SPECTRAL THEORY FOR ORDERED SPACES A Thesis Submitted in Partial Fulfilment of the Requirements for the Degree of Master of Science by G. PRIYANGA to the School of Mathematical Sciences National Institute of Science Education and Research Bhubaneswar 16-05-2017 DECLARATION I hereby declare that I am the sole author of this thesis, submitted in par- tial fulfillment of the requirements for a postgraduate degree from the National Institute of Science Education and Research (NISER), Bhubaneswar. I authorize NISER to lend this thesis to other institutions or individuals for the purpose of scholarly research. Signature of the Student Date: 16th May, 2017 The thesis work reported in the thesis entitled Spectral Theory for Ordered Spaces was carried out under my supervision, in the school of Mathematical Sciences at NISER, Bhubaneswar, India. Signature of the thesis supervisor School: Mathematical Sciences Date: 16th May, 2017 ii ACKNOWLEDGEMENTS I would like to express my deepest gratitude to my thesis supervisor Dr. Anil Karn for his invaluable support, guidance and patience during the two year project period. I am particularly thankful to him for nurturing my interest in functional analysis and operator algebras. Without this project and his excellent courses, I might have never appreciated analysis so much. I express my warm gratitude to my teachers at NISER and summer project supervisors, for teaching me all the mathematics that I used in this thesis work. Ahugethankstomyparentsfortheircontinuoussupportandencouragement.I am also indebted to my classmates who request the authorities and extend the deadline for report submission, each time! iii ABSTRACT This thesis presents an order theoretic study of spectral theory, developed by Alfsen and Shultz, extending the commutative spectral theorem to its non- commutative version. In this project, we build a spectral theory and functional cal- culus for order unit spaces and explore how this generalised spectral theory fits in the scheme of commutative case (function spaces) as well as the non-commutative case (Jordan Algebras). Amotivatingexampleforthistheorycomesfromthespectraltheoremfor monotone complete CR(X) spaces, which we study first. The goal is to extend the order theoretic ideas involved here, to a more general setting, namely to or- dered spaces. So, we begin with an order unit space A, which is in separating order and norm duality with a base norm space V. Here, we define maps called compressions, which give rise to projective units in A. Using projective units and projective faces, we develop various notions like comparability, orthogonality, com- patibility and also obtain certain lattice structures, under an assumption called standing hypothesis.Then,weconstructanabstractnotionofrangeprojection in A, which leads to a spectral decomposition result and functional calculus, for spaces satisfying a spectral duality condition. In the last section, we focus onto the spectral theorem for JBW-algebras and understand how the spectral theory works in non-commutative framework. Along the way, we also see concrete example of the order theoretic constructions developed above. iv Contents Introduction 1 1 Spectral Theory for Function Spaces 5 1.1 Range Projections . 7 1.2 Spectral Theorem . 11 2 Order Structure 15 2.1 Ordered Vector Spaces . 15 2.1.1 OrderUnitSpaces ....................... 17 2.1.2 BaseNormSpaces ....................... 22 2.2 Duality between Order Unit Space and Base Norm Space . 24 3 Spectral Theory for Ordered Spaces 30 3.1 Projections . 31 3.1.1 TangentSpacesandSemi-exposedFaces . 31 3.1.2 Smooth Projections . 35 3.2 Compressions . 41 3.2.1 ProjectiveUnitsandProjectiveFaces . 47 3.3 Relation between Compressions . 54 3.3.1 Comparability . 54 3.3.2 Orthogonality . 55 3.3.3 Compatibility . 57 v CONTENTS 3.4 The Lattice of Compressions . 65 3.4.1 The lattice of compressions when A = V ⇤ ........... 81 3.5 Range Projections . 87 3.6 SpacesinSpectralDuality . 91 3.7 Spectral Theorem . 107 3.7.1 Functional Calculus . 111 4 Spectral Theory for Jordan Algebras 114 4.1 OrderUnitAlgebras .......................... 115 4.1.1 CharacterisingOrderUnitAlgebras . 122 4.1.2 Spectral Result . 127 4.2 JordanAlgberas............................. 129 4.2.1 The Continuous Functional Calculus . 136 4.2.2 Triple Product . 140 4.2.3 Projections and Compressions in JB-algebras. 145 4.2.4 Orthogonality . 150 4.2.5 Commutativity . 151 4.3 JBWalgebras ............................. 158 4.3.1 Range Projections . 166 4.3.2 Spectral Resolutions . 172 5 Appendix 176 vi Introduction The theory of von-Neumann algebras can be seen as a non-commutative generali- sation of integration theory. More precisely, as the lebesgue integral of a function is approximated by a sequence of simple functions, every self-adjoint element of a von-Neumann algebra can be written as a limit of finite linear sum over orthogonal projections. This idea is known as Spectral Decomposition Theory.Theobjective of this thesis work is to understand the order theoretic aspect of spectral theory, in the commutative case as well as non-commutative case. The generalised spectral theory for ordered spaces, presented in this thesis, was originally developed by Erik M. Alfsen and Frederic W. Shultz, in the late 20th century. Motivated by structures in quantum mechanics, Alfsen and Shultz were interested in characterising the state space of operator algebras, particularly Jordan algebras and C⇤-algebras. As part of this larger project, they established aspectraltheoryandfunctionalcalculusfororderunitspaces,whichgeneralised the corresponding results for von-Neumann algebras and JBW-algebras. This generalised theory seems to have applications in quantum mechanics. For instance, in the standard algebra model of quantum mechanical measurement, observables are represented by self-adjoint elements of a von Neumann algebra M and the states are represented by the elements in the normal state space K of M. Under the new generalisation, the basic concepts of this theory can be studied in a broader order-theoretic context: by representing states as elements in the distinguished base K of a base norm space V and observables by elements in the 1 CONTENTS order unit space A = V ⇤. However, in my thesis work, the interest is not in state space characterisation or investigating its applications in physics. The focus of this project is developing ageneralspectraltheoryforsuitableorderedspacesandunderstandinghowthis extends the commutative spectral theorem to its non-commutative version. In this project, we mainly work with an order unit space (A, A+, e)whichisin separating order and norm duality with a base norm space (V,V+,K).Oninves- tigating the spectral theorem for monotone complete CR(X) spaces, which is one of the primary motivating example for this theory, one learns that projections and their orthogonality is the fundamental object involved in developing the spectral result. In order to construct these notions in a more general setting, we define maps, called compressions, on the order unit space A. The image of the order unit e under these compressions (known as projective unit) behave like projections in A. We then explore various properties of the projective units and projective faces and develop notions of comparability, orthogonality and compatibility between compressions. We further find that under an assumption called standing hypothesis,thesetof compressions on A form a lattice. Using the properties of this lattice structure, we then construct order theoretic tools, called range projections,analogoustothose in CR(X) spaces. Finally, we specialise to spaces in spectral duality,wherethese range projections lead to a spectral decomposition theory and functional calculus on A. So, we obtain an abstract spectral theorem for order unit spaces. But how do these order theoretic constructions look in particular cases? A commutative example of this theory is seen in the case of monotone complete CR(X) spaces. An- other concrete example of this theory appears in the non-commutative framework of Jordan algebras. So, the last part of this thesis is devoted to understanding 2 CONTENTS the spectral theory for JBW-algebras. Here, we study about JB-algebras through order unit algebras, see examples of concrete compression (Up)andlearnhowthe spectral theory works in a non-commutative setting. In summary, we develop an order theoretic model of spectral theory for order unit spaces and explore how this generalised spectral theory fits in the scheme of commutative case (function spaces) as well as the non-commutative case (Jordan Algebras). We now briefly describe the contents of each chapter. Chapter 1 presents the order theoretic study of spectral theorem for monotone complete CR(X) spaces (continuous functions on a compact Hausdor↵space X). The spectral result and range projections constructed in this chapter are again used in Chapter 4, while developing the spectral theory for JBW-algebras. This material is mostly based on Alfsen and Shultz’s first book [2] (chapter 1). Chapter 2 introduces order structure on vector spaces. Here, we explore vector order, cones, order unit space and base norm space. This is followed by a discussion on duality: dual pair, order duality, norm duality and then we show that order unit spaces and base norm spaces are dual to each other. Chapter 3 focusses on the construction and development of the main spectral theorem. It begins with definitions and results on tangent spaces, semi-exposed faces and smooth projections in cones. This