *-algebras, . like tensor products (see [5, 36, 37, 41, 43, 52, 53, 59]). In [2] we construct the tensor product of two Banach quasi *-algebras in order to obtain again a Banach quasi *-algebra tensor
tra le Università degli Studi del Molise, Università degli Studi di Ferrara e Università della Tuscia ***** REGOLAMENTO GENERALE DI ORGANIZZAZIONE, AMMINISTRAZIONE, CONTABILITÀ E FINANZA ***** Sede Legale: Via Ravenna, 8 00161 Roma - Italia Tel: 39 06 4451707 - Fax: 39 06 44360433 amministrazione
7.4 Sizing Step-Up Transformers Analysis 100 7.4.1 Sizing Step-up transformers for Wind Farm plants without ESS 101 7.4.2 Sizing Step-up transformers for Wind Farm plants with ESS 104 7.5 Conclusion 108 References 109 . vii Nomenclature List of Abbreviat
UNIVERSITÀ DEGLI STUDI DI CAGLIARI DIPARTIMENTO DI PEDAGOGIA, PSICOLOGIA, FILOSOFIA VERBALE DELLA COMMISSIONE INCARICATA DELL'AFFIDAMENTO DEGLI INCARICHI DI INSEGNAMENTO V ACANTI PER L' A.A. 2013/14 SSD M-PSI/01, M-PED/01, M-PED/03 La Commissione, nominata dal Dipartimento di Pedagogia, Psicologia, Filosofia, è composta da: Pro f.
cagliari, 10/10/2014 . universitÀ degli studi di cagliari . dipartimento di sanita’ pubblica, medicina clinica e molecolare . scuola di specializzazione di medicina del lavoro . documento di valutazione dei rischi per la salute e la sicurezza degli studenti della facolta’ di medicina e chirurgia
UNIVERSITA’ DEGLI STUDI DI CAGLIARI DIPARTIMENTO DI STUDI STORICI GEOGRAFICI E ARTISTICI DOTTORATO DI RICERCA IN STORIA MODERNA E CONTEMPORANEA XXIII CICLO L’INFANZIA ABBANDONATA NELLA SARDEGNA MODERNA: IL PADRE D’ORFANI Settore scientifico disciplinare di afferenza: M-STO/2 – Storia moderna
Le Collane di Rhesis Quaderni camilleriani 3 Il cimento della traduzione Comitato Scientifico MASSIMO ARCANGELI (Università di Cagliari), ANTONIO ÁVILA MUÑOZ (Universidad de Málaga), LORENZO BLINI (Università degli Studi Internazionali di Roma), FRANCESCA BOARINI (Università di Cagliari), PAOLA CADEDDU (Università di
BIOTECNOLOGIE MOLECOLARI E INDUSTRIALI (CLASSE LM-8- D.M. 2004 n. 270 e successivi adeguamenti) Manifesto degli Studi a.a. 2016/2017 È istituito presso l’Università degli Studi di Perugia il Corso di Studio Magistrale in Biotecnologie Molecolar
SI V236 UNIVERSITÀ DEGLI STUDI DI PADOVA Selezione per l'ammissione ai Corsi di formazione per il conseguimento della specializzazione per le attività di
