UUnniivveerrssiittyy ooff NNeeww HHaammppsshhiirree UUnniivveerrssiittyy ooff NNeeww HHaammppsshhiirree SScchhoollaarrss'' RReeppoossiittoorryy Doctoral Dissertations Student Scholarship Spring 2010 VVoonn NNeeuummaannnn aallggeebbrraass,, aaffiffilliiaatteedd ooppeerraattoorrss aanndd rreepprreesseennttaattiioonnss ooff tthhee HHeeiisseennbbeerrgg rreellaattiioonn Zhe Liu University of New Hampshire, Durham Follow this and additional works at: https://scholars.unh.edu/dissertation RReeccoommmmeennddeedd CCiittaattiioonn Liu, Zhe, "Von Neumann algebras, affiliated operators and representations of the Heisenberg relation" (2010). Doctoral Dissertations. 601. https://scholars.unh.edu/dissertation/601 This Dissertation is brought to you for free and open access by the Student Scholarship at University of New Hampshire Scholars' Repository. It has been accepted for inclusion in Doctoral Dissertations by an authorized administrator of University of New Hampshire Scholars' Repository. For more information, please contact [email protected]. VON NEUMANN ALGEBRAS, AFFILIATED OPERATORS AND REPRESENTATIONS OF THE HEISENBERG RELATION BY ZHE LIU B.S., Hebei Normal University, China, 2003 DISSERTATION Submitted to the University of New Hampshire in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy in Mathematics May 2010 UMI Number: 3470108 All rights reserved INFORMATION TO ALL USERS The quality of this reproduction is dependent upon the quality of the copy submitted. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if material had to be removed, a note will indicate the deletion. UMT Dissertation Publishing UMI 3470108 Copyright 2010 by ProQuest LLC. All rights reserved. This edition of the work is protected against unauthorized copying under Title 17, United States Code. ProQuest LLC 789 East Eisenhower Parkway P.O. Box 1346 Ann Arbor, Ml 48106-1346 This dissertation has been examined and approved. FU^lu**! IS. KJOLAJLAJTYL Dissertation Director, Richard V. Kadison Professor of Mathematics Dissertation Director, Liming Ge Professor of Mathematics Don W. Hadwin Professor of Math-ematics, Rita Hibschweiler Prof^j§sor/6| Mathematics Eric Nordgren Professor of Mathematics ^ Junhao Shen Associate Professor of Mathematics '+( o2 I 9LO \Q Date DEDICATION To my Parents. in ACKNOWLEDGEMENTS It is a pleasure to express my gratitude to a number of people from whom I have received invaluable assistance in the form of mathematical inspiration, mathematical wisdom, encouragement and sustaining kindness. Among these people, are Professor Liming Ge, my advisor and teacher, who gave me the opportunity to come to the United States and who showed me many fruitful paths in the gardens of mathematics, Professor Don Hadwin and Professor Eric Nordgren from whom I learned mathemat ical subjects important to me, Professor Junhao Shen who was always ready to share his mathematical insight and ingenuity with me, Professor Maria Basterra, my minor advisor, and Professor Rita Hibschweiler for her constant helpfulness and friendly encouragement. I especially want to thank Professor Richard Kadison, who suggested the topic and the central problem of this thesis, for his patient guidance and kindness throughout my work on this project. Without his guidance, this thesis would not have been written. Finally, I would like to thank the institutes and the faculties, students, and staffs of the mathematics departments at which I worked on the project of this thesis; they were all unstintingly kind and helpful to me. The first of these is the University of New Hampshire at which I studied with my advisor Professor Ge, the second was Louisiana State University where I worked with Professor Kadison while he was visiting there during the spring semester of 2008, and the last has been the University iv of Pennsylvania, Professor Kadison's home university, where I have been a guest for the academic year 2009 - 2010 while completing this first phase of our project. TABLE OF CONTENTS DEDICATION Hi ACKNOWLEDGEMENTS iv ABSTRACT viii INTRODUCTION 1 1 VON NEUMANN ALGEBRAS 6 1.1 Preliminaries 6 1.2 Factors 10 1.3 The trace 14 2 UNBOUNDED OPERATORS 17 2.1 Definitions and facts 17 2.2 Spectral theory 20 2.3 Polar decomposition 24 3 OPERATORS AFFILIATED WITH FINITE VON NEUMANN ALGEBRAS . 28 3.1 Finite von Neumann algebras 28 3.2 The algebra of affiliated operators 31 vi 4 REPRESENTATIONS OF THE HEISENBERG RELATION . .. 40 4.1 In B(H) 41 4.2 The classic representation 47 4.3 In &/{M) with M a factor of type Hi 72 BIBLIOGRAPHY 76 vn ABSTRACT VON NEUMANN ALGEBRAS, AFFILIATED OPERATORS AND REPRESENTATIONS OF THE HEISENBERG RELATION by Zhe Liu University of New Hampshire, May, 2010 Von Neumann algebras are self-adjoint, strong-operator closed subalgebras (contain ing the identity operator) of the algebra of all bounded operators on a Hilbert space. Factors are von Neumann algebras whose centers consist of scalar multiples of the identity operator. In this thesis, we study unbounded operators affiliated with finite von Neumann algebras, in particular, factors of Type Hi. Such unbounded opera tors permit all the formal algebraic manipulations used by the founders of quantum mechanics in their mathematical formulation, and surprisingly, they form an alge bra. The operators affiliated with an infinite von Neumann algebra never form such an algebra. The Heisenberg commutation relation, QP — PQ = —ihl, is the most fundamental relation of quantum mechanics. Heisenberg's encoding of the ad-hoc quantum rules in this simple relation embodies the characteristic indeterminacy and uncertainty of quantum theory. Representations of the Heisenberg relation in var ious mathematical structures are discussed. In particular, we answer the question - whether the Heisenberg relation can be realized with unbounded operators in the algebra of operators affiliated with a factor of type Hi. viii