Lecture Notes ni Mathematics Edited yb ~, Dold dna .B nnamkcE 486 ~erban &lit&rtS naD Voiculescu Representations of AF-Algebras dna of the Group U )oo( r galreV-regnirpS Berlin. Heidelberg-NewYork 5791 Authors Dr. Serban-Valentin &lit&rtS Dr. Dan-Virgil Voiculescu Academie de la Republique Socialiste de Roumanie Institut de Math@matique Calea Grivitei 12 Bucuresti 21 Roumania Library of Congress Cataloging in Publication Data Stratila, Serban-Valentin~ 1943 - Representations of iF-al~ebras and of the 6roup (Lecture notes in mathematics ; 486) Bibliography: p. Includes indexes. i. Operator algebras. 2. Representations of alge- bras. 3. Locally compact groups. 4. Representations of groups. I. Voiculescu~ Dan-~-irgil, 1949- joint authoz II. Title. III. Series: Lecture notes in mathematics (Berlin); 486. QA3~ no. 486 QA326 510'.8s 512'.55 7~-26896 AMS Subject Classifications (1970): 22D10, 22D25, 46 L05, 46L10 ISBN 3-540-07403-1 Springer-Verlag Berlin (cid:12)9 Heidelberg (cid:12)9 NewYork ISBN 0-387-07403-1 Springer-Verlag New York (cid:12)9 Heidelberg (cid:12)9 Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photo- copying machine or similar means, and storage in data banks. Under w 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. ~ by Springer-Verlag Berlin - Heidelberg 1975 Printed in Germany Offsetdruck: Julius Beltz, Hemsbach/Bergstr. INTRODUCTION Unitary representations of the group of all unitary opera- tors on an infinite dimensional Hilbert space endowed with the StTong-operator topology have been studied by I.E.Segsl (30) in connection with quantum physics . In2~ A.A.Kirillov classified all irreducible unitary representations of the group of those unl- tary operators which are congruent to the identity operator modulo compact operators , endowed with the norm-topology . Also , in 2~ the representation problem for the unitary group U(oo) , together with the assertion that U(OO) is not a type I group , is mentio- ned . The group U(oo) , well known to topologists , is in a cer- tain sense a smallest ~finite dimensional unitary group , being for instance a dense subgroup of the "classical" Banach-Lie groups of unitary operators associated to the Schatten -von Neumann classes of compact operators (~8 S) . Also , the restriction of representations from U(n+~) to U(n) has several nice features which make the study of the representations of U(~) somewhat easier than that of the analogous groups SU(~) , 0(oo) , SO(~) , Sp(~) . ~.hT study of factor representations of the non locally compact group U(OO) required some associated C ~- algebra . The C*- algebra we associated to a direct limit of compact separable groups , G = aril G n , has the property that its factor repre- VI sentations correspond either to factor representations of Gee , or to factor representations of some G n and , since the distinc- tion is easy between these two classes , it is of effective use . This C*- algebra is an AF - algebra , i.e. a direct limit of finite-dimensional C~- subalgebras . AF - algebras , introdu- ced and studied by O.Bratteli (~) , are a generalization of UHF - algebras . For the UHF - algebra of the canonical anticommutation relations of mathematical physics there is the general method of L.Garding and A.Wightman (12S) for studying factor representations and , in particular , the cross-product construction which yields factor representations in standard form . So we had to give an extension of this method to AF - algebras (Chapter I) . For U(~) this amounts to a certain desintegration of the representations with respect to a commutative C - algebra , the spectrum of which is an ~nfinite analog of the set of indices for the Gelfand - Zeit- lin basis (37) (cid:12)9 For U(oO) in this frame-work , a complete classification of the primitive ideals of the associated O*- alge- bra , in terms of a upper signature and a lower signature , is possible (Chapter III). Simple examples of irreducible represen- tations for each primitive ideal are the direct limits of irredu- cible representations of the U(n)'s , but there are many other irreducible representations (cid:12)9 Using the methods of Chapter I , we study (Chapter IV) certain class of factor representations of U(oo) which restricted to the U(n)'s contain only irreducible representations in anti- v s~etric tensors . This yields in particular an 4nfinity of non- equivalent type III factor representations , the modular group in the sense of Tcmita's theory (~32) with respect to a certain cyclic and separating vector having a natural group interpretation. Analogous results are to be expected for other types of tensors . The study of certain infinite tensor products (Chapter V) gives rise to a class of type II~ factor representations . As in the classical theory for U(n) , the ccmmutant is generated by a representation of a permutation group . In fact it is the regular representation of the ~nfinite prmutation group S(oo) which generates the hyperfimite type II~ factor . Other examples of type lloo factor representations are given in w 2 of Chapter V Type II~ factor representations of U(oo) were studied in (E3@,E35 ) and the results of the present work were announced in ( 38 Concluding , from the point of view of this approach , the representation problem for U(oo) seems to be of the same kind as that of the infinite anticommutation relations , though "combinatoriall~' more complicated . Of course , a more group - theoretical approach to the representations of U(~) would be of much imterest . Thamks are due to our colleague Dr. H.Moscovici for drawing our attention on E2~S and for useful discussions . The authors would like to express their gratitude to Mrs. lV Sanda Str~til~ for her kind help in typing the manuscript . The group U(~) is the direct limit of the unitary groups U(~) c U(2) c ... c U(n) c ... , endowed with the direct limit topology . Let H be a complex separable Hilbert space and enl an orthonormal basis . Then U(oo) can be realized as the group of unitary operators V on H such that Ve n = e n excepting only a finite number of indices n . Similarly , we consider GL(oo) the direct limit of the GL(n) ' s . By U&(~o) we denote the group of unitaries V on H such that V - I be nuclear , endowed with the topology derived from the metric d(V',V") = Tr(IV' - V" I ) . Also , by U(H) and GL(H) we denote all unitary , respectively all invertible , operators on the Hilbert space H . As usual , wo - topology means weak-operator topology and so - topology means strong-operator topology. Since it might be useful for the reader to have at hand certain classical facts concerning the irreducible representations of U(n), especially in view of Chapters IV and V, there is an Appendix about these representations. vMI The bibliography listed at the end contains, besides references to works directly used, also references to works we felt related to our subject. We apologize for possible omissions. Bucharest, March 12 th 1975. The Authors. CONTENTS CHAPTER I . On the structure of AF - al~ebras and their representations ........................... I w I . Diagonalization of AF - algebras ............... 3 w 2 . Ideals in AF - algebras ........................ 20 w 3 (cid:12)9 Some representations of AF - algebras .......... 31 CHAPTER II . Th e C*- algebra associated to a direct limit of compact ~ ......................... 75 w I . The L - algebra associated to a direct limit of compact groups .............................. 78 w 2 . The AF - algebra associated to a direct limit of compact groups and its diagonalisation ..... 62 CHAPTER III. The primitive ideals of A(U(oo)) .......... 81 w I . The primitive spectrum of A(U(| )) ............. 81 w 2 . Direct limits of irreducible representations ... 39 CHAPTER IV . Type III factor ,rep,resentations of U(oo) in antis vmmetric tensors .................. 79 CHAPTER V . Some type IIco factor ,rePresentations of U(o0 ) ................................. (cid:12)9 721 w 1 , Infinite tensor product representations ...... ,. 721 w 2 , Other type IIoo factor representations ...... ,., 146 APPENDIX : Irreducible ,representati0n ~ of U(n) ..... ,. 155 NOTATION INDEX ...................................... , ~ 160 SUBJECT INDEX ..................................... , .... 164 BIBLIOGRAPHY ........................................ o~ 166 CHAPTER I ON THE STRUCTURE OF AF - ALGEBRAS AND THEIR REPRESENTATIONS The uniformly hyperfinite C*- algebras (UHF - algebras) , which appeared in connection with some problems of theoretical physics , were extensively studied , important results concerning their structure and their representations being obtained by J. Gl~mm (15) and R. Powers (Z4) . They are a particular case of approximately finite dimensional C ~- algebras (AF - algebras) considered by O.Bratteli ( i ) , who also extended to this more general situation some of the results of J. Gl~mm and R. Powers . Our approach to the representation problem of the unitary group U(~) required some other developments , also well known for the UH~ - algebra of canonical anticommutation relations . Chapter I is an exposition of the results we have obtained in this direction , treated in the general context of AF - algebras. We shall use the books of J. Di~nier (~ 6 , T ) as refe- rences for the concepts and results of operator algebras . If MT , M 2 , ... are subsets of the C*- algebra A , then we shall denote by < M~ , M 2 , ...> or the smallest C - subalgebra of A containing ~_~ M n and by n l.m.(M~ , M 2 , ... ) (reap. c.l.m.(M~ , M 2 , -.. )) 2 the linear manifold (resp. the closed linear manifold) spanned by ~_~ M n . Also , for any subset M of A , we shall denote n by M' the commutant of M in A : M' = {xE A ; xy = yx (~) y ~ M} . A maximal abelian subal~ebra (abreviated m.a.s.a.) of a C*- algebra A is an abelian C ~- subalgebra C of A such that C' = C . A conditional expectation of a C - algebra A with respect to a C*- subalgebra B in A is a linear mapping P : A ~B such that : 2) li)x(Pll ~ IIxll for all x ~ A ; 3) P(x) ~> 0 for all x ~ A , x ~ 0 ; #) P(x)~P(x) ~ P(x*x) for all x e A ; 5) P(yxz) = yP(x)z for all x (cid:12)9 A , y,z ~ B . Obviously , a conditional expectation of A with respect to B is a (linear) projection of norm one of A onto B . Conversely , J. Tomiyama (33) has proved that any projection of norm one of A onto B is a conditional expectation . In what follows we shall use the result of J. Tomiyama only in order to avoid some rather tedious verifications . An approximately finite dimensional C ~- algebra (abrevia- ted AF - algebra) is a C - algebra A such that there exists an ascending sequence l&n} n >Io of finite dimensional C ~- sub- algebras in A with