Selected Title s i n Thi s Serie s 30 Robert o Longo , Editor , Mathematica l physic s i n mathematics an d physics : Quantu m and operato r algebrai c aspects , 200 1 29 Teres a Fari a an d Pedr o Freitas , Editors , Topic s i n functiona l differentia l an d difference equations , 200 1 28 Davi d R . McDonal d an d Stephe n R . E . Turner , Editors , Analysi s o f communication networks : Cal l centres , traffi c an d performance , 200 0 27 Shu i Fen g an d Ann a T . Lawniczak , Editors , Hydrodynami c limit s an d relate d topics , 2000 26 Nea l Madras , Editor , Monte . Carlo methods , 200 0 25 A . G . Ramm , P . N . Shivakumar , an d A . V . Strauss , Editors , Operato r theor y an d its applications , 200 0 24 Edwar d Bierstone , Bori s Khesin , Askol d Khovanskii , an d Jerrol d E . Marsden , Editors, Th e Arnoldfest : Proceeding s o f a conferenc e i n honou r o f V. I . Arnol d fo r hi s sixtieth birthday , 199 9 23 Myrn a H . Wooders , Editor , Topic s i n mathematica l economic s an d gam e theory . Essay s in hono r o f Robert J . Aumann , 199 9 22 Graciel a Chichilnisky , Editor , Topolog y an d markets , 199 9 21 Shigu i Ruan , Gai l S . K . Wolkowicz , an d Jianhon g Wu , Editors , Differentia l equations wit h application s t o biology , 199 8 20 Pete r A . Fillmor e an d Jame s A . Mingo , Editors , Operato r algebra s an d thei r applications II , 199 8 19 Willia m G . Dwyer , Stephe n Halperin , Richar d Kane , Stanle y O . Kochman , Mark E . Mahowald , an d Pau l S . Selic k (Editor-in-Chief) , Editors , Stabl e an d unstable homotopy , 199 8 18 Pano s M . Pardalo s an d Henr y Wolkowicz , Editors , Topic s i n semidefinit e an d interior-point methods , 199 8 17 Joachi m J . R . Cunt z an d Masou d Khalkhali , Editors , Cycli c cohomolog y an d noncommutative geometry , 199 7 16 Victo r P . Snaith , Editor , Algebrai c K-theory , 199 7 15 Stephe n P . Braham , Jac k D . Gegenberg , an d Rober t J . McKellar , Editors , Sixt h Canadian conferenc e o n genera l relativit y an d relativisti c astrophysics , 199 7 14 Moura d E . H . Ismail , Davi d R . Masson , an d Miza n Rahman , Editors , Specia l functions, g-serie s an d relate d topics , 199 7 13 Pete r A . Fillmor e an d Jame s A . Mingo , Editors , Operato r algebra s an d thei r applications, 199 7 12 Dan-Virgi l Voiculescu , Editor , Fre e probability theory , 199 7 11 Collee n D . Cutle r an d Danie l T . Kaplan , Editors , Nonlinea r dynamic s an d tim e series: Buildin g a bridg e betwee n th e natura l an d statistica l sciences , 199 7 10 Jerrol d E . Marsden , Georg e W . Patrick , an d Willia m F . Shadwick , Editors , Integration algorithm s an d classica l mechanics , 199 6 9 W . H . Kliemann , W . F . Langford , an d N . S . Namachchivaya , Editors , Nonlinea r dynamics an d stochasti c mechanics , 199 6 8 Larr y M . Bate s an d Davi d L . Rod , Editors , Conservativ e system s an d quantu m chaos, 199 6 7 Willia m F . Shadwick , Perinkula m Sambamurth y Krishnaprasad , an d Tudo r Stefan Ratiu , Editors , Mechanic s day , 199 6 6 Ann a T . Lawnicza k an d Raymon d Kapral , Editors , Patter n formatio n an d lattic e gas automata, 199 6 5 Joh n Chadam , Marti n Golubitsky , Willia m Langford , an d Bria n Wetton , Editors, Patter n formation : Symmetr y method s an d applications , 199 6 (Continued in the back of this publication) This page intentionally left blank Mathematical Physic s i n Mathematics an d Physic s This page intentionally left blank http://dx.doi.org/10.1090/fic/030 FIELDS INSTITUT E COMMUNICATIONS THE FIELD S INSTITUT E FO R RESEARC H I N MATHEMATICA L SCIENCE S Mathematical Physic s i n Mathematics an d Physic s Quantum an d Operato r Algebraic Aspect s Roberto Long o Editor American Mathematica l Societ y Providence, Rhod e Islan d The Field s Institut e for Researc h i n Mathematical Science s The Field s Institut e i s named i n honou r o f the Canadia n mathematicia n Joh n Charle s Fields (1863-1932) . Field s wa s a visionar y wh o receive d man y honour s fo r hi s scientifi c work, includin g election to the Royal Society o f Canada i n 190 9 and to the Royal Society o f London i n 1913 . Amon g othe r accomplishment s i n th e servic e o f the internationa l math - ematics community , Field s wa s responsibl e fo r establishin g th e world' s mos t prestigiou s prize fo r mathematic s research—th e Field s Medal . The Field s Institute fo r Researc h i n Mathematical Science s i s supported b y grants fro m the Ontari o Ministr y o f Education an d Trainin g and the Natural Science s and Engineerin g Research Counci l o f Canada . Th e Institut e i s sponsore d b y McMaste r University , th e University o f Toronto, th e Universit y o f Waterloo, an d Yor k University , an d ha s affiliate d universities i n Ontari o an d acros s Canada . 2000 Mathematics Subject Classification. Primar y 81-06 , 46-06 ; Secondary 81T05 , 47L90 . Library o f Congres s Cataloging-in-Publicatio n Dat a Mathematical physic s i n mathematic s an d physic s : quantu m an d operato r algebrai c aspect s / Roberto Longo , editor . p. cm . — (Field s Institute communications , ISS N 1069-526 5 ; v, 30 ) Includes bibliographica l references . ISBN 0-8218-2814- 2 (acid-fre e paper ) 1. Operato r algebras—Congresses . 2 . Quantu m theory—Congresses . 3 . Mathematica l physics—Congresses. I . Longo , Roberto. II . Series . QC20.7.O65M37 200 1 530.12—dc21 200104598 9 Copying an d reprinting . Materia l i n thi s boo k ma y b e reproduce d b y an y mean s fo r edu - cational an d scientifi c purpose s withou t fe e o r permissio n wit h th e exceptio n o f reproductio n b y services that collec t fee s fo r deliver y o f documents an d provide d tha t th e customar y acknowledg - ment o f the sourc e i s given . Thi s consen t doe s no t exten d t o othe r kind s o f copyin g fo r genera l distribution, fo r advertisin g o r promotiona l purposes , o r fo r resale . Request s fo r permissio n fo r commercial us e o f materia l shoul d b e addresse d t o th e Assistan t t o th e Publisher , America n Mathematical Society , P . O. Bo x 6248 , Providence , Rhod e Islan d 02940-6248 . Request s ca n als o be mad e b y e-mai l t o reprint-permissionOams.org . Excluded fro m thes e provision s i s material i n articles fo r whic h the autho r hold s copyright. I n such cases , requests fo r permissio n to us e or reprint shoul d b e addressed directl y to the author(s) . (Copyright ownershi p i s indicated i n the notic e i n the lowe r right-hand corne r o f the firs t pag e o f each article. ) © 200 1 by the America n Mathematica l Society . Al l rights reserved . The America n Mathematica l Societ y retain s al l right s except thos e grante d t o th e Unite d State s Government . Printed i n the Unite d State s o f America . @ Th e pape r use d i n this boo k i s acid-free an d fall s withi n th e guideline s established t o ensur e permanenc e an d durability . This publicatio n wa s prepared b y The Field s Institute . Visit th e AM S hom e pag e a t URL : http://www.ams.org / 10 9 8 7 6 5 4 3 2 1 0 6 0 5 04 0 3 02 0 1 This volume contains the proceedings o f the conference "Mathematica l Physic s in Mathematic s an d Physics . Quantu m an d Operato r Algebrai c Aspects " hel d i n Siena o n Jun e 20-24 , 2000 . W e dedicated thi s conferenc e t o Sergi o Dopliche r an d John E . Robert s o n th e occasio n o f thei r sixtiet h birthda y wit h admiratio n an d friendship. C. D'Antoni, W . Bergmann , F. Fidaleo , D. Guido , T. Isola , C . Pinzari , R. Longo , L . Zsid o This page intentionally left blank Contents Preface xii i Programme o f the Conferenc e x v Photograph - Participant s t o the Conferenc e xvi i Letters xi x An Applicatio n o f the DR-Dualit y Theor y fo r Compac t Group s to Endomorphis m Categorie s o f C*-Algebras wit h Nontrivia l Cente r 1 HELLMUT BAUMGARTE L AN D FERNAND O LLED O Modular Invariant s an d Subfactor s 1 1 JENS BOCKENHAUE R AN D DAVI D E .