Graph Algebra s This page intentionally left blank http://dx.doi.org/10.1090/cbms/103 Conference Boar d o f the Mathematical Science s CBMS Regional Conference Series in Mathematics Number 10 3 Graph Algebra s Iain Raebur n Published fo r th e Conference Boar d of the Mathematica l Science s by the v^^c^ America n Mathematica l Societ y Providence, Rhod e Islan d with support fro m th e National Scienc e Foundatio n NSF-CBMS Regiona l Researc h Conferenc e o n Grap h Algebras : Operator Algebra s W e Ca n See , hel d a t th e Universit y o f Iowa , May 31-Jun e 4 , 200 4 Partially supporte d b y th e Nationa l Scienc e Foundatio n Research partiall y supporte d b y the Australia n Researc h Counci l through th e AR C Centr e fo r Comple x Dynami c System s an d Contro l at th e Universit y o f Newcastle . 2000 Mathematics Subject Classification. Primar y 46L05 ; Secondary 46L08 , 46L35 , 46L55, 46L80 , 22D35 . For additiona l informatio n an d update s o n thi s book , visi t www.ams.org/bookpages/cbms-103 Library o f Congres s Cataloging-in-Publicatio n Dat a Raeburn, Iain , 1949- Graph algebra s / Iai n Raeburn . p. cm . — (CBMS regiona l conferenc e serie s i n mathematics, ISS N 0160-764 2 ; no. 103 ) Includes bibliographica l reference s an d index. ISBN 0-8218-3660- 9 (alk . paper) 1. Algebra—Graphi c methods—Congresses . 2 . Grap h theory—Congresses . I . Title . II. Regiona l conferenc e serie s i n mathematics ; no. 103 . QA1.R33 no . 103 [QA219] 510 s—dc22 [512] 200504120 6 Copying an d reprinting . Individua l reader s o f thi s publication , an d nonprofi t librarie s acting fo r them, ar e permitted t o make fai r us e of the material, suc h a s to copy a chapter fo r use in teachin g o r research . Permissio n i s granted t o quot e brie f passage s fro m thi s publicatio n i n reviews, provide d th e customary acknowledgmen t o f the source i s given. Republication, systemati c copying , o r multiple reproduction o f any material i n this publicatio n is permitte d onl y unde r licens e fro m th e America n Mathematica l Society . Request s fo r suc h permission shoul d b e addressed t o the Acquisitions Department , America n Mathematica l Society , 201 Charle s Street , Providence , Rhod e Islan d 02904-2294 , USA . Requests ca n also b e made b y e-mail t o reprint-permissionOams.org . © 200 5 by the American Mathematica l Society . Al l rights reserved . The America n Mathematica l Societ y retain s al l rights except thos e grante d t o the United State s Government . Printed i n the United State s o f America. @ Th e paper use d i n this boo k i s acid-free an d falls withi n the guidelines established t o ensure permanenc e an d durability. Visit th e AMS hom e pag e at http: //www. ams. org/ 10 9 8 7 6 5 4 3 2 1 1 0 09 08 07 06 0 5 Contents Preface vi i Introduction 1 Chapter 1 . Directe d graph s an d Cuntz-Kriege r familie s 5 Chapter 2 . Uniquenes s theorem s fo r graph algebra s 1 5 Chapter 3 . Proof s o f the uniquenes s theorem s 2 5 Chapter 4 . Simplicit y an d idea l structure 3 3 Chapter 5 . Arbitrar y graph s 4 1 Chapter 6 . Application s t o non-abelia n dualit y 5 1 Chapter 7 . /^-theor y o f graph algebra s 6 1 Chapter 8 . Cuntz-Pimsne r algebra s 7 1 Chapter 9 . Topologica l graph s 7 9 Chapter 10 . Higher-ran k graph s 8 9 Appendix A . Backgroun d materia l 9 9 A.l. Projection s an d partia l isometrie s 9 9 A.2. Matri x algebra s an d direc t sum s 10 2 Bibliography 10 5 Index 11 1 This page intentionally left blank Preface These note s wer e mainl y writte n whil e I wa s preparin g m y lecture s fo r th e CBMS-NSF conferenc e o n Graph Algebras: Operator Algebras We Can See, whic h was hel d a t th e Universit y o f Iow a fro m 3 0 May to 4 June 2004 . Th e te n chapter s roughly correspon d t o the ten lectures , though fo r logica l reasons they appea r her e in a slightl y differen t order . I a m ver y gratefu l t o thos e wh o organise d th e conference , thos e wh o cam e t o the conference , an d thos e wh o hav e helpe d m e try t o ge t th e glitche s ou t o f thes e notes. I n particular , I thank : • Pau l Muhly, David Pask and Mark Tomforde, wh o did a marvellous job of organising the conference: th e informa l atmospher e wa s great, an d every - thing ra n smoothly . (Well , everythin g whic h didn' t g o through Chicag o airport.) Pau l especiall y di d a n enormous amoun t o f work o n the details , always i n his inimitably cheerfu l way . Thanks , Paul . • Natha n Brownlowe , Tyron e Crisp , James Foster , Danie l Go w an d Rishn i Ratnam, wh o helpe d m e organis e lectur e note s fo r a n honour s cours e I gave i n 2002 , which wer e my starting poin t fo r thes e notes . • Astri d a n Huef , Marcel o Laca , Pau l Muhly , Aida n Sims , Mar k Tomford e and Trent Yeend, who provided m e with (sometime s embarrassingly long ) lists o f corrections o n parts o f the notes . Th e remainin g mistake s ar e no t Dana Williams ' fault 1. Iain Raebur n Newcastle November 200 4 See [114 , pag e xiv]. This page intentionally left blank Bibliography S. Adji , M . Laca , M . Nilse n an d I . Raeburn , Crossed products by semigroups of endomor- phisms and the Toeplitz algebras of ordered groups, Proc . Amer . Math . Soc . 12 2 (1994) , 1133-1141. S. Allen , D . Pas k an d A . Sims , A dual graph construction for higher-rank graphs, and K-theory for finite 2-graphs, Proc . Amer . Math . Soc , t o appear ; arXiv.math.OA/0402126 . 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Math . 6 (2000) , 307-324 . B. Blackadar , A'-Theor y fo r Operato r Algebras , MSR I publications , vol . 5 , Springer-Verlag , New York , 1986 ; second edition , Cambridg e Univ . Press , 1998 . O. Bratteli , Inductive limits of finite-dimensional C*-algebras, Trans . Amer . Math . Soc . 171 (1972) , 195-234 . C. Consan i an d M . Marcolli , Spectral triples from Mumford curves, Internat . Math . Res . Not. 3 6 (2003) , 1945-1972 . T. Cris p and D . Gow , Contractible subgraphs and Morita equivalence of graph C* -algebras, Proc. Amer . Math . Soc , t o appear ; arXiv.math.OA/0404542 . J. Cuntz , Simple C* -algebras generated by isometries, Comm . Math . Phys . 5 7 (1977) , 173 - 185. J. Cuntz , A class of C*-algebras and topological Markov chains II : Reducible chains and the Ext-functor for C*-algebras, Invent . Math . 6 3 (1981) , 25-40 . J. Cunt z an d W . Krieger , A class of C* -algebras and topological Markov chains, Invent . Math. 5 6 (1980) , 251-268 . R.E. Curto , P.S . Muhl y an d D.P . Williams , Cross products of strongly Morita equivalent C*-algebras, Proc . Amer . Math . So c 9 0 (1984) , 528-530 . K.R. Davidson , C*-Algebra s b y Example , Field s Inst . Monographs , vol . 6 , Amer . Math . Soc, Providence , 1996 . V. Deaconu , Groupoids associated with endomorphisms, Trans . Amer . Math . Soc . 34 7 (1995), 1779-1786 . V. Deaconu, Generalized Cuntz-Krieger algebras, Pro c Amer . Math . Soc 12 4 (1996) , 3427- 3435. V. Deacon u an d P.S . Muhly , C*-algebras associated with branched coverings, Proc . Amer . Math. Soc .