Translations o f

MATHEMATICAL MONOGRAPHS

Volume 22 6

Hilbert C*-Module s

V. M. Manuilo v E. V. Troitsk y

Translated b y th e author s

American Mathematica l Societ y Providence, Rhod e Islan d

" Q v D E D ^

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Translations o f

MATHEMATICAL MONOGRAPHS

Volume 22 6

Hilbert C*-Module s

V. M. Manuilo v E. V. Troitsk y

Translated b y th e author s

American Mathematica l Societ y Providence, Rhod e Islan d

" Q v D E D ^

10.1090/mmono/226

E D I T O R I A L C O M M I T T E E

A M S S u b c o m m i t t e e

Robert D . MacPherso n Grigori i A . Marguli s Jame s D . Stashef f (Chair )

A S L S u b c o m m i t t e e Steffe n Lemp p (Chair )

I M S S u b c o m m i t t e e Mar k I . Freidli n (Chair )

B . M . MaHyMjiOB , E . B . TpoHirKH H

C* r M J I b B E P T O B b l M O H y J I H

SAKTOPHAJI, MOCKBA , 200 1

This wor k wa s originall y publishe d i n Russia n b y Faktoria l Pres s unde r th e titl e "C *

rnjib6epTOBt>i MO^yJin " ©2001 . Th e presen t translatio n wa s create d unde r licens e fo r

the America n Mathematica l Societ y an d i s publishe d b y permission .

Translated fro m th e Russia n b y V . M . Manuilo v an d E . V . Troitsky .

2000 Mathematics Subject Classification. Primar y 46L08 ; Secondar y 46Lxx .

For additiona l informatio n an d update s o n thi s book , visi t w w w . a m s . o r g / b o o k p a g e s / m m o n o - 2 2 6

Library o f Congres s Cataloging-in-Publicatio n D a t a

Manuilov, V . M. (Vladimi r Markovich) , 1961 -[C* Gil'bertov y moduli . English ] Hilbert C*-module s / V.M . Manuilov, E.V . Troitsky ; translated b y V.M. Manuilo v an d E.V.

Troitsky. p. cm . (Translation s o f mathematical monographs , ISS N 0065-928 2 ; v. 226)

Includes bibliographica l reference s an d indexes . ISBN 0-8218-3810- 5 (acid-fre e paper ) 1. C*-algebras . 2 . Hilber t algebras . I . Troitskii , E . V . (Evgeni i Vadimovich ) II . Title .

III. Series .

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Contents

Preface vii

Chapter 1 . Basi c Definition s 1 1.1. C*-algebra s 1 1.2. Pre-Hilber t module s 3 1.3. Hilber t C*-module s 4 1.4. Th e standar d Hilber t modul e HA 8 1.5. Hilber t C*-bimodule s an d stron g Morit a equivalenc e 1 1

Chapter 2 . Operator s o n Hilber t Module s 1 5 2.1. Bounde d an d adjoin t able operator s 1 5 2.2. Compac t operator s i n Hilber t module s 1 8 2.3. Complemen t able submodule s an d projection s i n Hilber t C*-module s 2 2 2.4. Ful l Hilber t C*-module s 2 4 2.5. Dua l modules . Self-dualit y 2 7 2.6. Banach-compac t operator s 3 2 2.7. C*-Fredhol m operator s an d index . Mishchenko' s approac h 3 3 2.8. Representation s o f groups o n Hilber t module s 4 2 2.9. Equivarian t Fredhol m operator s 5 2

Chapter 3 . Hilber t Module s ove r W*-Algebra s 5 5 3.1. VK*-algebra s 5 5 3.2. Inne r produc t o n dua l module s 5 8 3.3. Hilber t W*-module s an d dua l Banac h space s 6 1 3.4. Propertie s o f Hilber t VF*-module s 6 2 3.5. Topologica l characterizatio n o f self-dua l Hilber t W* -modules 6 5 3.6. Fredhol m operator s ove r W* -algebras 6 6 3.7. Dupr e - Fillmor e theore m fo r Hilber t module s ove r finit e W*-algebra s 6 9

Chapter 4 . Reflexiv e Hilber t C*-Module s 7 5 4.1. Inne r produc t o n bidua l module s 7 5 4.2. Ideal s an d bidua l module s 7 9 4.3. Reflexivit y o f Hilber t module s ove r /C + 8 2 4.4. Reflexivit y o f modules ove r C{X) 8 4 4.5. Hilber t module s relate d t o conditiona l expectation s o f finit e inde x 8 6

Chapter 5 . Multiplier s o f .A-Compac t Operators . Structur e Result s 9 9 5.1. Extensio n o f a Hilber t C*-modul e b y th e envelopin g VF*-algebr a 9 9 5.2. Multiplier s an d centralizer s 10 1 5.3. Multiplier s o f A-compac t operator s 10 7 5.4. Quasi-multiplier s o f ^-compac t operator s . 11 0

vi C O N T E N T S

5.5. Stric t topolog y 11 4 5.6. Multiplier s an d Hilber t modules . Th e commutativ e cas e 11 8 5.7. Inne r product s o n Hilber t C*-module s 12 6

Chapter 6 . Diagonalizatio n o f Operator s ove r C*-Algebra s 13 3 6.1. Proble m o f diagonalizin g operator s i n Hilber t C*-module s 13 3 6.2. Quadrati c form s o n H' A relate d t o selfadjoin t operator s 13 6 6.3. Diagonalizin g operator s i n th e V^*-cas e 13 8 6.4. Continuit y o f "eigenvalues " 14 3 6.5. Cas e o f infinit e W*-algebra s 14 5 6.6. Cas e o f C*-algebra s o f rea l ran k zer o 14 6 6.7. Cas e o f continuous fields o f trace C*-algebra s 14 8 6.8. Schrodinge r operato r a s a n operato r actin g o n a Hilber t C*-modul e 15 4 6.9. Example : A continuous field o f rotation algebra s 15 7

Chapter 7 . Homotop y Trivialit y o f Group s o f Invertibl e Operator s 15 9 7.1. Technica l lemma s 15 9 7.2. Proo f o f the Cuntz-Higso n theore m 16 4 7.3. Th e cas e AcK 16 6 7.4. Som e othe r case s 16 9 7.5. Dixmier-Douad y Theore m fo r h(A) 17 3 7.6. Som e generalization s 17 5 7.7. Neubaue r typ e homotop y 17 6

Chapter 8 . Hilber t Module s an d KK-Theory 18 1 8.1. Tenso r product s 18 1 8.2. Mai n definition s 18 2 8.3. Cuntz' s approac h 18 4 8.4. Generalize d Kasparo v bimodule s 18 8 8.5. Classifyin g space s fo r som e K- an d KX-group s 18 9

Bibliography 19 3

Notation Inde x 19 9

Index 201

Preface

Hilbert C*-module s provid e a natura l generalizatio n o f Hilber t space s arisin g when the field of scalars C i s replaced b y a n arbitrar y C*-algebra . Thi s generaliza -tion, i n th e cas e o f commutative C*-algebras , appeare d i n th e pape r [59 ] o f I . Ka -plansky; howeve r th e noncommutativ e cas e seeme d to o complicate d a t tha t time . The genera l theory o f Hilbert C* -modules (i.e. , for a n arbitrar y C*-algebr a servin g as 'scalars' ) appeare d 2 5 years ago in the pioneering papers of W. Paschke [100 ] an d M. Rieffel [108] . Thi s theory has proved to be a very convenient too l in the theory of operator algebras , allowin g on e to obtai n informatio n abou t C*-algebra s b y study -ing Hilber t C*-module s ove r them . I n particular , a serie s o f result s abou t som e classes o f C*-algebra s (lik e AVK*-algebra s an d monoton e complet e C*-algebras ) was obtaine d i n thi s wa y [38] . A n importan t notio n o f Morit a equivalenc e fo r C*-algebras wa s als o formulate d i n term s o f Hilber t C*-module s [109 , 18] . Thi s notion als o ha s application s i n th e theor y o f grou p representations . I t turne d ou t to b e possibl e t o extrac t informatio n o n grou p action s fro m Hilber t C*-module s arising fro m thes e action s [102 , 110] . Som e result s abou t conditiona l expectation s of finite inde x [5 , 133 ] an d abou t completel y positiv e map s o f C*-algebras [2 ] were also obtaine d usin g Hilber t C*-modules .

The theory of Hilbert C* -modules ma y also be considered a s a noncommutativ e generalization o f the theory o f vector bundle s [33 , 69]. Thi s was the reaso n Hilber t modules becam e a too l i n topologica l application s — namel y i n inde x theor y o f elliptic operators , i n K- an d KK-theory [92 , 90 , 62 , 63 , 64 , 65 , 128 ] an d i n noncommutative geometr y a s a whole [24 , 29] .

Among other applications , one should emphasiz e the theory o f quantum group s [135, 136] , unbounded operator s a s a too l fo r Kasparov' s i f if-theory [3 , 4 ] (als o Section 8.4 ) an d som e physica l application s [72 , 80] .

Alongside thes e applications , th e theor y o f Hilber t C*-module s itsel f ha s bee n developed a s well. A number o f results abou t th e structur e o f Hilber t module s an d about operator s o n the m hav e bee n obtaine d [74 , 42 , 88 , 77 , 81 , 127] . Beside s these results , a n axiomati c approac h i n the theor y o f Hilbert module s base d o n th e theory o f operato r space s an d tenso r product s wa s develope d [12 , 11] .

A detaile d bibliograph y o f th e theor y o f Hilber t C*-module s ca n b e foun d in [43] .

Some results presented her e were only announced i n the literature o r the proof s were discussed rathe r briefly . W e have trie d t o fill such gaps . W e could no t discus s all th e aspect s o f th e theor y o f Hilber t C*-module s here , bu t w e trie d t o explai n in detai l th e basi c notion s an d theorem s o f thi s theory , a numbe r o f importan t examples, an d als o som e result s relate d t o th e authors ' interest .

vii

viii PREFAC E

A majo r par t o f thi s boo k forme d th e conten t o f th e lectur e cours e presente d by the author s a t th e Departmen t o f Mechanics an d Mathematic s o f Moscow Stat e University i n 1997 .

We ar e gratefu l t o A . S . Mishchenko fo r introducin g th e theor y o f Hilber t C* -modules to us . Togethe r wit h Yu. P. Solovyov, he has acquainted u s with the circl e of problems relate d t o it s application s i n topology .

While workin g i n thi s field an d i n th e proces s o f writin g th e presen t text , a significant influenc e o n u s was mad e b y ou r frien d an d co-autho r M . Frank .

We hav e discusse d a numbe r o f problem s o f th e Hilber t C*-modul e theor y with L . Brown , A . A . Irmatov , G . G . Kasparov , R . Nest , G . K . Pederse n an d W. Paschke . Som e application s wer e considere d als o with J . Cuntz , A . Ya . Helem -skii, J . Kaminker , V . Nistor , J . Rosenberg , K . Thomsen , B . L . Tsyga n an d others .

Our researc h wa s partiall y supporte d b y a serie s o f subsequen t Russia n Foun -dation o f Basi c Researc h grants .

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Notation Inde x

aV2 1 a > 0 2 4 a > 0 1 A+ 1 A** 5 6 A0 15 5 Ag° 15 5 A * £ 18 5 £ ( # ) 1 /3-lim 11 4 C 0 (X) 8 0 DC(A) 10 3 GL 5 0 GL* 5 2 E : A—>B 6 End A(M) 1 5 End*A(M) 1 6 / * 2 9 HA 6 tf<^ 2 9 #A4 6 UB 4 2 H o n ^ O ^ A / " ) 1 5

ftom A {M,N) 1 6 index F 3 6 K 1 9 /C(M) 1 8 K(M,N) 1 8 /CA 12 3 &K(M,N) 3 2 K ( £ ) 8 7 K(A) 3 3 Xo(A) 3 3 Kn(A) 3 3 KK{A,B) 18 2 k (A) 6 Z2(.M) 6

M ^ ) 4 LC(A) 10 5 LM(A) 10 5 M' 2 7 A4 + 4 3 A4# 9 9 M 7 5 (A4,M> 1 1 M n 1 9 M(A) 10 2 M n (A) 1 9 Af1- 6 (AA)B 4 3 N^ 2 9 p V g 7 1 p A g 7 1 P+{A) 1 P(A) 3 3 qA 18 5 tf! 5 5 RC(A) 10 5 RM(;4) 10 5 RR{A) 14 7 s-lim 5 5 Sp(A) 1 tsr(A) 14 9 TI 6 5 T2 6 5 ^x,y 1 8 w-lim 5 5 x 2 7 x • a 3 [XJp 11 4 Z(A) 5 7

e 5 e 5

199

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Index

A-orthobasis, 8 A-valued product , 3

equivalent, 12 6 A, B-bimodule, 18 2 A-B-equivalence bimodule , 1 1 algebra

C*- 1 divisibly infinit e (DI) , 3 1 module-infinite (MI) , 3 0 cr-unital, 2

universal enveloping , 5 6 von Neumann , 5 6 W*- 5 6

continuous, 5 7 finite, 5 7 of typ e I , 5 7 of typ e II , 5 7 of typ e Hi , 5 7 of typ e IIoo , 5 7 of typ e III , 5 7 properly infinite , 5 7 semifmite, 5 7 cr-finite, 5 7

approximate unit , 2 bicommutant, 5 5 Cauchy-Bunyakovsky inequality , 4 centralizer

double, 10 3 left, 10 5 quasi-, 10 5

commutant, 5 5 compatible pair , 11 8 conditional expectation , 6

faithful, 6 of finit e index , 8 7

algebraically, 8 6 characteristic numbe r of , 8 7

element o f C*-algebr a hermitian, 1 positive, 1 strictly positive , 2

element o f modul e nonsingular, 8

e-orthogonal submodules , 17 1

equivalent projections , 5 8 essential ideal , 10 1 F-topology, 12 6 factor, 5 8 functional, 2 7

nonsingular, 13 0 normal, 5 6 positive, 1 represent able, 13 0

G-£-module, 4 2 G-C*-algebra, 4 2 GNS-construction, 2 group,

Grothendieck, 3 3 linear,

full genera l GL , 5 0 general GL* , 5 2

inner product , equivalent, 12 6

Kasparov bimodule , 18 2 unbounded, 18 9

KK-groups, Cuntz definition , 18 8 Kasparov definition , 18 2

lemma of Dixmier-Douady , 17 3

module dual, 2 7 Hilbert, 4

count ably generated , 8 free, 4 full, 1 1 reflexive, 7 9 self-dual, 2 7 standard, 6

pre-Hilbert, 3 projective finitel y generated , 9

multiplier, 10 2 left, 10 5 quasi-, 10 5 right, 10 5

operator homotopy , 18 4

202 INDEX

operator o n module , 1 5 A-Fredholm, 3 4

index of , 3 6 adjoint, 1 5 Banach-compact, 3 2 compact, 1 8 elementary, 1 8

orthogonal complement , 6 polarization equality , 4 positive squar e root , 1 pre-Hilbert A-J3-bimodule , 1 1 property (E) , 16 9 property (K) , 16 9 representation o f a n algebra , 1

nondegenerate, 10 2 universal, 5 6

set A-precompact, 3 3

space pre-dual, 5 6

spectrum o f a n element , 1 state, 1 strong Morit a equivalence , 1 2 submodule,

complemented, 2 2 symmetrization, 3 3 tensor produc t

exterior, 18 1 interior, 18 1

theorem of Cuntz-Higson , 16 6 of Dixmier-Douad y fo r modules , 17 4 of Dupre-Fillmore , 1 0 of vo n Neuman n abou t bicommutant , 5 5

topology, left strict , 11 6 quasi-strict, 11 8 strict, 11 4 strong, 5 5 strong*, 5 5 strong module , 11 6 strong* module , 11 6 weak ,5 5 cr(E\E)-, 5 6 a(E,E*)-, 5 6 <j-strong 5 5 cr-strong* 5 5 cr-weak, 5 5

trace, 5 7 faithful, 5 7 finite, 5 7 normal, 5 7 semifinite, 5 7

triple, degenerated, 18 2

ultra wea k direc t sum , 6 3 unitalization, 1 vector

cyclic, 1 separating, 1

Titles i n Thi s Serie s

226 V . M . Manuilo v an d E . V . Troitsky , Hilber t C*-modules , 200 5

225 S . M . Natanzon , Modul i o f Rieman n surfaces , rea l algebrai c curves , an d thei r

superanaloges, 200 4

224 Ichir o Shigekawa , Stochasti c analysis , 200 4

223 M a s a t o s h i N o u m i , Painlev e equation s throug h symmetry , 200 4

222 G . G . Magaril-Il'yae v an d V . M . Tikhomirov , Conve x analysis : Theor y an d

applications, 200 3

221 K a t s u e i Kenmotsu , Surface s wit h constan t mea n curvature , 200 3

220 I . M . Gelfand , S . G . Gindikin , an d M . I . Graev , Selecte d topic s i n integra l

geometry, 200 3

219 S . V . Kerov , Asymptoti c representatio n theor y o f th e symmetri c grou p an d it s

applications t o analysis , 200 3

218 Kenj i Ueno , Algebrai c geometr y 3 : Further stud y o f schemes , 200 3

217 Masak i Kashiwara , D-module s an d microloca l calculus , 200 3

216 G . V . Badalyan , Quasipowe r serie s an d quasianalyti c classe s o f functions , 200 2

215 Tatsu o Kimura , Introductio n t o prehomogeneou s vecto r spaces , 200 3

214 L . S . Grinblat , Algebra s o f set s an d combinatorics , 200 2

213 V . N . Sachko v an d V . E . Tarakanov , Combinatoric s o f nonnegativ e matrices , 200 2

212 A . V . Mel'nikov , S . N . Volkov , an d M . L . Nechaev , Mathematic s o f financia l

obligations, 200 2

211 Take o Ohsawa , Analysi s o f severa l comple x variables , 200 2

210 Toshitak e Kohno , Conforma l field theor y an d topology , 200 2

209 Yasumas a Nishiura , Far-from-equilibriu m dynamics , 200 2

208 Yuki o M a t s u m o t o , A n introductio n t o Mors e theory , 200 2

207 Ken'ich i Ohshika , Discret e groups , 200 2

206 Yuj i Shimiz u an d Kenj i Ueno , Advance s i n modul i theory , 200 2

205 Seik i Nishikawa , Variationa l problem s i n geometry , 200 1

204 A . M . Vinogradov , Cohomologica l analysi s o f partia l differentia l equation s an d

Secondary Calculus , 200 1

203 T e Su n Ha n an d King o Kobayashi , Mathematic s o f informatio n an d coding , 200 2

202 V . P . Maslo v an d G . A . Omel'yanov , Geometri c asymptotic s fo r nonlinea r PDE . I ,

2001

201 Shigeyuk i Morita , Geometr y o f differentia l forms , 200 1

200 V . V . Prasolo v an d V . M . Tikhomirov , Geometry , 200 1

199 Shigeyuk i Morita , Geometr y o f characteristi c classes , 200 1

198 V . A . Smirnov , Simplicia l an d opera d method s i n algebrai c topology , 200 1

197 Kenj i U e n o , Algebrai c geometr y 2 : Sheave s an d cohomology , 200 1

196 Yu . N . Lin'kov , Asymptoti c statistica l method s fo r stochasti c processes , 200 1

195 Minor u Wakimoto , Infinite-dimensiona l Li e algebras , 200 1

194 Valer y B . Nevzorov , Records : Mathematica l theory , 200 1

193 Toshi o Nishino , Functio n theor y i n severa l comple x variables , 200 1

192 Yu . P . Solovyo v an d E . V . Troitsky , C*-algebra s an d ellipti c operator s i n differentia l

topology, 200 1

191 Shun-ich i Amar i an d Hirosh i Nagaoka , Method s o f informatio n geometry , 200 0

190 Alexande r N . Starkov , Dynamica l system s o n homogeneou s spaces , 200 0

189 Mitsur u Ikawa , Hyperboli c partia l differentia l equation s an d wav e phenomena , 200 0

TITLES I N THI S SERIE S

188 V . V . Buldygi n an d Yu . V . Kozachenko , Metri c characterizatio n o f rando m variable s and rando m processes , 200 0

187 A . V . Fursikov , Optima l contro l o f distribute d systems . Theor y an d applications , 200 0

186 Kazuy a Kato , Nobushig e Kurokawa , an d Takesh i Saito , Numbe r theor y 1 :

Fermat's dream , 200 0

185 Kenj i Ueno , Algebrai c Geometr y 1 : Fro m algebrai c varietie s t o schemes , 199 9

184 A . V . Mel'nikov , Financia l markets , 199 9

183 Haj im e Sato , Algebrai c topology : a n intuitiv e approach , 199 9

182 I . S . Krasil'shchi k an d A . M . Vinogradov , Editors , Symmetrie s an d conservatio n

laws fo r differentia l equation s o f mathematica l physics , 199 9

181 Ya . G . Berkovic h an d E . M . Zhmud' , Character s o f finite groups . Par t 2 , 199 9

180 A . A . Milyut i n an d N . P . Osmolovskii , Calculu s o f variation s an d optima l control ,

1998

179 V . E . Voskresenskii , Algebrai c group s an d thei r birationa l invariants , 199 8

178 Mi t su o Morimoto , Analyti c functional s o n th e sphere , 199 8

177 Sator u Igari , Rea l analysis—wit h a n introductio n t o wavele t theory , 199 8

176 L . M . Lerma n an d Ya . L . Umanskiy , Four-dimensiona l integrabl e Hamiltonia n

systems wit h simpl e singula r point s (topologica l aspects) , 199 8

175 S . K . Godunov , Moder n aspect s o f linea r algebra , 199 8

174 Ya-Zh e Che n an d Lan-Chen g Wu , Secon d orde r ellipti c equation s an d ellipti c

systems, 199 8

173 Yu . A . Davydov , M . A . Lifshits , an d N . V . Smorodina , Loca l propertie s o f

distributions o f stochasti c functionals , 199 8

172 Ya . G . Berkovic h an d E . M . Zhmud' , Character s o f finit e groups . Par t 1 , 199 8

171 E . M . Landis , Secon d orde r equation s o f ellipti c an d paraboli c type , 199 8

170 Vikto r Prasolo v an d Yur i Solovyev , Ellipti c function s an d ellipti c integrals , 199 7

169 S . K . Godunov , Ordinar y differentia l equation s wit h constan t coefficient , 199 7

168 Junjir o Noguchi , Introductio n t o comple x analysis , 199 8

167 Masay a Yamaguti , Masayosh i Hata , an d Ju n Kigami , Mathematic s o f fractals , 199 7

166 Kenj i Ueno , A n introductio n t o algebrai c geometry , 199 7

165 V . V . Ishkhanov , B . B . Lur'e , an d D . K . Faddeev , Th e embeddin g proble m i n

Galois theory , 199 7

164 E . I . Gordon , Nonstandar d method s i n commutativ e harmoni c analysis , 199 7

163 A . Ya . Dorogovtsev , D . S . Silvestrov , A . V . Skorokhod , an d M . I . Yadrenko ,

Probability theory : Collectio n o f problems , 199 7

162 M . V . Boldin , G . I . Simonova , an d Yu . N . Tyurin , Sign-base d method s i n linea r

statistical models , 199 7

161 Michae l Blank , Discretenes s an d continuit y i n problem s o f chaoti c dynamics , 199 7

160 V . G . Osmolovskii , Linea r an d nonlinea r perturbation s o f th e operato r div , 199 7

159 S . Ya . Khavinson , Bes t approximatio n b y linea r superposition s (approximat e nornography), 199 7

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