Von Neumann algebras, affiliated operators and representations of the Heisenberg relation

88 Pages·2017·3.01 MB·English

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