Non-Commutative Analysis

562 Pages·2017·7.67 MB·English

by  Palle Jorgensen

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Published by
World Scientific Publishing Co. Pte. Ltd.
5 Toh Tuck Link, Singapore 596224
USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601
UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
Library of Congress Cataloging-in-Publication Data
Names: Jørgensen, Palle E. T., 1947– | Tian, Feng (Mathematics professor)
Title: Non-commutative analysis / by Palle Jorgensen (The University of Iowa, USA),
Feng Tian (Hampton University, USA).
Description: New Jersey : World Scientific, 2017. |
Includes bibliographical references and index.
Identifiers: LCCN 2016047613| ISBN 9789813202115 (hardcover : alk. paper) |
ISBN 9789813202122 (pbk : alk. paper)
Subjects: LCSH: Functional analysis. | Hilbert space. | Mathematical physics. | Stochastic processes.
Classification: LCC QA320 .J638 2017 | DDC 515/.7--dc23
LC record available at https://lccn.loc.gov/2016047613
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
Copyright © 2017 by World Scientific Publishing Co. Pte. Ltd.
All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means,
electronic or mechanical, including photocopying, recording or any information storage and retrieval
system now known or to be invented, without written permission from the publisher.
For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance
Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy
is not required from the publisher.
Printed in Singapore
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Dedicated to the memory of William B. Arveson (see [MJD+15])
(22 November 1934 – 15 November 2011).
“What goes around has come around, and today quantum informa-
tion theory (QIT) has led us back into a finite-dimensional context.
Completelypositivemapsonmatrixalgebrasaretheobjectsthatare
dual to quantum channels; in fact, the study of quantum channels
reducestothestudyofcompletelypositivemapsofmatrixalgebras
that preserve the unit. This is an area that is still undergoing vig-
orous development in efforts to understand entanglement, entropy,
and channel-capacity in QIT.”
— William B. Arveson (written around 2009.)1,2
1Forasummaryofthetermsusedintheabovequote(QIT,entanglement,andrelated),
seeAppendixB.
2Thecoverimageillustratesageometricaspectofnon-commutativity. ItisfromFigure10.3
(pg. 371) which deals with certain implications for Riemann surfaces of non-commutativity
forunboundedoperators.
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Foreword
by Wayne Polyzou, Professor of Physics, University of Iowa
IfonehadtosynthesizethenoveltyofPhysicsoftheXXcentury
with a single magic word, one possibility would be “Noncommuta-
tivity”.
— Sergio Doplicher and Roberto Longo. From [CCG+04]
Progress in science, engineering and mathematics comes fast and it often
requires a significant effort to keep up with the advances in other fields that
impact applications. Functional analysis (especially operators in Hilbert space,
unitaryrepresentationsofLiegroups,andspectraltheory)isonedisciplinethat
impacts my physics research. Bringing my students up to speed with the sub-
ject facilitates their ability to efficiently perform research, however the typical
curriculum in functional analysis courses is not directed to practitioners whose
primary objective is applications. This is also reflected in the many excellent
available texts on the subject, which primarily focus on the mathematics, and
are directed at students aspiring to a career in mathematics.
IhavebeenfortunatetohavePalleJorgensenasacolleague. Heparticipates
inaweeklyjointmathematicalphysicsseminarthatisattendedbyfacultyand
students from both departments. It provides a forum to address questions re-
lated to the role of mathematics in physics research. Professor Jorgensen has
a healthy appreciation of applications of functional analysis; in these seminars
he has been at the center of discussions on a diverse range of applications in-
volving wavelets, reflection positivity, path integrals, entanglement, financial
mathematics, and algebraic field theory.
A number of the mathematically inclined students in my department have
benefited from taking the functional analysis course taught by Professor Jor-
gensen. These students are motivated to enroll in his class because the course
material includes a significant discussion of applications of functional analysis
to subjects that interest them.
This book is based on the course that Professor Jorgensen teaches on func-
tional analysis. It fills in a gap that is not addressed by the many excellent
available texts on functional analysis, by using applications to motivate basic
vii
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viii NON-COMMUTATIVE ANALYSIS
results in functional analysis. The way that it uses applications makes the ma-
terial more accessible to students; particularly for students who will eventually
find careers in related disciplines. The book also points to additional reference
material for students who are motivated to learn more about a specific topic.
W. Polyzou
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Preface
There are already many books in Functional Analysis, so why another?
Themainreasonisthatwefeelthereisaneed: intheteachingatthebegin-
ning graduate level; more flexibility, more options for students and instructors
in pursuing new directions. And aiming for a book which will help students
with primary interests elsewhere to acquire a facility with tools of a functional
analyticflavor,sayinspectraltheoryforoperatorsinHilbertspace,incommu-
tative and non-commutative harmonic analysis, in PDE, in numerical analysis,
in stochastic processes, or in physics.
Thebook Overthedecades,FunctionalAnalysis,andthetheoryofoperators
in Hilbert space, have been enriched and inspired on account of demands from
neighboringfields,withinmathematics,harmonic analysis (waveletsandsignal
processing), numerical analysis (finite element methods, discretization), PDEs
(diffusionequations,scatteringtheory),representation theory;iterated function
systems (fractals, Julia sets, chaotic dynamical systems), ergodic theory, opera-
tor algebras, and many more. And neighboring areas, probability/statistics (for
example stochastic processes, It¯o and Malliavin calculus), physics (representa-
tion of Lie groups, quantum field theory), and spectral theory for Schrödinger
operators.
Thebookisbasedonacoursesequence(two-semesters313-314)taught,over
theyearsattheUniversityofIowa, bythefirst-namedauthor. Thestudentsin
the course made up a mix: some advanced undergraduates, but most of them,
firstorsecondyeargraduatestudents(frommath,aswellassomefromphysics
and stats.)
Wehavesubsequentlyexpandedthecoursenotestakenbythesecond-named
author: we completed several topics from the original notes, and we added a
numberofothers,sothatthebookisnowself-contained,anditcoversaunified
theme; and yet it stresses a multitude of applications. And it offers flexibility
for users.
A glance at the table of contents makes it clear that our aim, and choice
oftopics,isdifferentfromthatofmoretraditionalFunctionalAnalysiscourses.
This is deliberate. For example, in our choice of topics, we placed emphasis on
the use of Hilbert space techniques which are then again used in our treatment
of central themes of applied functional analysis.
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