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On X-Normed Spaces and Operator Theory on C 0 Over a Field with A ON X-NORMED SPACES AND OPERATOR THEORY ON c0 OVER A FIELD WITH A KRULL VALUATION OF ARBITRARY RANK by Angel Barría Comicheo A thesis submitted to the Faculty of Graduate Studies of The University of Manitoba in partial fulfillment of the requirements of the degree of Doctor of Philosophy Department of Mathematics University of Manitoba Winnipeg Copyright c 2018 by Angel Barría Comicheo Abstract Between 2013 and 2015 Aguayo et al. developed an operator theory on the space c0 of null sequences in the complex Levi-Civita field C by defining an inner product on c0 that induces the supremum norm on c0 and then studying compact and self- adjoint operators on c0, thus presenting a striking analogy between c0 over C and the Hilbert space `2 over C. In this thesis, we try to obtain these results in the most general case possible by considering a base field with a Krull valuation taking values in an arbitrary commutative group. This leads to the concept of X-normed spaces, which are spaces with norms taking values in a totally ordered set X not necessarily embedded in R. Two goals are considered in the thesis: (1) to present and contribute to a theory of X-normed spaces, and (2) to develop an operator theory on c0 over a field with a Krull valuation of arbitrary rank. In order to meet the goal (1), a systematic study of valued fields, G-modules and X-normed spaces is conducted in order to satisfy the generality of the settings required. For the goal (2), we identify the major differences between normed spaces over fields of rank 1 and X-normed spaces over fields of higher rank; and we try to find the right conditions for which the techniques employed in the rank-1 case can be used in the higher rank case. For (1) the author develops a new tool to work with transfinite induction simplifying the techniques employed in X-normed spaces, thus accomplishing a Generalized Baire Category Theorem that allows the proof of an Open Mapping theorem for X-normed spaces. Regarding (2), we show that an operator can be identified as i compact with adjoint by studying the behavior of the image of any base of c0. Although characterizations are obtained for some linear operators on c0, it is still unknown whether the spectral theorem holds for compact self-adjoint operators in the non-Archimedean case. ii Acknowledgements I want to thank my wife Marianela Oliva Contreras for her support and encourage- ment, and my parents and siblings for their confidence on me. Also I want to thank my thesis advisor Dr. Khodr Shamseddine who has been an invaluable source of wisdom and trust during my years as a doctoral student and that kindly helped me to overcome the difficulties that arose in this journey. I express my gratitude to the members of the Examiner Committee for their insights that helped me to improve this thesis. This thesis was supported by a doctoral scholarship for the author from the Formation of Advanced Human Capital Program of the Chilean National Commis- sion for Scientific and Technological Research (CONICYT-PFCHA/Doctorado en el Extranjero/72130551). To my wife. Contents Abstract i List of Tables v Introduction 1 List of contributions 5 List of symbols 9 1 Non-Archimedean valued fields 11 1.1 Ultrametric Spaces . 11 1.2 Non-Archimedean valued fields . 16 1.3 Completion of valued fields . 18 1.4 Ordered fields . 23 Formally real fields . 23 General Hahn fields and the Embedding theorem . 25 1.5 Hahn Fields and Levi-Civita fields . 27 1.6 Real-closed field extensions of R ..................... 30 1.7 Algebraic closure of valued fields and their completions . 32 1.8 General valuations and the higher rank case . 36 1.9 Catalog of fields . 41 ii 1.10 Classification of fields . 44 2 Banach spaces over fields with rank-1 valuations 46 2.1 Preliminaries . 46 2.2 Linear Operators . 52 2.3 Orthogonality . 57 2.4 Spaces of Countable type . 61 3 Normed spaces over fields with valuations of higher rank 67 3.1 Scaled Spaces . 68 3.2 Valued fields with valuations of higher rank . 72 3.3 G-modules . 75 3.4 Continuous linear operators and Dedekind completions . 79 3.4.1 Dedekind completion of an ordered set . 79 3.4.2 The Dedekind completion X# as a G-module . 82 3.4.3 Quotient spaces of X-normed spaces . 84 3.4.4 Coinitiality and Cofinality of G-modules . 89 3.4.5 Equivalence of norms in Lip(E, F ) and LipY (E, F ) ...... 92 3.4.6 Banach spaces and linear operators with finite rank . 93 3.4.7 Generalizing the Open Mapping Theorem . 95 3.5 Spaces of countable type and orthogonality . 97 4 Operator theory on c0 over a field of arbitrary rank 99 4.1 Inner product in c0 ............................ 100 4.2 Normal complement subspaces of c0 ................... 104 4.3 Operators on c0 admitting an adjoint operator . 109 4.4 Compact operators on c0 ......................... 112 4.5 Aspects of non-Archimedean Spectral Theory . 121 iii Diagonal Operators . 127 Conclusions . 132 5 Future work 134 5.1 Is there a spectral theorem for compact and self-adjoint operators on c0?..................................... 134 5.2 Characterization of Banach spaces with the complementation property 135 5.3 Is being of countable type an inheritable property for normed spaces? 136 5.4 Is the quotient space of a spherically complete normed space spheri- cally complete? . 136 5.5 Topological approach to Hilbert-like spaces . 137 Bibliography 143 iv List of Tables 1.1 Catalog of fields . 43 1.2 Classification of valued fields . 45 v Introduction In non-Archimedean Analysis, i.e. when we consider a non-Archimedean valued field as our base field to define normed spaces, we obtain new results that do not hold in classical Functional Analysis (i.e. when our base field is R or C), however we lose some other results, thus producing a whole new theory that sometimes demands a totally new intuition. In this new setting, the results that are possible to obtain depend greatly on the non-Archimedean valued field we choose to work with and also depend on the structure of the normed space that we consider. For example, let’s consider the concept of orthomodularity. Let K be a field with an involution a 7→ a∗. A vector space E over K with a Hermitian form (·, ·): E ×E → K (linear in the first variable and (x, y) = (y, x)∗ for all x, y ∈ E) is called orthomodular if the projection theorem holds: every subspace H of E satisfies: H = H⊥⊥ ⇒ E = H ⊕ H⊥. Notice that Hilbert spaces can be characterized in terms of orthomodularity. In fact, if E is an orthomodular space over K admitting an orthonormal sequence, then K = R or K = C and E is linearly homeomorphic to a Hilbert space [42]. In general, if a non-Archimedean valued field K has finite rank, then there is no infinite-dimensional Banach space over K that is orthomodular [30, 4.4.6]. However, if the non-Archimedean valued field K has infinite rank, then we can find an infinite- dimensional Banach space E over K that is orthomodular (where the norm satisfies ||x||2 = (x, x) for all x ∈ E) [30, 4.4.9]. In the first chapter of the thesis, a deep analysis of the metric and algebraic 1 2 structures of non-Archimedean valued fields is presented, providing among other things, descriptions of the most commonly used non-Archimedean valued fields. Ad- ditionally, the relationship between the order structures of ordered fields and their structure as valued fields is explained. The culmination of this introductory chapter is the classification of all the valued fields in a meaningful manner that is beneficial to non-experts and experts in the field alike. In the Chapters 2–4 of this thesis, we systematically study several aspects of non- Archimedean Functional Analysis in order to determine the conditions for which we can conceive compact and adjoint operators in a Banach space equipped with a Schauder basis in the most general settings possible. Since every Banach space (over a non-Archimedean valued field K) with a Schauder basis is linearly homeomorphic to the space c0 of null sequences in K ([32, 2.3.9]), we will focus on the study of operators defined on c0. As we will see, the properties of c0 vary depending on whether the valuation takes values in R or in an arbitrary commutative ordered group; those two cases will be referred to as the rank-1 case and the higher-rank case, respectively. In order to understand the impact of the choice of the valued field we will begin in Chapter 1 with an introduction to ultrametric spaces, where we will study the metric and topological properties of a non-Archimedean valued field. Then, a variety of non-Archimedean valued fields will be presented and their properties collected in an extensive catalog (1.9). We will review the basic structures of valued fields and will develop the tools that will allow us to fully classify them in 6 distinctive categories (1.10). Among other results (see List of contributions), we will prove that the order topology of any non-Archimedean ordered field coincides with the topology induced by their ‘natural’ valuation (1.8.9). Note that most of Chapter 1 (with the exception of a couple of new results) is a selection of results which have recently been published in a refereed paper [8] in Contemporary Mathematics of the American Mathematical 3 Society.
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