Graph Algebra s

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Conference Boar d o f the Mathematica l 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 o f th e Mathematica l Science s

by th e v^^c^ America n Mathematica l Societ y

Providence, Rhod e Islan d with suppor t fro m th e

National Scienc e Foundatio n

http://dx.doi.org/10.1090/cbms/103

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 th e 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 ; Secondar y 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 . — (CBM S 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

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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 grap h algebra s 1 5

Chapter 3 . Proof s o f th e uniquenes s theorem s 2 5

Chapter 4 . Simplicit y an d idea l structur e 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

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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 Ma y t o 4 Jun e 2004 . Th e te n chapter s roughly correspon d t o th e te n lectures , thoug h fo r logica l reason s the y 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 tr y t o ge t th e glitche s ou t o f thes e notes. I n particular , I thank :

• Pau l Muhly , Davi d Pas k an d Mar k Tomforde , wh o did a marvellous job of organising th e 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 throug h Chicag o airport.) Pau l especiall y di d a n enormou s amoun t o f work o n th e details , always i n hi s inimitabl y cheerfu l way . Thanks , Paul .

• Natha n Brownlowe , Tyron e Crisp , Jame s 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 m y startin g 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 Tren t Yeend , who provide d m e with (sometime s embarrassingl y long ) lists o f corrections o n part s 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] .

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I. Raebur n an d D. R Williams , Morit a Equivalenc e an d Continuous-Trac e C* -Algebras, Math. Survey s an d Monographs , vol . 60 , Amer . Math . Soc , Providence , 1998 . J. Renault , The ideal structure of groupoid crossed product C* -algebras, J . Operato r Theor y 25 (1991) , 3-36 . J. Renault , Cuntz-like algebras, Operato r Theoretica l Method s (Timi§oara , 1998) , Thet a Foundation, Bucharest , 2000 , page s 371-386 . M.A. Rieffel , Applications of strong Morita equivalence to transformation group algebras, Operator Algebra s an d Applications , Proc . Symp . Pur e Math. , vol . 38 , Par t I , Amer . Math . Soc. Providence , 1982 , page s 299-310 . M.A. Rieffel , Proper actions of groups on C* -algebras, Mapping s o f Operato r Algebras , Progress i n Math. , vol . 84 , Birkhauser , Boston , 1988 , page s 141-182 . M.A. Rieffel , Integrable and proper actions on C* -algebras, and square-integrable represen-tations of groups, Expositione s Math . 2 2 (2004) , 1-53 . G. Robertso n an d T . Steger , C * -algebras arising from group actions on the boundary of a triangle building, Proc . Londo n Math . Soc . 7 2 (1996) , 613-637 . G. Robertso n an d T . Steger , Affine buildings, tiling systems and, higher-rank Cuntz-Krieger algebras, J . rein e angew . Math . 51 3 (1999) , 115-144 . G. Robertso n an d T . Steger , Asymptotic K-theory for groups acting on Ai buildings, Canad . J. Math . 5 3 (2001) , 809-833 . M. R0rdam, Classification of nuclear simple C * -algebras, Encyclopaedi a Math . Sci. , vol. 126 , Springer-Verlag, Berlin , 2002 , page s 1-145 . M. R0rdam , F . Larse n an d N.J . Laustsen , A n Introductio n t o iC-Theor y fo r C* -Algebras, London Math . Soc . Studen t Texts , vol . 49 , Cambridg e Univ . Press , 2000 . J. Rosenber g an d C . Schochet , The Kilnneth theorem and the universal coefficient theorem for Kasparov's generalized K-functor, Duk e Math . J . 5 5 (1987) , 431-474 . J. Schweizer , Crossed products by C* -correspondences and Cuntz-Pimsner algebras, C*-alg -ebras (Munster , 1999) , Springer-Verlag , Berlin , 2000 , page s 203-226 . J. Schweizer , Dilations of C* -correspondences and the simplicity of Cuntz-Pimsner algebras, J. Funct . Anal . 18 0 (2001) , 404-425 . A. Sims , Relative Cuntz-Krieger algebras of finitely aligned higher-rank graphs, Indian a Univ. Math . J. , t o appear ; arXiv.math.OA/0312152 . A. Sims , Gauge-invariant ideals in the C* -algebras of finitely aligned higher-rank graphs, preprint, arXiv.math.OA/0406592 . B. Solel , You can see the arrows in a quiver algebra, J . Austral . Math . Soc . 7 7 (2004) , 111-122. J.S. Spielberg , Free-product groups, Cuntz-Krieger algebras, and covariant maps, Internat . J. Math . 2 (1991) , 457-476 . J.S. Spielberg , A functorial approach to the C*-algebras of a graph, Internat . J . Math . 1 3 (2002), 245-277 . J. Stillwell , Classica l Topolog y an d Combinatoria l Grou p Theory , Graduat e Text s i n Math. , vol. 72 , Springer-Verlag , Ne w York , 1980 . W. Szymahski , Bimodules for Cuntz-Krieger algebras of infinite matrices, Bull . Austral . Math. Soc . 6 2 (2000) , 87-94 . W. Szymahski , Simplicity of Cuntz-Krieger algebras of infinite matrices, Pacifi c J . Math . 199 (2001) , 249-256 . W. Szymahski , The range of K-invariants for C* -algebras of infinite graphs, Indian a Univ . Math. J , 5 1 (2002) , 239-249 . W. Szymahski , General Cuntz-Krieger uniqueness theorem, Internat . J . Math . 1 3 (2002) , 549-555. W. Szymahski , On semiprojectivity of C* -algebras of directed graphs, Proc . Amer . Math . Soc. 13 0 (2002) , 1391-1399 . H. Takai , On a duality for crossed products of C* -algebras, J . Funct . Anal . 1 9 (1975) , 25-39 . M. Tomforde , A unified approach to Exel-Laca algebras and C* -algebras associated to graphs, J . Operato r Theor y 5 0 (2003) , 345-368 . M. Tomforde , Simplicity of ultragraph algebras, Indian a Univ . Math . J . 5 2 (2003) , 901-925 . M. Tomforde , The ordered Ko-group of a graph C*-algebra, C . R . Math . Rep . Acad . Sci . Canada 2 5 (2003) , 19-25 .

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Index

action, 15 , 5 1 proper, 5 9

adding a hea d t o a source , 1 8 adding a tai l t o a sink , 1 8 adjacency matrix , 6 adjoint able operator , 7 2 AF-algebra, 2 0 aperiodicity condition , 9 4 approximately finite-dimensional, 2 0

bootstrap class , 3 4 Bratteli diagram , 2 0

of a n AF-algebra , 2 0 breaking vertex , 4 7 Bunce-Deddens algebra , 9 7

C*-algebra o f a directe d graph , 1 3 O*-algebra o f a /e-graph , 9 3 C*-algebra o f a topologica l graph , 7 9 C*-correspondence, 7 2 category, 8 9 circle group , 1 5 coaction, 5 2 coaction identity , 5 2 coassociative, 5 2 Coburn's theorem , 12 , 2 3 codomain o f a morphism , 8 9 cofinal, 3 3 collapsible path , 4 2 compact operato r o n a Hilber t module , 7 2 composition o f morphisms , 8 9 comultiplication, 5 2 Condition (K) , 3 5 Condition (L) , 2 3

for a topologica l graph , 8 1 connected directe d graph , 5 6 continuity o f i\-theory , 6 3 core, 2 5 corner, 1 9 correspondence, 7 2

graph, 7 2 covariant representatio n

for a coaction , 5 3 for a n action , 5 1

covering, 5 6 crossed produc t

for a coaction , 5 4 for a n action , 5 1 reduced, 5 1

Cuntz algebra , 2 3 infinite, 4 1

Cuntz-Krieger algebr a of a {0 , l}-matrix, 1 7 of a directe d graph , 1 3

Cuntz-Krieger famil y for a /c-graph , 9 1 for a row-finit e graph , 6 for a n arbitrar y graph , 4 1 in a C*-algebra , 7 on Hilber t space , 6

Cuntz-Krieger relation , 7 at a vertex , 7

Cuntz-Krieger uniquenes s theorem , 1 6 for a topologica l graph , 8 1 for a n arbitrar y graph , 4 5

Cuntz-Pimsner algebra , 7 5 Cuntz-Pimsner covariant , 7 5 cycle, 11 , 1 6

in a topologica l graph , 8 1

degree map , 9 0 fo,52 desingularisation, 4 4 dichotomy, 34 , 45 , 9 7 direct limit , 6 3

of C*-algebras , 10 3 direct su m o f C* -algebras, 27 , 10 2 directed graph , 5 domain o f a morphism , 8 9 Drinen-Tomforde desingularisation , 4 4 dual action , 5 4 dual coaction , 5 2 dual graph , 1 7 dual group , 5 2 dual Pimsner-Voiculesc u sequence , 6 4 dual system , 5 2 duality

Imai-Takai, 5 4

112 INDEX

Katayama, 5 4 Takesaki-Takai, 5 2

dynamical system , 5 1

En, 9 £ * , 9 E^k, 2 8 E°°, 3 3 £^°° , 33 edge, 5 edge matrix , 1 8 Elliott's classificatio n theorem , 6 3 emit a n edge , 5 entry t o a cycle , 1 6 equivalent projections , 6 1 essential correspondence , 7 3 exact sequence , 6 4 Exel-Laca algebra , 5 0

F<k, 2 8 factorisation property , 9 0 Fell bundle , 5 3 final projection , 10 1 finite graphs , 6 finitely aligned , 9 5 fixed-point algebra , 2 5 free semigroupoi d algebra , 7 7 full corner , 1 9 functor, 8 9 functoriality o f iC-theory , 6 2 fundamental group , 5 6

gap projection , 4 8 gauge action , 1 5

on a /c-grap h algebra , 9 4 gauge-invariant ideal , 3 8 gauge-invariant uniquenes s theorem , 1 6

for a fc-graph, 9 4 for a Cuntz-Pimsne r algebra , 7 6 for a topologica l graph , 8 0

generalised fixed-poin t algebra , 5 9 grading, 5 3 graph, 5 graph algebr a

of a row-finit e graph , 1 3 of a n arbitrar y graph , 4 2

graph correspondence , 7 2 of a topologica l graph , 7 9

graph o f ran k k, 9 0 graph C* -algebra, 1 3 Gross-Tucker theorem , 5 5 group C* -algebra, 5 2 groupoid mode l

for C*(A) , 94 , 9 6 for C*(E), 31 , 3 9

head, 1 8 hereditary set , 3 4 higher-order dua l o f a graph , 1 8

higher-rank graph , 9 0 Hilbert bimodule , 7 2 Hilbert module , 7 1 homogeneous space , 5 7

identity morphism , 8 9 Imai-Takai dualit y theorem , 5 4 imprimitivity bimodule , 5 8 in-delay, 4 4 induced homomorphism s i n /^-theory , 6 2 infinite Cunt z algebra , 4 1 infinite pat h i n a fc-graph, 9 3 infinite receiver , 4 1 initial projection , 10 1 inner product , C*-algebra-valued , 7 1 integrated form , 5 1 interior tenso r product , 7 3 irrational rotatio n algebra , 9 7 isomorphic graphs , 5

K(X), 7 2 K0l 6 1 Ki, 6 2 Katayama dualit y theorem , 5 4 Katsura's uniquenes s theorem , 8 1 &-graph, 9 0 Kirchberg-Phillips theorem , 7 0 K-theory

of /c-grap h algebras , 9 7 of a direc t limit , 6 3 of a direc t sum , 63 , 6 7 of AF-algebras , 6 3 of grap h algebras , 6 9 of matri x algebras , 61 , 62 of th e compac t operators , 6 4

£(X) , 72 labelling, 53 , 55 , 6 4 lattice, 4 8 length o f a path , 9 locally convex , 9 5 loop, 5 It: lef t translation , 5 1

M(A), 1 8 matrix unit s

for K(H), 10 3 for a matri x algebra , 10 2

Morita equivalence , 5 8 morphism i n a category , 8 9 multiplier algebra , 1 8 mutually orthogona l projections , 7

in a C*-algebra , 7 , 10 0

non-commutative sphere , 1 4 non-degenerate Cuntz-Kriege r family , 7 non-returning, 29 , 8 3 non-trivial saturate d hereditar y set , 3 5

OA, 1 7

INDEX 113

O n , 2 3

0oo ,41 O x , 7 5 object i n a category , 8 9 1-skeleton, 9 0 ordered KQ, 6 3 orthogonal projection , 9 9

partial isometry , 10 0 in a C*-algebra , 10 1

path category , 8 9 path o f degre e n i n a fc-graph, 9 0 path o f lengt h n , 9 7T X C/ , 5 1

TTI(E, w), 5 6

Pimsner-Voiculescu sequence , 6 4 preorder o n vertices , 3 3 Prim A, 4 8 primitive ideal , 4 8 primitive idea l space , 4 9 product syste m o f correspondences , 9 6 projection, 9 9 proper, 56 , 5 9 ( ^ T T ) ^ ) , 7 5

purely infinite , 3 4

quotient graph , 5 5

range ma p i n a graph , 5 receive a n edge , 5 reduced crosse d product , 5 1

by a homogeneou s space , 5 7 reduced walk , 5 6 regular covering , 5 6 regular representation , 51 , 54 relative ske w product , 5 7 restriction o f a coaction , 5 8 return path , 3 5 right-Hilbert bimodule , 7 2 row-finite graph , 6 rt: righ t translation , 5 1

saturated se t in a row-finit e graph , 3 4 in a n arbitrar y graph , 4 7

simple C* -algebra, 2 3 simple loop , 1 1 sink, 5 six-term exac t sequence , 6 4 skew-product graph , 55 , 6 4

relative, 5 7 small category , 8 9 source ma p i n a graph , 5 source vertex , 5 spanning tree , 5 7 spectral subspace , 5 3 stable isomorphism , 5 2

T Y , 7 3 tail, 1 8 Takesaki-Takai duality , 5 2 tensor algebra , 7 7 tensor power , 7 4 tensor produc t o f grap h algebras , 9 7 ®x,y, 72

Toeplitz algebra , 1 2 Toeplitz algebr a o f a correspondence , 7 3 Toeplitz representation , 7 3 Toeplitz-Cuntz-Krieger E-family , 7 4 topological graph , 7 9 topologically free , 8 1 transitive graph , 33 , 7 0

ultragraph, 5 0 ultragraph algebra , 5 0

vertex, 5

vertex matrix , 6 , 6 8

walk, 5 6

X ® n , 7 4

T, 1 2

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Titles i n Thi s Serie s

103 Iai n R a e b u r n , Grap h algebras , 200 5

102 K e n Ono , Th e we b o f modularity : Arithmeti c o f th e coefficient s o f modula r form s an d

g-series, 200 4

101 Henr i D a r m o n , Rationa l point s o n modula r ellipti c curves , 200 4

100 A lexande r Volberg , Calderon-Zygmun d capacitie s an d operator s o n nonhomogeneou s

spaces, 200 3

99 Alai n Lascoux , Symmetri c function s an d combinatoria l operator s o n polynomials , 200 3

98 Alexande r Varchenko , Specia l functions , K Z typ e equations , an d representatio n theory ,

2003

97 Bern d Sturmfels , Solvin g system s o f polynomia l equations , 200 2

96 Nik y K a m r a n , Selecte d topic s i n th e geometrica l stud y o f differentia l equations , 200 2

95 Benjami n Weiss , Singl e orbi t dynamics , 200 0

94 Davi d J . Sal t man, Lecture s o n divisio n algebras , 199 9

93 Gor o Shimura , Eule r product s an d Eisenstei n series , 199 7

92 Fa n R . K . Chung , Spectra l grap h theory , 199 7

91 J . P . Ma y e t a l . , Equivarian t homotop y an d cohomolog y theory , dedicate d t o th e

memory o f Rober t J . Piacenza , 199 6

90 Joh n R o e , Inde x theory , coars e geometry , an d topolog y o f manifolds , 199 6

89 Cliffor d H e n r y Taubes , Metrics , connection s an d gluin g theorems , 199 6

88 Crai g Huneke , Tigh t closur e an d it s applications , 199 6

87 Joh n Eri k Fornaess , Dynamic s i n severa l comple x variables , 199 6

86 Sori n Popa , Classificatio n o f subfactor s an d thei r endomorphisms , 199 5

85 Michi o J i m b o an d Tetsuj i Miwa , Algebrai c analysi s o f solvabl e lattic e models , 199 4

84 H u g h L . M o n t g o m e r y , Te n lecture s o n th e interfac e betwee n analyti c numbe r theor y an d harmonic analysis , 199 4

83 Carlo s E . Kenig , Harmoni c analysi s technique s fo r secon d orde r ellipti c boundar y valu e

problems, 199 4

82 Susa n M o n t g o m e r y , Hop f algebra s an d thei r action s o n rings , 199 3

81 Steve n G . Krantz , Geometri c analysi s an d functio n spaces , 199 3

80 Vaugha n F . R . Jones , Subfactor s an d knots , 199 1

79 Michae l Frazier , Bjor n Jawerth , an d Guid o Weiss , Littlewood-Pale y theor y an d th e

study o f functio n spaces , 199 1

78 Edwar d Formanek , Th e polynomia l identitie s an d variant s o f n x n matrices , 199 1

77 Michae l Chris t , Lecture s o n singula r integra l operators , 199 0

76 Klau s Schmidt , Algebrai c idea s i n ergodi c theory , 199 0

75 F . T h o m a s Farrel l an d L . Edwi n Jones , Classica l aspherica l manifolds , 199 0

74 Lawrenc e C . Evans , Wea k convergenc e method s fo r nonlinea r partia l differentia l

equations, 199 0

73 Walte r A . Strauss , Nonlinea r wav e equations , 198 9

72 Pete r Orlik , Introductio n t o arrangements , 198 9 71 Harr y D y m , J contractiv e matri x functions , reproducin g kerne l Hilber t space s an d

interpolation, 198 9 70 Richar d F . G u n d y , Som e topic s i n probabilit y an d analysis , 198 9 69 Fran k D . Grosshans , Gian-Carl o Rota , an d Joe l A . Ste in , Invarian t theor y an d

superalgebras, 198 7

68 J . Wi l l ia m He l ton , Josep h A . Ball , Charle s R . Johnson , an d Joh n N . Palmer , Operator theory , analyti c functions , matrices , an d electrica l engineering , 198 7

TITLES I N THI S SERIE S

67 Haral d Uprneier , Jorda n algebra s i n analysis , operato r theory , an d quantu m mechanics , 1987

66 G . Andrews , g-Series : Thei r developmen t an d applicatio n i n analysis , numbe r theory , combinatorics, physic s an d compute r algebra , 198 6

65 Pau l H . Rabinowitz , Minima x method s i n critica l poin t theor y wit h application s t o differential equations , 198 6

64 Donal d S . Passman , Grou p rings , crosse d product s an d Galoi s theory , 198 6

63 Walte r Rudin , Ne w construction s o f function s holomorphi c i n th e uni t bal l o f C n , 198 6

62 Bel a Bollobas , Extrema l grap h theor y wit h emphasi s o n probabilisti c methods , 198 6

61 Mogen s Flensted-Jensen , Analysi s o n non-Riemannia n symmetri c spaces , 198 6

60 Gille s Pisier , Factorizatio n o f linea r operator s an d geometr y o f Banac h spaces , 198 6

59 Roge r How e an d Alle n Moy , Harish-Chandr a homomorphism s fo r p-adi c groups , 198 5

58 H . Blain e Lawson , Jr. , Th e theor y o f gaug e fields i n fou r dimensions , 198 5

57 Jerr y L . Kazdan , Prescribin g th e curvatur e o f a Riemannia n manifold , 198 5

56 Har i Bercovici , Cipria n Foia§ , an d Car l Pearcy , Dua l algebra s wit h application s t o

invariant subspace s an d dilatio n theory , 198 5

55 Wil l ia m Arveson , Te n lecture s o n operato r algebras , 198 4

54 Wil l ia m Fulton , Introductio n t o intersectio n theor y i n algebrai c geometry , 198 4

53 Wi lhe l m Klingenberg , Close d geodesie s o n Riemannia n manifolds , 198 3

52 Ts i t -Yue n Lam , Orderings , valuation s an d quadrati c forms , 198 3

51 Masamich i Takesaki , Structur e o f factor s an d automorphis m groups , 198 3

50 Jame s Eell s an d Lu c Lemaire , Selecte d topic s i n harmoni c maps , 198 3

49 Joh n M . Franks , Homolog y an d dynamica l systems , 198 2

48 W . Stephe n Wilson , Brown-Peterso n homology : a n introductio n an d sampler , 198 2

47 Jac k K . Hale , Topic s i n dynami c bifurcatio n theory , 198 1

46 Edwar d G . Effros , Dimension s an d C*-algebras , 198 1

45 Ronal d L . Graham , Rudiment s o f Ramse y theory , 198 1

44 Phil l i p A . Griffiths , A n introductio n t o th e theor y o f specia l divisor s o n algebrai c curves ,

1980

43 Wil l ia m Jaco , Lecture s o n three-manifol d topology , 198 0

42 Jea n Dieudonne , Specia l function s an d linea r representation s o f Li e groups , 198 0

41 D . J . N e w m a n , Approximatio n wit h rationa l functions , 197 9

40 Jea n Mawhin , Topologica l degre e method s i n nonlinea r boundar y valu e problems , 197 9

39 Georg e Lusztig , Representation s o f finite Chevalle y groups , 197 8

38 Charle s Conley , Isolate d invarian t set s an d th e Mors e index , 197 8

37 Masayosh i Nagata , Polynomia l ring s an d affin e spaces , 197 8

36 Car l M . Pearcy , Som e recen t development s i n operato r theory , 197 8

35 R . Bowen , O n Axio m A diffeomorphisms , 197 8

34 L . Auslander , Lectur e note s o n nil-thet a functions , 197 7

33 G . Glaube r man, Factorization s i n loca l subgroup s o f finite groups , 197 7

32 W . M . Schmidt , Smal l fractiona l part s o f polynomials , 197 7

31 R . R . Coifma n an d G . Weiss , Transferenc e method s i n analysis , 197 7

30 A . Pelczyriski , Banac h space s o f analyti c function s an d absolutel y summin g operators , 1977

For a complet e lis t o f t i t le s i n th i s series , visi t th e AMS Bookstor e a t w w w . a m s . o r g / b o o k s t o r e / .