Hilbert Spaces

C*-ALGEBRAS

VOLUME 4: HILBERT SPACES

C*-ALGEBRAS

VOLUME 4: HILBERT SPACES

North-Holland Mathematical Library Board of Honorary Editors:

M. Artin, H. Bass, J. Eells, W. Feit, P.J. Freyd, F.W. Gehring, H. Halberstam, L.V. Hormander, J.H.B. Kemperman, W.A.J.Luxemburg, F. Peterson, I.M. Singer and A.C. Zaanen Board of Advisory Editors: A. Bjomer, R.H. Dijkgraaf, A. Dimca, A.S. Dow, J.J. Duistermaat, E. Looijenga, J.P. May, I. Moerdijk, S.M. Mori, J.P. Palis, A. Schrijver, J. Sjostrand, J.H.M. Steenbrink, F. Takens and J. van Mill

VOLUME 61

ELSEVIER Amsterdam - London - New York - Oxford - Paris - Shannon - Tokyo

C*-Algebras Volume 4: Hilbert Spaces

Corneliu Constantinescu Departement Mathematik, ETH Zurich CH-8092 Zurich Switzerland

ELSEVIER Amsterdam - London - New York - Oxford - Paris - Shannon - Tokyo

ELSEVIER SCIENCE B.V. Sara Burgerhartstraat 25 P.O. Box 2 1 1, 1000 A E Amsterdam, The Netherlands

0 2001

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Preface Funct,ional analysis plays an important role in the program of studies a t the Swiss Federal Instit,ute of Technology. At present, courses entitled F~lnctional .4nalysis I and 11 arc taken during the fifth and sixth semester, rcspectivcly. I have taught these courses several times and after a while typewritten lecture notes resulted that were distributed to the students. During the academic year 1987/88: I was fortunate enough to have an eager enthusiastic group of students that I had already erlcoul~teredpreviously in other lecture courses. These stutlt>rltswanted to learn rrlore in the area and asked rrie to design a cont,inuation of' the courses. Accordingly. I proceeded during the academic year, following, wit,h a series of special lectures, Functional Analysis I11 and IV, for which I again distrih~~tcd typewritten Icct,~lrcnotes. At the end I found that t,hcrc had accum~llatcda mass of text,ual material, and I asked myself if I should not publish it in the form of a book. Cnfortunately, I realized that the two special lecture series (they had been given only once) had been badly organized and contained ~rlaterialthat should have been included in the first two portions. And so I came to the conclusion that I should write everything anew - and if a t all then preferably in English. Little did I realize what I was letting myself in for! The number of pages grew almost impcrccptihly and at t,hc end it had more than do~lhlcd.Also: the English language turned o ~ to~ be t a st~~mhling 1)lock for me; I would likc to take this opport~lnityto thank Prof. Imre Bokor and Prof. Edgar Reich for their help in this regard. Above all I must thank Mrs. Barbara Aqnilino, who wrote, first a b \ ~ o r d ~ 4 A R Cand ' ~ ~ then , a Lq$J'r'M v e r s i o ~with ~ great co~r~petence, a ~ ~ g e l patience, ic a11d utter devotion, in spite of illness. My thanks also go to the Swiss Federal Inst,it,utcof Tc,chnology that. genero~lslyprovided the infrast,ructure for this extensive enterprise and to my c~ollcagncswho showed their understanding for i t -

Corneliu Constantinescu

This Page Intentionally Left Blank

Table of Contents of Volume 4

Introduction 5

...............................

Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Pre-Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 General Results . . . . . . . . . . . . . . . . . . . . . . . 5.1.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.3 Hilbert sums . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Orthogonal Projections of Hilbert Space . . . . . . . . . . . . . 5.2.1 Projections onto Convex Sets . . . . . . . . . . . . . . . 5.2.2 Orthogonality . . . . . . . . . . . . . . . . . . . . . . . . 5.2.3 Orthogonal Projections . . . . . . . . . . . . . . . . . . . 5.2.4 Mean Ergodic Theorems . . . . . . . . . . . . . . . . . . 5.2.5 The Frechet-Riesz Theorem . . . . . . . . . . . . . . . . 5.3 Adjoint Operators . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 General Results . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Supplementary Results . . . . . . . . . . . . . . . . . . . 5.3.3 Selfadjoint Operators . . . . . . . . . . . . . . . . . . . . 5.3.4 Normal Operators . . . . . . . . . . . . . . . . . . . . . . 5.4 Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 Cyclic R.epresentation . . . . . . . . . . . . . . . . . . . 5.4.2 General Representations . . . . . . . . . . . . . . . . . . 5.4.3 Example of Representations . . . . . . . . . . . . . . . . 5.5 Orthonormal Bases . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.1 General Results . . . . . . . . . . . . . . . . . . . . . . . 5.5.2 Hilbert Dimension . . . . . . . . . . . . . . . . . . . . . 5.5.3 Standard Examples . . . . . . . . . . . . . . . . . . . . . 5.5.4 The Fourier-Plancherel Operator . . . . . . . . . . . . . 5.5.5 Operators and Orthonormal Bases . . . . . . . . . . . . 5.5.6 Self-.normal Compact Operators . . . . . . . . . . . . . . 5.5.7 Examples of R.eal C*-Algebras . . . . . . . . . . . . . .

xix

3 3 3 14 19 24 24 29 33 54 63 72 72 86 108 123 130 130 146 156 166 166 191 206 218 223 243 258

...

v1l1

Table of Contents

5.6 Hilbert right C*-Modules . . . . . . . . 5.6.1 Some General Results . . . . . . 5.6.2 Self-duality . . . . . . . . . . . . 5.6.3 Von Neumann right W*-modules 5.6.4 Examples . . . . . . . . . . . . . 5.6.5 K E . . . . . . . . . . . . . . . . . 5.6.6 Matrices over C'-algebras . . . . 5.6.7 Type I W'-algebras . . . . . . . Name Index Subject Index

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

286 286 310 341 373 430 477 515

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 539

Symbol Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 567

Contents of All Volumes

Table of Contents of Volume 1

Some Notation and Terminology

. . . . . . . . . . . . . . . . . . . .

1 Banact1 Spaces . . . . . . . . . . . . . . . . . . 1.1 Xormed Spaces . . . . . . . . . . . . . . . . 1.1.1 Gencral Results . . . . . . . . . . . . 1.1.2 Somc Standard Examples . . . . . . 1.1.3 A.linkowski's Thcorcm . . . . . . . . . 1.1.4 Locallv Compact Norn~edSpaces . . 1.1.5 Prodncts of Normed Spaccs . . . . . 1.1.6 S n m n ~ a l ~Families le . . . . . . . . . . Exercises. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.2 Oprrators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 General Resnlts . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Standard Exarrlples . . . . . . . . . . . . . . . . . . . . . 1.2.3 Infinit.e Matricrs . . . . . . . . . . . . . . . . . . . . . . 1.2.4 Qnotient Spaces . . . . . . . . . . . . . . . . . . . . . . . 1.2.5 Cornplernented Sut~spaces . . . . . . . . . . . . . . . . . 1.2.6 Tht! 'Topology of Pointwise Convergence . . . . . . . . . 1.2.7 Convex Sets . . . . . . . . . . . . . . . . . . . . . . . . .

1

7 7 7 12 31 33 37 40 58 61 61 74 92 113 120 134 138 148 150 153 139 159 171

The Alaoglu Bor1rt)aki Theorcrn . . . . . . . . . . . . . . Rilir~carMaps . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Thc llahn-Banad1 Theorem . . . . . . . . . . . . . . . . . . . . 1.3.1 Thc Banach Theorern . . . . . . . . . . . . . . . . . . . . 1.0.2 Exarnples in Measure Theory . . . . . . . . . . . . . . . I .3.3 'I'llc Hahn Ranach Thcorcrn . . . . . . . . . . . . . . . . 180 1.3.4 'The Transposc of an Operator . . . . . . . . . . . . . . . 191 1.2.8 1.2.9

1

Table of Contents

Polar Sets . . . . . . . . . . . . The Bidual . . . . . . . . . . . The Krein-~mulian Theorem . Reflexive Spaces . . . . . . . . . Completion of Normed Spaces . Analytic Functions . . . . . . . Exercises . . . . . . . . . . . . . 1.4 Applications of Baire's Theorem . . . . 1.4.1 The Banach-Steinhaus Theorem 1.4.2 Open Mapping Principle . . . . Exercises . . . . . . . . . . . . . 1.5 Banach Categories . . . . . . . . . . . 1.3.5 1.3.6 1.3.7 1.3.8 1.3.9 1.3.10

1.5.1 Definitions . . . 1.5.2 Functors . . . . 1.6 Nuclear Maps . . . . . 1.6.1 General Results 1.6.2 Examples . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

1.7 Ordered Banach spaces . . . . . . 1.7.1 Ordered normed spaces . . 1.7.2 Order Continuity . . . . . Name Index Subject Index

. . . . . . . . . . . . . . 199 . . . . . . . . . . . . . . 211 . . . . . . . . . . . . . . 228

. . . . . . . . . . . . . . 240 . . . . . . . . . . . . . . 245 . . . . . . . . . . . . . . 246 . . . . . . . . . . . . . . 254 . . . . . . . . . . . . . . 256 . . . . . . . . . . . . . . 256 . . . . . . . . . . . . . . 264 . . . . . . . . . . . . . . 280 . . . . . . . . . . . . . . 281 . . . . . . . . . . . . . . . . . 281 . . . . . . . . . . . . . . . . . 288 . . . . . . . . . . . . . . . . . 308 . . . . . . . . . . . . . . . . . 308 . . . . . . . . . . . . . . . . . 322 . . . . . . . . . . . . . . . . . 334 . . . . . . . . . . . . . . . . . 334 . . . . . . . . . . . . . . . . . 340

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359

Symbol Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371

Table of Contents of Volume 2

Introduction 2

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xix

Bartach Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.1

2.1.1 2.1.2 2.1.3 2.1.4 2.1.5

2.2

2.3

2.4

3 3

General Results . . . . . . . . . . . . . . . . . . . . . . . 3 Invert.iblcElcmcnts . . . . . . . . . . . . . . . . . . . . . 13 Tho Spectrum . . . . . . . . . . . . . . . . . . . . . . . . 17 Standard Examples . . . . . . . . . . . . . . . . . . . . . 32

Con~~~lexification of Algebras . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . Norrnod Algcl, ras . . . . . . . . . . . . . . . . . . . . . . . . . .

51 65 69

. . . . . . . . . . . . . . . . . . . . . . . 2.2.1 GortoralRes~~lts 2.2.2 The Standard Examples . . . . . . . . . . . . . . . . . . 2.2.3 'The Kxponcntial Function and the Neumann Series . . . 2.2:1 Invertible Elements of IJnit.al Ranach Algc.11ras . . . . . . 2.2.5 Thc Theorems of Ricsz and Gclfand . . . . . . . . . . . . 2.2.6 Poles of Rosolvtrnts . . . . . . . . . . . . . . . . . . . . . 2.2.7 Mod~llcs . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . .

69 82 114 125 153 161 174

197 Invol~itiveBanach .i\lgrbras . . . . . . . . . . . . . . . . . . . . 201 2.3.1 Involntivc Algebras . . . . . . . . . . . . . . . . . . . . . 201 2.3.2 Invo11it.ivc~Banach Algebra. . . . . . . . . . . . . . . . . 241 2.3.3 Scsql~iliriearForms . . . . . . . . . . . . . . . . . . . . . 275 2.3.4 I'ositive I,irtear Forms . . . . . . . . . . . . . . . . . . . 287

2.3.5 2.3.6

The State Space . . lnvol~itivcMotiules Exercises . . . . . . Gelfarltl Algebras . . . . .

. . . . . . . . . . . . . . . . . . . . . 305 . . . . . . . . . . . . . . . . . . . . . 322

2.4.1 2.4.2

. . . . . . . . . . . . . . . . . . . . . 328 . . . . . . . . . . . . . . . . . . . . . 331 The Gclfand Transform . . . . . . . . . . . . . . . . . . . 331 Invol~~tive Gelfand Algebras . . . . . . . . . . . . . . . . 343

Table of Contents

xii

Examples . . . . . . . . . . . . . . . . . . . . . . . . . . Locally Compact Additive Groups . . . . . . . . . . . . . Examples . . . . . . . . . . . . . . . . . . . . . . . . . The Fourier Transform . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . .

358 365 378 390 396

3 Compact Operators . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 The General Theory . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 General Results . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.3 Fredholm Operators . . . . . . . . . . . . . . . . . . . . 3.1.4 Point Spectrum . . . . . . . . . . . . . . . . . . . . . . . 3.1.5 Spectrum of a Compact Operator . . . . . . . . . . . . . 3.1.6 Integral Operators . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Linear Differential Equations . . . . . . . . . . . . . . . . . . . . 3.2.1 Boundary Value Problems for Differential Equations . . . 3.2.2 Supplementary Results . . . . . . . . . . . . . . . . . . . 3.2.3 Linear Partial Differential Equations . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . .

399 399 399 419 437 468 477 489 517 518 518 530 549 563

2.4.3 2.4.4 2.4.5 2.4.6

Name Index Subject Index

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 565

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 568

Symbol Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 588

...

Xlll

Table of Contents of Volume 3

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xix

C* -Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 3

Introduction 4

4.1

Tlic General Thcory 4.1.1 4.1.2

General Rc?sults . . . . . . . . . . . . . . . . . . . . . . . The Symnlctry of C*-Algebra . . . . . . . . . . . . . . .

Functional c a l c ~ ~ l uins C*-Algebras . . . The Theorem of Fuglede-P~itnam . . . . T h e Order Relation . . . . . . . . . . . . . . . . 4.2.1 Definition and General Propcrt.ies . . . . 4.2.3 Exa~nples . . . . . . . . . . . . . . . . . 4.1.3 4.1.4

4.2

. . . . . . . . . . . . . . . . . . . . . . . .

4 30

. . . . . . . . . 56 . . . . . . . . .

75

. . . . . . . . .

92

. . . . . . . . .

92 . . . . . . . . . 116 4.2.4 Powers of Positive Elen~cnts . . . . . . . . . . . . . . . . 123 4.2.5 Thc Mod1111ls . . . . . . . . . . . . . . . . . . . . . . . . 143 4.2.6 Ideals and Quotients of C*-Algebras . . . . . . . . . . . 150 4.2.7 T h e Ordered Set of Orthogonal Projectior~s . . . . . . . 162 4.2.8 Approximate Unit . . . . . . . . . . . . . . . . . . . . . 178 4.3 Snpplemcntary Rcsnlt.s on C* Algebras . . . . . . . . . . . . . . 208 4.3.1 Tho Exterior Mllltiplication . . . . . . . . . . . . . . . . 208 4.3.2 Order Complet.e C*--Algebra5 . . . . . . . . . . . . . . . 215 4.3.3 The Carrier . . . . . . . . . . . . . . . . . . . . . . . . . 243 4.3.4 Heretiitilry C*-S11ba1gebra.s . . . . . . . . . . . . . . . . 263 4.3.5 Simple C*-algebras . . . . . . . . . . . . . . . . . . . . . 276 4.3.6 Sl~pplement. ary Res~iltsConcerning Complcxifiratior~ . . 286 4.4 IV*-A1get)ras . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297 4.4.1 General Properties . . . . . . . . . . . . . . . . . . . . . 297 4.4.2 F as arl E s u b m o d ~ ~of l e E' . . . . . . . . . . . . . . . 309 4.4.3 Polar Rrpresent ation . . . . . . . . . . . . . . . . . . . . 335 4.4.4 W*-Homorr~orphisms. . . . . . . . . . . . . . . . . . . . 361

xiv

Table of Contents

Subject Index

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Symbol Index . . . . .

.

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.

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.

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. . 388

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.

411

Table of Contents of Volume 4

Introduction 5

...............................

Hilbcrt Spaces . . . . . . . 5.1 Pre-Hilbert Spaces . . 5.1.1 General Results 5.1.2 Examples . . . 5.1.3 Hilbert sums . 5.2

5.3

5.4

5.5

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Orthogonal Projections of Hilbert Space . . . . . . . . . . . . . 5.2.1 Projections onto Convex Sets . . . . . . . . . . . . . . . 5.2.2 Orthogonality . . . . . . . . . . . . . . . . . . . . . . . . 5.2.3 Orthogonal Projections . . . . . . . . . . . . . . . . . . . 5.2.4 Mean Ergodic Theorems . . . . . . . . . . . . . . . . . . 5.2.5 Tho Frbchet.. Riesz Theorem . . . . . . . . . . . . . . . . Adjoint Operators . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 General Res~llts . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Supplementary Results . . . . . . . . . . . . . . . . . . . 5.3.3 Selfadjoint Operators . . . . . . . . . . . . . . . . . . . . 5.3.4 Normal Operators . . . . . . . . . . . . . . . . . . . . . . R.epresentations . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 Cyclic: Representation . . . . . . . . . . . . . . . . . . . 5.4.2 General Representations . . . . . . . . . . . . . . . . . . 5.4.3 Example of Representations . . . . . . . . . . . . . . . . Orthonormal Bases . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.1 General Results . . . . . . . . . . . . . . . . . . . . . . . 5.5.2 Hilhert Dimension . . . . . . . . . . . . . . . . . . . . . 5.5.3 Standard Examples . . . . . . . . . . . . . . . . . . . . . 5.5.4 The Fourier-Plancherel Operator . . . . . . . . . . . . . 5.5.5 Operators and Orthonormal Bases . . . . . . . . . . . . 5.5.6 Self -normal Cornpac:t Operators . . . . . . . . . . . . . . 5.5.7 Examples of Real C*-Algebras . . . . . . . . . . . . . .

xix

3 3 3 14 19 24 24 29 33 54 63 72 72 86 108 123 130 130 146 156 166 166 191 206 218 223 243 258

xvi

Table of Contents

5.6 Hilbert right C* -Modules . . . . . . . . 5.6.1 Some General Results . . . . . . 5.6.2 Self-duality . . . . . . . . . . . . . 5.6.3 Von Neumann right W* -modules 5.6.4 Examples . . . . . . . . . . . . . 5.6.5 ICE . . . . . . . . . . . . . . . . . 5.6.6 Matrices over C*-algebras . . . . 5.6.7 Type I Lit*-algebras . . . . . . . Name Index Subject Index

. . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . .......... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

286 286 310 341 373 430 477 515

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 539

Symbol Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 567

xvii

Table of Contents of Volume 5

...............................

xix

6 Selected Chapters of C*-Algebras . . . . . . . . . . . . . . . . . . . 6.1 LP-Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.1 Characteristic Families of Eigenvalues . . . . . . . . . . . 6.1.2 Characteristic Sequences . . . . . . . . . . . . . . . . . . 6.1.3 Properties of the LP-spaces . . . . . . . . . . . . . . . . 6.1.4 Hilbert-Schmidt Operators . . . . . . . . . . . . . . . . . 6.1.5 The Trace . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.6 Duals of LP-spaces . . . . . . . . . . . . . . . . . . . . . 6.1.7 Exterior Mi~ltiplicationand LP-Spaces . . . . . . . . . . 6.1.8 The Canonical Projection of the Tridual of K . . . . . . 6.1.9 Integral Operators on FIilbert Spaces . . . . . . . . . . . 6.2 Selfadjoint Linear Differential Equations . . . . . . . . . . . . . 6.2.1 Selfadjoint Boundary Value Problems . . . . . . . . . . . 6.2.2 The Reglilar Sturm-Liouville Theory . . . . . . . . . . . 6.2.3 Selfadjoint Linear Differential Equations on T . . . . . . 6.2.4 Associated Parabolic and Hyperbolic Evolution Equations 6.2.5 Selfadjoint Linear Partial Differential Equations . . . . . 6.2.6 Associated Parabolic arid Hyperbolic Evolution Equations 6.3 Von Neumann Algebras . . . . . . . . . . . . . . . . . . . . . . 6.3.1 The Strong Topology . . . . . . . . . . . . . . . . . . . . 6.3.2 Bidual of a C*-algebra . . . . . . . . . . . . . . . . . . . 6.3.3 Extensiorl of the Functional Calculus . . . . . . . . . . . 6.3.4 Von Neumann-Algebras . . . . . . . . . . . . . . . . . . 6.3.5 The Commlrt.ants . . . . . . . . . . . . . . . . . . . . . . 6.3.6 Irredllcible Representations . . . . . . . . . . . . . . . . 6.3.7 Commutative von Neumann Algebras . . . . . . . . . . . Algebras . . . . . . . . . . . . . 6.3.8 Representations of W*-. 6.3.9 Finite-dimensional C*-algebras . . . . . . . . . . . . . .

3 3 3 10 21 46 56 72 79 102 116 124 125 139 150 153 184 192 202 203 218 263 283 293 299 320 325 334

Introduction

xviii

Table of Contents

6.3.10 A generalization . . . . . . . . . . . . . . . . . . . . . . . 355 7

C' algebras Generated by Groups . . . . 7.1 Projective Representations of Groups 7.1.1 Schur functions . . . . . . . . 7.1.2 Projective Representations . . 7.1.3 Supplementary Results . . . . 7.1.4 Examples . . . . . . . . . . . 7.2 Clifford Algebras . . . . . . . . . . . 7.2.1 General Clifford Algebras . . . . . . . . . . . . . . . . 7.2.2 7.2.3 CL(IN) . . . . . . . . . . . . . Namc Index S~lbjcctIndex

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

369 369 369 404 431 466 492 492 518 538

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 559 . . . . . . . . . . . . . . . . . . . . . ... . . . . . . . 563

Symbol Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 592

xix

Introduction This book has evolved from the I(~c:tnrccourso on F~~rrctiorral Analysis I had givc,rr sc~vt,raltirrres at thc! ETH. 'fht, text has a strict logical order, in the style of "D(+irriton Theorern I'roof - Example Excrcisrs". Thc proofs arc9rathcr -

thorough and there

-

many rxanrplcs.

The, first part of the t~ook(the first three chapters. resp. the first two vol~l-

rnes) is devotc.d to the theory of Banach spaccs in th(>most general sense of the t,errrr. The pllrpose of the first chaptcr (resp. first v o l ~ ~ n ris e )to introduce those rcs~rltso n Banach spaccs which are used later or which are closely conrrc,ctcd with the book. It therefore only contains a small part of the theory, and several rcsults ilte stated (and proved) in ;I dilutod form. The second chaptvr (which t,ogethc,r with Chapter 3 rriakes thc second volume) deals with Barrach algcbras (anti invol~itivcnanach algebras). which constitute t h main ~ topic of the first part of the t~ook.T l ~ ethird c:lrapter deals with conrpact operators 011 Banach spaces i111tllirrear (ordinary and partial) differential eq~lations- applications of tht, thc~)l.yof Rarrach algct)ras. The sc~c,ondpart of the t~ook(the last four chapters. resp. the last three, vol~~rrres) is d(~vot.c!tlto the theory of Hilhcrt sl)ac:t!s. oncc again in tht. gc!r~c!riil scrrsr of the, ttrrr~.It 1)egirrswith a chapter (Chapter 4 , rrsp. Volnrr~e3) on the theor.v of C* iilgct)ras and I,I''- algebras which arc essentially the focns of thc I)ook. Chapter 5 (resp. Volurne 4) t.rt!ats Hilbc'rt spacvs for which wc had no ~rccdcarlic,r. It contains t h rcprosentation ~ theorc:ms, i.e. the theorems on isomclt rifs hetnrclen a1)strac.t C*-;rlgcl)ras and the c,oncrc'tc (i" -algebras of operators on Hilbert spaccs. Clraptcr G (which together with Chapter 7 rnakes Vol~irr~o 5) presents tht! tlic,ory of CP spaccs of operators, its application to the self achoirrt linear (ordinary and partial) diftercrrtial c'cl~~atiorrs,and the von N(:~imannillgcbras. Firrallj~:Chapter 7 present,s examples of C* algc1)ras tl(f~nc!dwith the, aid of gl.o~lps,in partic~rlarthe Clifford algebras. Many important domains of C* algr1)ras arc igrrored in the present 1)ook. It stro~lltlt ~ ecmpha.sized that t . h ~ whole theory is constr~lc~ted in parallcl for the real and for the c o n ~ l ~ l en~~rnt)rrs. x i.c. t,hc. C* algel~rasare real or complex.

In addition to the a1)ovt: (vcrtic:al) str~lcturcof t,ho t~ook:thcrc, is also i1 second (horizontal) division. It c,onsists of a main st.rand, eight l)ri~nc.h(,s. alltl ad(litionii1 material. The rcsnlts belonging to the main strand arc rnarked wit11 ( 0 ) . 1,ogically sprakirrg. a reader coultl rcst.rict hirnself/ht!rsclf t o thcsc anti igrrort, the rest. R c s ~ ~ lon t s the eight sl~bsitliarybranches are rnarkecl with ( 1 ) , (2). (3) . (4) . (5) . ( 6 ) . (7) , and (8) . The key is

Infinite Matrices Banach Categories Nl~rlcarMaps Locally Compact Groups Differential Equations Laurent Series Clifford Algebras Hilbert C*-Modules These arc (logically) independent of each other, hut all depend on the main strand. Finally, the results which belong to the additional material have no marking and from a logical perspective may he ignored. So the reader can shorten for himselflherself this very long book using the above marks. Also, si~lcethe proofs are given with almost all refererices, it is possible to get into the book a t any level and not to read it linearly. U'r assume that the reader is familiar with classical analysis and has rudimentary knowledge of set theory, linear algebra, point set topology, and integration t,hcory. The hook addrcssrs itself mainly to mathematicians, or to physicists interested in C* algebras. I would like to apologize for any omissions in citations occasioned by the fact that mv acquaintance with the history of functional analysis is, unfortunately, very ~estricted.For this history we recorrirr~endthe following texts. 1 . BIRKHOFF: G. and KREYSZIG, E.: The Establishment of Functional Analysis, Historia Mathcmatica 11 (1984), 238 321.

2. BOURBAKI, N., Elements of the History of Mathematics, (21. Topological Vector Spaces). Springer-Verlag (1994). 3. DIEUDONYB, .I.. History of Functional Analysis, North Holland (1981). 4. DIEUDONNE, J.: A Panorama of Pure Mathematics (Chapter C 111: Spectral Theory of Operators), Academic Press (1982).

5. HEUSER., H., Funktionalanalysis, 2. Auflage (Kapitel XIX: Ein Blick auf die wcrdcndc Rlnctionalanalysis), Tcubner (1986); 3. Auflagc (1992). G . KADISON, R.V., Operator Algebras, the First Forty Years, in: Proceedings of Syrriposia in Pure Mathe~natics38 1 (1982). 1-18.

7. MONNA, A.F., Fi~nctionalAnalysis in Historical Pcrspcctivc, .John WhiIcy k Sons (1973).

8. STEEN, L.A., Highlights in the History of Spectral Theory, Amer. Math. Monthly 80 (1973), 359 382. Thcrc is no shortagc of cxcellcnt books on C* algebras. Neverthclcss, we hope that this book will be also of some utility to the mathematics community.

This Page Intentionally Left Blank

VOLUME 4: HILBERT SPACES

This Page Intentionally Left Blank

5 . Hilbert Spaces

Most books on functional analysis treat the theory of Hilbert spaces before the theory of C*-algebras if they treat the latter at all. This is didactically justifiable since the former is simpler and is the main source of examples for the latter. But in a strictly logical sense, the theory of C*-algebras has priority. The development of this theory does not require even the definition of a Hilbert space. Conversely the theory of Hilbert spaces benefits from the theory of C*-algebras because the set of operators on a Hilbert space is a C*-algebra in a natural way. There is a didactic principle which says that in a book on mathematics the degree of difficulty should be roughly speaking an increasing function of the page number. While we do not wish to underestimate the importance of this principle, we have chosen in this instance to let logical priority be our guide and to rely on the reader not to completely ignore Hilbert spaces. This inversion of the customary order does have substantial benefits, for when dealing with the algebra of operators on a Hilbert space we may avail ourselves of all the results from the theory of C'-algebras.

5.1 Pre-Hilbert Spaces A number of general results, constructions, and examples from the theory of Hilbert spaces are pesented i r ~this first section.

5.1.1 General R e s u l t s Definition 5.1.1.1 ( 0 ) (von Neumann, 1928) Let E be a vector space. A scalar product on E is a positive sesquilinear form f on E (Propositzon 2.3.3.3 e)), such that for every x E E ,

4

5. Hilbert Spaces

A pre-Halbert space is a vector space endowed with a scalar product. A preHilbert space is called real (complex) if the ground field i n R (C). W e shall usually denote a scalar product i n a pre-Hzlbert space E by

Ex E

+M ,

( x ,y ) H

(XI

y)

(scalar product of x and y )

and for x E E , we define

J. von Neumann defined only spearable Hilbert spaces. Non separable Hilbert spaces wrre considered by H. Lowing (1934) and F. Rellich (1935). Proposition 5.1.1.2

(0)

If E is a pre-Hilbert space, then the m a p

zs a n o r m and

for all x , y E E . T h e above n o r m is called the canonical norm on the preHilbert space E . Unless otherwise stated, we always regard a preHzlbert space as endowed with zts canonical norm. A Halbert space is a complete pre-Hilbert space. A real (complex) Hilbert space is a complete real (complex) pre-Hilbert space. The assertion follo~vsimmediately from Proposition 2.3.3.9.

Corollary 5.1.1.3

(0)

If E is a pre-Hilbert space, then the m a p

is contznuous By Schwarz's Inequality!

for all ( x , y ) 6 E x E and the assertion now follows frorn Proposition 1.2.9.2 c + a.

5.1 Pre-Hzlbert Spaces

5

Corollary 5.1.1.4 ( 0 ) Let E , F be pre-Hilbert spaces and u : E -t F a bijective lznear map. u is a n isometry of the n o n e d spaces E and F iff

( ~ x I ~= Y( 5) 1 ~ ) for all x, y E E

.

The necessity follows from Propositiori 2.3.3.7 a =. b and Proposition 2.3.3.8

a

+ c & d . The sufficiency follows from ll~x11~ = (ux1ux) =

(215) =

llx112

for every x E E .

Corollary 5.1.1.5 Let E # (0) be a pre-Hilbert space. Take u E C ( E ) , a E K , and x E E # . If

then x

E

Ker (a1 - u ) and

CY

E up(u)

< IaI2 - 2 a h + llu112 = 0 (Corollary 2.3.3.4). Hence

x E Ker (a1 - u )

If u

# 0 , the11 a # 0 and x # 0 . Thus a E u p ( u ) .If u = 0 , then CY

= 0 E up(u).

Theorem 5.1.1.6 ( 0 ) (Jordan, von Neumann) Let E be a normed space. T h e following are equivalent: a ) There is a scalar product o n E generating the n o r m of E

5. Hilbert Spaces

6

If these conditions are fi~lfillled, then the scalar product i n a) is unique. The ~lniquencssof the scalar product follows from Proposition 2.3.3.7 a + b and Proposition 2.3.3.8 a + c & d . a + b & c follows from the parallelogram law (Proposition 2.3.3.2 b)). b + c (resp. c + b). Put

Then

1

2 (RSP. 5 ) j ( l l a + bl12 + lla - bl12) = 2(llxl12+ IIYII~). b&c+a. Case1

IK=R

Given x , y E E : put

I

=

I

1

+l 2

-

Ilx

-

Y112).

Then, for every x , y E E ,

( 4 ~= )( ~ 1 2 ) and

(ZIT)

= 11x1I2.

Thus only linearity in the first variable needs further proof. Step 1 Now

x,y, z

E

E

+ ( x + ylz) = ( x ( z )+ ( y ( z )

5.1 Pre-Hilbert Spaces

Step2

nEINU{O},x,y~E+(nx~y)=n(x~y)

The assertion follows from Step 1, by complete induction. Step 3

X,

yE E

* (-xly)

= -(xly)

By Step 1, 0 = (01~)= (x

-

~ I Y=) ( 4 ~ + ) (-xIy).

Hence (-XIY) = -(XI?/).

Step 4

(Y

E Q , s,y E E

There are nl E Z , n E IX with

* (crxly) = cr(x1y)

(1 =

F . By Steps 2 arid 3,

TL(CYXIY) = (nosly) = (mxly) = m(x1y).

Hencr 711

(QXIY)= -(zly) n

-(xly) = (-xlv)

=4x1~).

5 II - xll llvll = llxll llvll

by the triangle inequality and Step 3. Thus

5. Hzlbert Spaces

Step 6 Let

R ,x, y E E 3 (axly) = a(xly)

cr E

be a sequence in

& converging to a . Then

I ( a x l ~) (~nxIy)l= I ( @ x l ~+) (-anxly) = I((@- an)xly)I I

I II(Q - an)xlI l l ~ l l= la - a n 1 IIxII IIYII by Steps 3, 1, and 5. Thus

(axJy) = lim (ansly) = lim an(xly) = cr(xly) n+m

n-cc

by Step 4.

Case 2

IK = C

Giver1 x, y E E , define 1

re(xlv) := 4 (llx + y1I2 - llx

Step 1

xEE

-

* (xlx) = 1 1 ~ 1 1 ~

Now

re(xlx) = l l ~ l l ~ ,

yIl2)

>

5.1 Pre-Halbert Spaces

=

1 4

(/I

-

-

ix

+ Y112

-

llix

+ Y112)

=

-rc(ylix) = -im(?/lx) .

Hencc ( ~ I Y= ) (~14.

a , P E a , X, y , z E E * ( a x + P y l z )

Step 3

= ~ ( x l +P(YIz) ~ )

By Case 1, (ax + Pylz) = re(ax

Step 4

+ PyIz) + i r e ( a x + P Y ~ ~=z )

x, y E E =+ (ixly) = i(x1y)

By Steps 2 and 3,

irn(ix1y) = -irn(yliz) = -re(yl

-

x) = -re(-x(y) = re(x1y).

Hencc (ixly) = re(ix1y)

Step 5

+ iirn(ix1y) = -irn(xly) + zrc(x1y) =

a , 0 E c , x, y, z E E 3 (xlay

+ Pz) = 6(xIy) + D(x)z).

By Steps 2: 3, and 4, (slay + 02) = (NY+ PzIx) = ~ ( Y I x+) P(zIx) = 6 (~1s+ )

=

p ( 2 1 ~ )= ~ ( X I Y+)~ ( x I Y ) .

Rerr~nrk. This theorem makes it possible to define pre-Hilbert (Hilbert) spaces as norrncd (Banach) spaces satisfying the parallelogram law.

5. Halbert Spaces

10

-

Corollary 5.1.1.7 ( 0 ) Let E be a pre-Hilbert space and E the completion of the associated normed space. There is a unique scalar product o n E generatzng the n o r m of E . E with this scalar product is called the completion of E . The scalar product of E is the restrzction to E of the scalar product of

E.

-

Take x , y E E . There are sequences ( x , ) , , ~ , ( Y ~ in ) E ~converging ~ ~ to x and y , respectively. By the parallelogram law (Proposition 2.3.3.2 b ) ) ,

+

) ) x y)I2 + ) ) x- y ) J 2= lim J J x n+ yn(I2+ lim IJxn - YnI12 = n+w

n+co

By Theorem 5.1.1.6, there is a unique scalar product on E generating its norm. The restriction to E of this scalar product generates the norm of E . Hence it coincides with the original scalar product of E (Theorem 5.1.1.6). Corollary 5.1.1.8 Let E , F be pre-Hilbert spaces and take u E L ( E ,F ) . T h e n the following are equivalent:

a ) lluxll = llxll for every x E E

b) ( u z l u y ) = ( x J y )for all x , y E E . a

* b.

The map E+ImE,

x-us

is an isometry and the assertion follows from Theorem 5.1.1.6. b a is trivial.

*

rn

Proposition 5.1.1.9 ( 0 ) Let E be a real (complez) pre-Hilbert space, f a sesquzlinear form o n E , and take a E IR+ . If

for every x E E , then

(IS(X,Y)I

for all x , y E E .

+ I ~ ( Y , X ) 1I 2 4 1 x 1 1I I Y I I )

5.1 Pre-Hilbert Spaces

We may assume without loss of generality that x

11

# 0 and y # 0 . Consider

By Proposition 2.3.3.2 b),c),

5

1

( I+ I

+I

( - Y)1)

<

4

+ v1I2 + 112 - vll2) = 01 (11x1I2+ llvl12) .

(111

Thus

Suppose IK = C . Take y,6 E IR with

f

( x ,Y ) = eZr7lf( x ,Y ) I ,

f ( y , x ) = e2'*1f(Y?x)l Then

I ~ ( Yx) ,I + I ~ ( YX ,) I =

Y ) + e - 2 f 6 f (X~) I1=

Proposition 5.1.1.10 Let E be a pre-Hzlbert space and take x E E . Let ( x , ) , , ~ be a sequence in E . The following are equivalent: a) b)

lim x, = x .

n+w

lim (xnlx)= nlim +m

n+m

1 1 ~ ~ =1 1 1 ~1 ~ 1 1 ~

5. Hilbert Spaces

a + b is trivial. b + a . By Corollary 2.3.3.4,

for every n

E

IN. Hence lim llxn - xll = 0 ,

n+m

Proposition 5.1.1.11 Put:

(0)

Then a o b o c + d o e . %

b . By Corollary 2.3.3.4,

from which the assertion follows. b + c . By Corollary 2.3.3.4,

Hence

c e

rn

Let E be a pre-Hilbert space and take x, y E E.

d ) :w x and y are lznearly dependent.

a

lim xn = x .

n+m

+ a & d and d + e are trivial. + d . Take 0 E IR with

5.1 Pre-Hilbert Spaces

By b + c , x and eZeyare linearly dependent. Hence x and y are lineraly dependent.

Remark. a ) It is easy to see that d) does not irnply b) (take y b) In e l ,

:= - x ) .

and so a ) does not imply c) in the case of arbitrary Banach spaces.

14

5. Hilbert Spaces

5.1.2 Examples Example 5.1.2.1

(0)

Let ( T , I , p ) be a measure space. The m a p

is a scalar product. This scalar product renders L Z ( p ) a Hilbert space with n o r m

By taking p t o be the counting measure in the above example, we get: Example 5.1.2.2

(0)

Let T be a set. T h e map

is a scalar product, which renders e2(T)a Hilbert space with n o r m

Example 5.1.2.3

The m a p

is a scalar product. e2 endowed with this scalar product is a Hilbert space (it is called the Hilbert space of square summable sequences). Its n o r m is

Example 5.1.2.4

Take n E I N . The map

is a scalar product (it is called the canonical scalar product of generates the Euclidean norm.

IKn). It

rn

Example 5.1.2.5 Let T be a locally compact group, M b ( T ) the involutive unital Banach algebra defined i n Proposition 2.3.2.28 a ) , X a left invariant Haar measure o n T , and take p E M b ( T ). For x , y E L Z ( X ), define

5.1 PreeHzlbert Spaces

(Proposition 2.2.2.14 a)), and

T h e n f is a scalar product i f l up(u) c 10, oo[. This condition is fulfilled whenever fi is invertible. By Proposition 2.3.3.10, f is a positive sesquilinear form and by Example 3.1.4.14, up(u) c R+. If 0 E up(u), then ux = 0 for some x E L2(X)\{O). Thus

f (x, x) = (uxlx) = 0 and f is not a scalar product. Assume that 0 !$ up(u). Take x E L2(X) with

f (x, x) = 0 Then 0 = (uxlx) = (p'

* p *XIS) = (p * x J p* x)

(Proposition 2.3.2.28 c), Example 2.3.2.27 b)). Thus

Hence f is a scalar product. If p is invertible, the11 p* is ir~vertible(Proposition 2.3.1.14), p * * p is invertible, and u is invertible (Proposition 2.2.2.14 e)). Hence 0 !$ up(u). W

Example 5.1.2.6 Take p E [l,co]\{2) and let T be a set. If the norm of P ( T ) is generated b y a scalar product then C a r d T 5 1 .

5. Hilbert Spaces

Assume C a r d T

> 1 . Let s , t be two distinct elements of T . Then

By the parallelogram law (Proposition 2.3.3.2 b)), the norm of P ( T ) is not generated by a scalar product.

Example 5.1.2.7 Let n E IN and take a E ReIK,,, . The map

is a scalar product iff u(a) c 10, m[ and e v e y scalar product on IKn is of this form. By Example 2.3.3.6, f is a positive sesquilinear form iff u(a) C Assume that 0 E ~ ( ( 1 )Then . there is an x E IKn\{O) with

R+.

and so

Hence f is not a scalar product. Now assume that 0 $ o ( a ) . Take x E IKn with

(1 . 11 is the Euclidean norm. Then, by Lagrange's Theorem, there is an a E IR with

where

for every i E N n . Thus a E u(a) and

It follows that

5.1 Pre-Hilbert Spaces

for every y E IKn\{O). Hence f is a scalar product. Let g be a scalar product on IKn . Given i , j E K, , put

4, := g(e,, e,) Then

for all x, y E IKn , i.e. g is of the above form.

Example 5.1.2.8 Every infinite-dimensional vector space E admits two scalar products, which generate non-equivalent norms. Let ( x , ) , be ~ ~an algebraic basis of E and f a strictly positive real function on I such that inf f ( ~ =) 0 , LEI

sup f ( ~ =) oc LEI

Then

are two scalar products on E which generate non-equivalent norms.

Proposition 5.1.2.9 ( 0 ) Let E be a vector space and f a positive sesquilinear form on E . Define

b)

F is a vector subspace of E . W e write u : E + E I F for the quotient map.

c) There is a unique map

such that

for every x , y E E

5. Halbert Spaces

18

d ) g is a scalar product, called the scalar product associated to f a) is a consequence of Schwarz's Inequality (Proposition 2.3.3.9). b) follows from a ) and Corollary 2.3.3.4. c) Take X , Y E E I F and x 1 , x 2 X~ , y , , y , ~ Y . T h e n

so that

by a ) . This proves the existence of g . The uniqueness is trivial. d ) is easy to check.

Example 5.1.2.10

The map

is a n injective homomorphism of unital real algebras. Identifyzng G with its image wzth respect to the above map, M becomes a two-dimensional complex vector space and the map

((aP , , y , 6 ) , (a',P', r ' , 6 ' ) )

* ( a + Pi)(crt - P'i) + (Y + 6 i ) ( ~-' b'i)

is a scalar product generating the euclidean n o r m o n M The proof is a straightforward verification.

5.1 Pre-Hilbert Spaces

5.1.3 Hilbert sums

(0)

Proposition 5.1.3.1 and put

O E := ~

LEI

Let ( E L ) r EbeI a family of pre-Hilbert spaces

I I X

E L E~ I

a) @EL is a vector subspace of LEI

, ~L El I l

E

1

xLl12<m

fl E,

.

rEI

b ) The map

(

) ( x

LEI

)

, ( x . Y ) H~

+

(

X

~

I

Y

~

)

LEI

is a scalar product with associated norm @EL + R+

x ++

1

LEI

(x

1

llxLl12)5

LEI

a)E, with this scalar product is called the Hilbert

sum of the family

LEI

(ELILEI.

c) If each

EL ( L

I) is complete then @EL is also complete.

E

LEI

d ) If each EL ( L

I) is separable and if I is countable, then a)E, is also

E

LEI

separable.

a) Take x , y E @EL and a, P E IK . Then LEI

(

(cIx

+

Py!Ll12)

LEI

Hence ax + fly

E

=

(x

lax

+

. Y L 2

LEI

@ E L ,i.e. a)E, is a vector subspace of LEI

LEI

n EL

LEI

5. Hilbert Spaces

b) Take

x ,

y , 2 E @EL and a , /J E IK . By Schwarz's Inequality, LEI

C

5

I(X~IY~)I

LEI

x

I I X ~ II II

Y ~ III

LEI

LE I

LEI

We have

+

= C ( a ( x L 1 z L ) /J(y,lzL))= a x ( x L l z L + ) LEI

LEI

LEI

LEI

x ( x , l x L )= LEI

PC(Y~I~L), LEI

LEI

E1 1 ~ ~ 1 1IR+~ E

I

LEI

and the assertion follows C) Let ( x , ) , , ~ be a Cauchy sequence in

@EL.For every

E

> 0 , there is

LEI

an n, E IN such that

>

for all m, n E N with m 2 n, , n n, . It follows that for every L E I , ( X , ( L ) ) , ~ N is a Cauchy sequence, and hence a convergent sequence. Define

Then

for every

E

> 0 and n

E IN, n

2 n, . Hence

5.1 Pre-Hilbert Spaces

and lim x , = x .

n+m

d) Given

L

E I , let A, be a countable dense set of El with 0 6 A , . Define

A

:=

in

A, ( { L

x 6

E I ( x , # 0) is finite}

'€1

It is easy to see that A is a countable dense set of @EL. LEI

Remark. If EL= IK for every

L

E I , then

Corollary 5.1.3.2 If (EL)LEI is a finite family ofpreeHilbert spaces, then the EL is generated by the scalar product Euclidean n o r m on

n

LEI

If each EL ( L E I ) is a Hilbert space, then

n E, is also a Hilbert space.

LEI

.

Proposition 5.1.3.3 ( 0 ) Let (El),,, , (F,),,, be two families of preHilbert spaces and take ( z L , ) ,E~ I C(EL,FL)wzth

n

LEI

W e define

5. Hzlbert Spaces

which proves a). Since O u , is linear we deduce further that LEI

Take X E I and let

XA E

E,# . Define

Then x E a)El and LEI

so that

Since

XA

is arbitrary,

Since X is also arbitrary,

and

Proposition 5.1.3.4 ( 0 ) Let ( E l ) , E I (FL)lEI, , and ( G L ) l Ebe~ families of pre-Hilbert spaces. Take a , E IK ,

5.1 Pre-Hilbert Spaces

(wL)LEI E

n

C(FL,G L )

LEI

with

sup l l ~ l l l< m, LEI

SUP I I v L I I LEI

<m

7

SUP llwcll LEI

Then

o .L, LEI

These relations are easy to check

= (ow,) LEI

0

(mu.) LEI

< m.

24

5. Halbert Spaces

5.2 Orthogonal Projections of Hilbert Space If F is a closed vector subspace of a Hilbert space, then we can define an orthogonal projection onto F , just as in the case of Euclidean spaces. This projection is used to prove the Frkchet-Riesz Theorem that every continuoris linear form on a Hilbert space can be written as (.lx) , with Il(.lx)ll = llxll . Thus the dual of a Hilbert space may be identified with the Hilbert space itself, whereby the isometry in question is a conjugate linear one.

5.2.1 Projections onto Convex Sets Proposition 5.2.1.1 convex set of E . a) y, 2 E A

* Ily

(0)

- 2112

Let E be a pre-Hilbert space, x E E , and A a

< 2\12 - ~ 1 1 ' + 2115 -

211' -

4d~(~)'

b) Every sequence ( x , ) , ~i n~ A with lim JIx - x,)) = dA(x)

n+m

is a Cauchy sequence. If it converges,then

a) i(y

+ 2)

belongs to A . By the parallelogram law (Proposition 2.3.3.2

b)),

5 2llx - y1I2 + 2112 - z\I2- 4 d ~ ( x ) ' b) follows from a ) .

Theorem 5.2.1.2 ( 0 ) (0.Nikodyrn, 1931; F. Riesz, 1934) Let E be a pre-Hilbert space. Take x E E and let A be a nonempty convex set of E . If

25

5.2 Orthogonal Projections of Hilbert Space

A is conlplete with respect to the induced metric, then there is a unique y

E

A

such that

y is characterized by the property z EA

r e ( x - yly - z ) 1 0

W e define

Step 1

Uniqueness

Take y, z E A with 112

- yll = llx - zll = d a ( x ) .

B y Proposition 5.2.1.1 a ) , Ily

-

z1I2

< 2112 - yll + 2112 - zll

-

4da(z) = 0

Hence

Step 2

Existence

Let ( x , ) , , ~ b e a sequence in A with lim Ilx, - xi1 = d ~ ( x ) .

n+m

By Proposition 5.2.1.1 b ) , ( x , ) , ~is~ a Cauchy sequence in A . Since A is ~ to some y E A . By Proposition 5.2.1.1 b ) , complete, ( x , ) , ~converges

Step 3

y, z E A , Ilx - yll = d A ( x ) + re(x - yly - z )

+ ( 1 - o ) y E A and so; by Corollary 2.3.3.4, YI12 = d ~ ( x<) IIx~ - ( o z + (1 - cr)Y)112=

Take o ~ ] 0 , l [T.h e n o z IIx

-

>0

5. Hilbert Spaces

Since a is arbitrary, re(x - yly - 2 ) Step 4

y E A,

(2

E

A

>0

+ re(x - yly - 2 ) > 0 ) 3 113: - yll

=~ A ( x )

By Corollary 2.3.3.4,

I"

for every

2

- 2112 = II(z

-

Y) + (Y - 2)112 =

E A . Hence -

yll = ~

A ( x )

Remark. The above theorem does not hold for ge,~eralBanach spaces. Let A be the closed convex hull of {%en In E I N ) in t " . Then d A ( 0 )= 1 but

11x11 > 1 for every x 6 A .

Proposition 5.2.1.3 ( 0 ) Let E be a pre-Hilbert space and A a nonempty convex set of E which is complete with respect to the induced metric.

b)

KA

is uniformly contznuous.

a ) By Theorem 5.2.1.2,

5.2 Orthogonal Projections of Hzlbert Space

re(y - ~

> 0,

IKA(Y)

-~

I~A(x)

-~ A ( Y ) )

A ( Y )

A ( X ) )

It follows t h a t ~ ~ ( T A (Y )Y

2 0,

and so

b) follows from a ) . C) T A ( X )

E A,

so that K A o K A ( x ) = T A ( K A ( 5 ) )= K A ( x )

for every x E E . Hence 71.4 0 T A

=7 r ~ .

Proposition 5.2.1.4 Let E be a pre-Hilbert space, Q a nonempty downward (upward) directed set of nonempty convex sets of E which are complete i n the induced metric, 5 the lower (upper) section filter of Q , and put

If B is nonempty and complete i n the induced metric, then lim r A( x ) = K B ( x ) A,$

for every x E E .

5. Hilbert Spaces

We first remark that B is convex. Consider

(p:%--+E,

A++TA(X).

We have llv(A) - cp(~')Il' = IITA(x)- ~ A ~ ( x ) III '

for all A, A' E 2l with A' c A (Proposition 5.2.1.1 a)). Sirice (dA(~))AEll is an increasing bounded (decreasing) family in R + ,it follows that p(3) is a Cauchy filter. Hence l i r n a ~ ( x )exists and belongs to B . Now A,3

<

dB(x) 5 llx - l i r n ~ ~ ( x ) I I lirn llx - rA(x)ll= limdA(X) I dg(x) , A,$ .4,3 43

x

-

1

lirn r A ( x ) = lirn Ilx - T A ( x )=~ l~i r n d ~ ( x = ) ~B(x) A-3 A,3

so that lirn r A ( x )= XB(X), A.3 by Theorem 5.2.1.2.

5.2 Orthogonal Projections of Hzlbert Space

5.2.2 Orthogonality

Definition 5.2.2.1 ( 0 ) Let E be a yre-Hilbert space. Given x, y E E and subsets A, B of E , define xly

:@

(xly) = O

(x and y are orthogonal),

A I B :e((2,y) E A x B + s l y ) (A and B are orthogonal), AL

:= { X E

E 1y E A

Proposition 5.2.2.2 A, B subsets of E . a) A'

*xly}

(0)

(the orthogonal set of A )

Let E be a pre-Hilbert space. Take x E E and

is a closed vector subspace of E

a) Given x E A , define

By Corollary 5.1.1.3, Z

E' for every x E A . Since

it follows that A' is a closed vector subspace of E . b) and c) follow from the definition. d) By b) and c),

30

5. Hilbert Spaces

Proposition 5.2.2.3 ( 0 ) (Pythagoras' Theorem) Let E be a pre-Hilbert space and ( x , ) , ~a~finite family of pairwise orthogonal elements of E . T h e n

LEI

LEI

First consider I = { l , 2 ) . By Corollary 2.3.3.4,

Now let I be arbitrary. We prove the relation by complete induction with respect to Card I . Take X E I and put J := I\(X). By Proposition 5.2.2.2 a ) ,

Hence, by the above proof and inductive hypothesis,

Corollary 5.2.2.4 ( 0 ) Let E be a Hilbert space and ( x , ) , ~a~family of pairwise orthogonal elements of E . Then (x,),,~ is summable iff ( / 1 ~ , 1 ) ~ )is, ~ 1 summable and i n this case

By Pythagoras' Theorem,

for every J E g f ( I ) , and the assertion now follows from Proposition 1.1.6.6.

Corollary 5.2.2.5

(0)

Let E be a Hilberl space and A a subset of E ,

such that given x, y E A ,

(xly) = a*,, . Take f E IK* . T h e n ( f ( z ) ~ ) , , ~is svmmable iff f E e2(A) and in this case

5.2 Orthogonal Projections of Halbert Space

Corollary 5.2.2.6 such that

(5)

31

Let E be a Hilbert space and A a subset of E

for all x , y E A . Take f E 1 2 ( A ) .Let further T be a topological space and take g E K T x Asuch that g(., x ) is continuous and

for every x E A . T h e n the map

(Corollary 5.2.2.5) is continuous Take to E T and

E

> 0 . There is a B E ?,(A) suc11 that

Further there is a neighbourhood U of t o such that for each t E U ,

It follows from Corollary 5.2.2.5, that

for every t E U .

Corollary 5.2.2.7 Let E be a pre-Hilbert space and ( X ~ ) , , a, family of pairwise orthogonal elements of E . Take x E E such that

for every y E E . Then ( x , ) , is ~ ~summable and

5. Hilbert Spaces

We rnay assurne that E is complete (Corollary 5.1.1.7). Given

and so

By Corollary 5.2.2.4, ( x , ) , ~is~summable. Since

for every y

E, x - xL E I' 2 =

-xx. LEI

E x , LEI

= 2 .

IX.XL) LEI

=

L

E I,

5.2 Orthogonal Projections of Hilbert Space

5.2.3 Orthogonal Projections

Proposition 5.2.3.1 ( 0 ) Let E be a pre-Hilbert space, F a complete subspace of E and take (x, y) E E x F . Then the following are equivalent:

If these conditions hold, then

a 3 b . Take z E F and cw E I K . Since y - cwz E F , it follows from Theorem 5.2.1.2 that

0

I re(x - yly - (y - a z ) ) = re(x - ylcwz) = reti(x - ylz)

If we set a := -(x - ~ I z ) then we get successively 0

I -I(x

-

ylz)I2,

( x - ylz) = 0 ,

X-YE

FI.

b + a . Take z ~ F . T h e ny - ~ E Fa n d s o

and Y = 71F(x) , by Theorem 5.2.1.2. Now we prove the last assertion. By b) and Pythagoras' Theorem (Proposition 5.2.2.3),

5. Hilbert Spaces

34

Corollary 5.2.3.2 ( 0 ) If E is a pre-Hilbert space and F a complete subspace of E , then n~ is a projection of E onto F with IlnFII 5 1 . It is called the orthogonal projection of E onto F .

Take x, y E E and a , /3 E lK . By Proposition 5.2.3.1 a -

+b,

~ F ( x ) Y, - ~ F ( YE) F'

It follows that = ~ ( -xTF(x)) + P(Y - TF(Y)) F L (02 + /3y) - (QxF(x) + /~TF(Y))

(Proposition 5.2.2.2 a)). Thus

by Proposition 5.2.3.1 b + a . Hence KF is linear. By Proposition 5.2.1.3 a), c), TF is a projection of E onto F with l l ~ ~ 5l l 1 . Remark. We show below (Theorem 5.3.1.13) that, if E is a Hilbert space, then t ( E ) is in a natural way a C*-algebra. In Proposition 5.3.2.8, we show further that the above notion of "orthogonal projection" coincides with the one introduced in Definition 4.1.2.18. Proposition 5.2.3.3 ( 0 ) Let E be a n o n e d space, F a pre-Hilbert space, and u : E + F a compact operator of the underlying real normed spaces. Let 6 be a n upward directed set of complete subspaces of F the union of which is dense i n F and let 3 be the upper section filter of 6 . T h e n

lirnnGou = u G.8 Put

K := u ( E # ) and given G E 6 , define

Then fc is a downward directed family of continuous functions with infimum 0 . Since K is compact, (fC)GECI converges to 0 uniformly (Dini's Theorem). Hence

1 1 U~ - ~uII 0

= SUP

I ~ K ~ ( u-x U)Z I ~

rEE#

for every G E 6 and the assertion now follows.

=

sup fc(y) YEK

5.2 Orthogonal Projections of Hilbert Space

Corollary 5.2.3.4 then

(0)

35

If E is a normed space and F a Hilbert space,

C j ( E ,F ) = K ( E ,F ) . By Proposition 3.1.1.4,

C f ( E ,F )

c K ( E ,F )

Take u E K ( E , F ) and let 6 be the set of finite-dimensional subspaces of F . Then 6 is upward directed and

By Proposition 5.2.3.3

where 5 denotes the upper section filter of 6 . Since for every G E 6 , ac 0 u belongs to L j ( E ,F ) ,

K ( E ,F ) = C j ( E ,F ) . Proposition 5.2.3.5 ( 0 ) Let E be a pre-Hilbert space and F a complete subspace of E . Let u be a projection of E onto F . Then the following are equivalent:

b) (uxlx) E

R+ for

c) Ker u

c F'

d ) Ker u

=

F'

every x E E

5. Halbert Spaces

a b

* b follows from Proposition 5.2.3.1. +c.

Take x L-E Keru and y E F . Then

o I(u(x + ay)lx + a y ) = (aylx + a y ) = ~ ( Y I X+) a21I~1l2 for every a E IR, and so

Keru c

c F'.

* d . Take x E F L . Then ( U X ~ X=) 0 , u x - X E K e r u c F'.

We deduce, successively, that 11~211' =

(uxIux)

-

(uxIx) = (uxIux

-

2)

= 0,

x E Keru,

FL d + e . Take

X E

c Keru.

E.Then x - ux E Keru

=

FL.

It follows that (x - us) l u x ,

11~1= 1 ~ 11(x - ux) + ~ (Pythagora' Theorem). Hence

~ = 111 2 - 1ux11' ~ + l l ~ x 1 1>~ llux11'

5.2 Orthogonal Projections of Hilbert Space

e from

+ f.

Assume that u

# 0 . Then F # ( 0 ) . Take x

I I X I I=

E F\{O). It follows

11~x11 I I I U I I11x11,

that

u(x-ux+(Yy)=ux-u2x+(Yuy=(Yy,

so that J C X ) ~ ) )= ~ ) ll0yJ1~ ) ~ =

JIu(x - ux

+ ( Y Y ) )<) ~l l ~ 1 1 ~ 1 1 2-- ux +

~ + 2reh(x 1 1 - uxly) ~ + la1211y(12

5 llx - ~

(Corollary 2.3.3.4). If we define (Y

where

:= p(x

-

ux1y),

p E IR, then 2pl(x

+ 112 - 11x11~> 0 .

-

Since /3 is arbitrary, we deduce that (x - uxlv) = 0 , and so

UX

(Proposition 5.2.3.1 b

3

= KFX

a). Hence u = a~ .

I

(YY))~

5. Halbert Spaces

38

Corollary 5.2.3.6 ( 0 ) Let E be a pre- Hilbert space and F, G, H complete subspaces of E . If XFXC is a projection of E onto H , then

By Corollary 5.2.3.2,

and the assertion follows now from Proposition 5.2.3.5 e

Proposition 5.2.3.7 subspace of E , then

(0)

+ a.

H

Let E be a pre-Hilbert space. If F is a complete

and the map

is an isometry. If E is also complete then

By Corollary 5.2.3.2, 5.2.3.5 a + d ,

KF

is a projection of E onto F and by Proposition

Ker T,V = F' By Murray's Theorem (Theorem 1.2.5.8 b

+ a),

E = F ~ F ~ . By Pythagoras' Theorem, the map FQF'+E, is an isometry. \?ie have F c F" (y, z ) E F x F L with

(X,Y)MX+Y

(Proposition 5.2.2.2 b)). Take x E F L L . There is a pair

z=y+z.

From

5.2 Orthogonal Projections of Hilbert Space

(Proposition 5.2.2.2 a)) we deduce that z = 0 . (Proposition 5.2.2.2 e)), and

Hence F L L c F , F" = F . By Murray's Theorem (Theorem 1.2.5.8) and the above considerations, l E - T,V is a projection of E onto F' with Ker ( I E - xF) = F Thus, by Proposition 5.2.3.5 d

=F ~ L

+a,

Hence 1~ = TF + K P L

Corollary 5.2.3.8 ( 0 a complemented subspace.

)

Every closed vector subspace of a Hilbert space is

Corollary 5.2.3.9 ( 0 ) Let E be a pre-Hilbert space and A a subset of E . If E is complete or if A is finite, then -4'' is the closed vector subspace of E generated by A . Let F be the closed vector subspace of E generated by A . Since A L L is a closed vector subspacc of E containing A (Proposition 5.2.2.2 a),b)), F cA~L. If E is complete, then F is complete as well. If A is finite, then F is finitedimensional and therefore complete. Take x E A L L . By Proposition 5.2.3.7, there is a pair ( y , z ) E F x F L with

By the above considerations, y E A". deduce further

By Proposition 5.2.2.2 a),b),e), we

5. Hilbert Spaces

Hence

Remark. The hypothesis " E complete or A finite" cannot be relinquished, as the following example shows. Put

and let E denote the vector subspace of C2 generated by A U {x} . Then A L = (0) , A L L = E , and x does not belong to the closed vector subspace of E generated by A . Corollary 5.2.3.10 ( 0 ) Let E be a Hilbert space and F a vector subspace of E . T h e n the following are equivalent: a) F = E .

From FLL=

F

(Corollary 5.2.3.9) we deduce that

E=FICBF (Proposition 5.2.3.7) and the assertion now follows. Definition 5.2.3.11 ( 0 ) Let E be a Hilbert space and take u E L ( E ) . A closed vector subspace F of E is said to reduce u if F and F L are U znvariant. Corollary 5.2.3.12 ( 0 ) Let E be a Hilbert space, F a closed vector subspace of E , and take u E L ( E ) .

5.2 Orthogonal P~ojectzonsof Hzlbert Space

a ) F is u-invariant iff

b) F reduces u iff

a) is easy to see. b) If F reduces u , then by a ) and Proposition 5.2.3.7,

Suppose that RFU

= UXF .

Take x E F and y E F' . Then

Ill;uy = u7rf.y = 0 (Proposition 5.2.3.5 a =+ d) , Uy E KernF = F' (Proposition 5.2.3.5 a =+ d ) , so that u(F)c F,

u ( F L ) C F1

Thus F reduces u . Proposition 5.2.3.13 ( 0 ) Let F, G be closed vector subspaces of the Hilbert space E . Then the following are equivalent:

5. Hilbert Spaces

42

f)

KG

- KF = 7 T c " p l .

g)

KG

- AF is a projection in E .

a @ b is trivial. a&b+c.Byb),

i.e. KFAG is a projection in E . By a), it is a projection onto F and so the assertion follows from Corollary 5.2.3.6. c + d . By Corollary 5.2.3.5 a + e ,

d

@

e . By Proposition 5.2.3.1,

= (ACXIX). I I ~ F X =~ ~( ~~ F x ~ x )llrcxll2 ,

d

=+

a . Take x E F . By Proposition 5.2.3.1,

1 1 ~ 1 =1 ~ l l ~ ~ xIl IlI ~~ G x I I ~ = 1 1 ~ 1 -1 ~ 115 - ~cx1I25 1 1 ~ 1 1 ~ Thus

and

FCC. a & b & c & e + f . Wehave (KG

i.e.

KG - K F

= K$

-

- AGKF

- AFKG

+

A;

= AG

- KF

is a projection in E . Take x E E . Then X - A G X E G ~ C

(Proposition 5.2.3.1 a

3

F ~ X , -AFXEFI

b). Hence

X G X - KFX

= (x

(Proposition 5.2.2.2 a)). Thus

-

rFx) -

(5 - KGX)

E

F'

,

5.2 Orthogonal Projections of Hilbert Space

Im(Tc-TF)

cGnF1

Since

whenever x E G n F L (Corollary 5.2.3.5 a =. d), it follows that projection of E onto G n F L . By e),

for every x E E . By Corollary 5.2.3.5 b

TG - x p

is a

+a,

f + g is trivial. g + a . Sirice

it follows that 1 T ~ T+ G r G n F ). 2

TF = - (

Take x E F . Then

IIxII

=

1

+ TGTFXI I

I I K ~ ~ l =l -I~TFTCX 2 I

(Corollary 5.2.3.5 a

3

1

1

I -2 (IITFRGXII+ IIT~XII)I

(II~TG~II + II~cxII)= ll~GxI1

e). Thus

(Proposition 5.2.3.1)

FcG. Proposition 5.2.3.14 ( 0 ) Let F, G be closed vector subspaces of a Hilbert space E and H := F n G . T h e n the following are equivalent:

44

5. Hzlbert Spaces

C)

7iGTF

= TH.

d ) (Fn H L ) 1 ( G n H L ) .

e) ( F n H L ) 1 G .

f) F = H @ ( F n G L ) . If these conditzons hold, then F + G is closed and

a

+b

b

3

TCTF

is trivial. c . Since

is a projection in E . Since X E

E===+K~T~X=KF ET FG nT G F= XH , XE

TGTF

H==+T~T~x=x,

is a projection onto H . By Corollary 5.2.3.6, TGTF

c

+d.

Take

x E

(Proposition 3.2.3.5 a

F n H I . Then

+ d), so that X E

(Proposition 5.2.3.5 a

= TH

KerrccGL c (GnHL)L

+ c). Hence (Fn H L ) 1 ( Gn H I ) .

d

+ e.

Take x E G . There is a (y, z ) E H x H L with x=y+z

(Proposition 5.2.3.7). By d ) ,

5.2 Orthogonal Projections of Halbert Space

Z = X - ~ E G ~ H ' C ( F ~ H ~ ) ~ . From F n H L c HL it follows that H C H L L c( F ~ H ' ) ' , x = ~ + z E ( F ~ H ~ ) ' (Proposition 5.2.2.2 a),b),c)) and so (Fn H L ) 1 G . e + f . Bye): FnH'c

F~G'.

Since

G' c

HI,

it follows that FnHL=FnGL. Take x E F . There is a unique (y, z ) E H x H L with x=y+z (Proposition 5.2.3.7). Now Z = X - Y E

F~H'=F~G',

so that

(Corollary 1.4.2.6). f + d . Take x E F n H I . By f), there is a pair (y,z) E H x (F n GL) with

46

5. Hilbert Spaces

From

it follows that y = 0 and

i.e.

( F n H1)1(GnH I ) . f

3

a . Take x E E . By f ) , there is a pair ( y , z ) E H x ( F n G L ) with T F X = Y + Z .

Then KGTFX

=Y=THTFX =THX

(Proposition 5.2.3.5 a =. d, Proposition 5.2.3.13 a TGTF = KH

+ c ) , i.e.

.

By d o f and by the symmetry of d), we deduce from the above equality that

TFTG

=T

~ K F .

We now prove the final assertion. Put := K F + K G - K f j .

U

By a),c), and Proposition 5.2.3.13 a 3 b & c, UTF =TF

,

UKc

= TG

,

UTH

= Tfj

.

Thus u2=u,

i.e. u is a projection in E . Take x r F

+ G . There is a

( y , z ) E F x G with

5.2 Orthogonal Projections of Halbert Space

so that

since the reverse inclusion is obvious, u is a projection of E onto F + G . Hence F + G is closed. Given x E E ,

(Proposition 5.2.3.13 a

+ e), so 71

(Proposition 5.2.3.3 b

= TF+G

+ a).

Corollary 5.2.3.15 ( 0 ) Let F , G be closed vector subspaces of the Hilbert space E . Then the following ale equivalent: a) F I G

b)

7 r F 7 r ~=

c) F

+G

0.

is closed and

a e b follows from Proposition 5.2.3.5 a b + c . Clearly, T ~ K F T G=

By Proposition 5.2.3.14, F

+d

0 =7l~Tc.

+ G is closed and

5. Hilbert Spaces

Thus

(Proposition 5.2.3.5 a

+ c) and FIG.

Corollary 5.2.3.16 Let space and

K

F be afinite set of orthogonal projections i n a Hilbert

is an orthogonal projectzon iff

whenever u , v E 3 are dzstinct. In this case,

The sufficiency of the condition and the last assertion follow from Corollary 5.2.3.15 b + c by complete induction. We now assume that K is an orthogonal projection. Let u , v be two distinct elements of 3 and x E 11nu . Then

1 1 ~ 1 >1 ~ (KxIx) = ~

>

+ 1 1 " J ~ 1 1 ~= 1 1 ~ 1 +1 ~ l l v ~ 1 1 ~

l l ~ X 1 1 ~ llu~11~

( W X I X= )

we3

we3

(Proposition 5.2.3.1), so

It follows vu=o. Proposition 5.2.3.17 Let F be a closed vector subspace of the Hilbert space E and take u E C ( E ). T h e n the following are equivalent:

a ) F L c K e r v and I m u c F .

5.2 Orthogonal Projections of Hilbert Space

a

+ b . It follows immediately from Imuc F,

that 7 r ~ U= 21.

Take x E E . Then

x - 7rpx E F J (Proposition 5.2.3.1 a

b

+a

* b), so that

is obvious.

Proposition 5.2.3.18 Let E be a Hilbert space, F a vector subspace of E , and G a Banach space. Take u L(F, G ) . T h e n there is a v E L ( E : G) such that

Since u is uniformly continuous, we may assume that F is closed. By Corollary 5.2.3.2, the map

has the required properties. Proposition 5.2.3.19 Let

(0)

0 := inf LEI

and take

< E t 2 ( I ) such that

Let ( x , ) , ~ , be a family i n the Hilbert space E .

llxlll,

P := sup llxLll< 00, LEI

5. Hzlbert Spaces

50

for distinct i n E and

1,

XE

I .

Then, given 77 E e 2 ( I ) , the family

Let J be a finite subset of

I .

( ~ ( L ) X , ) , , is ~

summable

Then

It follows t h a t

112v(l)xL1l'x 5

LEJ

Hence

( ~ ( L ) X , ) is , ~ summable ~

+

x

5

~ ~ ( 1 ) ~ 2 ~ ~ l x( ~~ ( ~ ) ~x t2 I v ( ~ ) ~ A ) l <. .#*AE l

(Proposition 1.1.6.6) and

Corollary 5.2.3.20 ( 0 ) Let ( x , ) , , , and ( Y ~ be) families ~ ~ ~ in the Hilbert spaces E and F , respectively. Take < , q E ('(1) such that

5.2 Orthogonal Projections of Hilbert Space

and

for distinct L, X E I . Then there is a unique u E C ( E ,F ) such that u = 0 on {XJL I)' and that U X , = YL

for every

If

L

E I . Moreover,

{ X , ~ LE

I ) L = 0 and

r := sup llxLll < m , LEI

6 := f:;

llvLll > llall ,

then

for every

3:

E E

Tho uniqueness follows from the fact that { X ~ LE I)" is the closed vector subspace of E generated by { x , I L E I ) (Corollary 5.2.3.9). Sow we prove the existence and the last assertion. By Proposition 5.2.3.19, the map

are well-defined and

Hence I m v is the closed vector subspace of E generated by { x , I L E I ) (Proposition 1.2.1.18 c)). Define

52

5. Hilbert Spaces

Then, by Proposition 1.2.1.18 b),

IIuxI12 = ll~v-'nIrnvx11~ I (rC2 + ~ ~ v J ~ ~ ) I I ~ -5~ ~ I T T ~ ~ x I ~ ~

for every x E E , so that

X E

If {x, 1 L E I)' E,

= (0)

, then Im v = E and by Proposition 1.2.1.18 b), given

Proposition 5.2.3.21 ( 0 ) Let E , F be infinite-dimensional Hilbert spaces. Take u E L ( E , F ) , and let u be the equivalence class of u i n L ( E , F ) / K ( E , F) . Then there is a sequence ( x , ) ~ , i~n E such that

for all m, n E IN and

llull =

II~xnll.

We construct the sequence ( x , ) , ~ inductively ~ such that for every n E IN

Take n E IIV and assume the sequence has been constructed up to n - 1 . Let G be the vector subspace of E generated by {x, ( m E INn-,) . Then u 0 n c E K ( E , F) , so that

Hence there is an x

E

E# such that

llull

-

, 1

< Il(u - u O .rrc)xll

5.2 Orthogonal Projections of Hilbert Space

By Proposition 5.2.3.1, r c l x E E# and

Hence

I uIG'I

> IIuT~LxII

=

I~UX

- ~ r c x l l=

II(u

1

.

- U 0 ~c)x11> llull

-

-n.

Then there is an xn E G I with

Proposition 5.2.3.22 ( 0 ) Let E be a Hilbert space and finite-dimensional vector subspaces of E . Then

5

the set of

for everg u E C ( E ). By Proposition 5.2.3.5 a

+ e,

r

sup I I ~ F U K F I I I I ~ I I . FE3

Take cu E ]0,11ull[. There is an x E E# such that

11ux11>0 . Let F be the vector subspace of E generated by { x ,u x ) . Then F E 5 and

I I ~ F ~ ~ F >I I ( I ~ F u ~ F x ~ =~ 1 1 ~ ~ ~ = x 1llu~ll 1 > 0. Since cu is arbitrary, it follows

54

5. Hilbert Spaces

5.2.4 Mean Ergodic Theorems

The main ideas of these results originate in a paper of J. von U I eumann.

Let E be a n unital Banach algebra. Take x E E with ( X " ) , ~ N bounded. Let A be the convex hull of {xn-' n E IN) i n E and let ( a n p ) n , p Ebe~ a famzly i n { x ) ' , such that is absolutely summable for every n E IN and

Proposition 5.2.4.1

Then for every n E I N , (anpxP-')pENis absolutely summable and for every V EA

,

Put

Then

Thus (anpxP-')p,, is absolutely summable for every n E IN Given n E I N , put

Then

5.2 Orthogonal Projections of Hzlbert Space

for every

7~

E

N and s o lim (x - l)x, = 0

n+w

Since

lim (xP- l)xn = 0

n+w

for any p E IN. There is a finite family

in IR+ with I C N ,

I t frdlows t h a t

for every n E IN. Hcncc lirn (y - l)x, = 0

n+cc

Lemma 5.2.4.2

There is a y €

for every n E N and t E For every p E N, put

Then

[&,$1

n+ with

5. Hilbert Spaces

56

p>tn+l*crp<

+

( n- t n - 1 l ) t (tn 1 ) ( 1 - t )

+

-

nt

nt

+ 1 < 1.

Since al =

there is a kn E INn-,

( y ) t l ( l - t)n-l - n t n l >n+l- 1, 1 t 1 ($''(I - t)" n+ 1

'

such t h a t Ikn - tn1 < 1 and

is an increasing function for p 5 kn and a decreasing function for p Hence this function takes its supremum a t k := kn E I N n - , , such t h a t

> k,.

By the Stirling inequalities (for n > 1 ),

=2

(;)

t k ( l - t1n-k =

2n!tk(l- t)n-k k !- k)!

&nneheken-ktk(~

-

t)n-k

-

- k)n-k

2 e n m k k J-(n

where

for every n E IN\{l). Given n E N \ { l ) , 2eh k + l hn5z(T)k+'(-)

n-k+l n-k

n-k+d

,

so that y := sup 6 < ntN

-

2eh

-s u p

6n E N

<m.

.

5.2 Orthogonal Projections of Halbert Space

Proposition 5.2.4.3

Let E be a Hilbert space. Take u E C ( E ) # . Put

F and let

be a family i n IR,

((Y,,),,,~N

for every n E

:= Ker

(1 - u )

, such that

W and

Then

and

for every x E E . In particular, 1

"

lirn - ):u p - ' x = R F X n+m n p=l lim ( t n l

n+m

+ (1

(Mean Ergodic Theorem),

-

t , ) ~ ) ~=xT F X

for every x E E and every sequence (tn)nEm i n ]0,1[ wath lim n t n ( l - t,) = GO.

n+m

We have

for every n E W . In particular, (anpuP-l)pENis absolutely surnmable and

Take z E E . Let A be the convex hull of {up-'x ( p E N ) and take y E A . There is a finite family (Pp)pEIin IR, such that I c N ,

5. Hilbert Spaces

Put

v belongs to the convex hull of

=

x

x

I p E Ih')

{up-'

Q ~ ~ u -~ - ~a x n

p ~ p=- l) ~:QnpUp-' ~ (1

PC r-4

PEN

and

[PEN

)

- V)X

for every n E IN . By Proposition 5.2.4.1,

Given n E N ,

Since

it follows

<

d ~ ( 0 ) lim inf n+m

Since y is arbitrary,

IpE1

):anpup-'x

I

I1

5 lirn sup ):anpuP-'x n+m IpEa

< Ilyll

5.2 Orthogonal Projectaons of Hilbert Space

is a Ca~ichysequence.

By Proposition 5.2.1.1 b), nEN

Define ID

:E

---+

E,

x ct lim

n+m

):anpup-'x. pcEi

By Corollary 1.4.1.3, w E L ( E ) and

We have uw = W U

and for n E IN and x E E ,

and so u~ux- wx = lim u n-tw

: a , , ~ P - ~-z lirn n+30

PEN

for every x E E by Proposition 5.2.4.1, i.c.

Hence

for every n E N and

for every x E E . Thus w is a projection in E

):anpup-'3:= PEN

5. Halbert Spaces

Take x E F . Then u x = x , and so unx = x for every n E N . We deduce that w x = x , i.e. x E Irn w , so that F c Irn w . h'ow take x E Irn w . There is a y E E with x = w y . Then

Thus x E F and so Inlw c F . Hence Imw = F and w is a projection of E onto F . By Proposition 5.2.3.5 e + a,

We now prove the final assertions. Put

Then

2

an1

+ )Ianp - h , p + l l = -n PEN

for every n E I N , so that

Put

Then

by Lemma 5.2.4.2

5.2 Orthogonal Projections of Hzlbert Space

61

Corollary 5.2.4.4 Let E be a Hilbert space. Take u E L ( E ) and a E K such that la1 = llull. Let F be a closed vector subspace of Ker ( a 1 - u ) . Then

In partzcular, F reduces u . We may assunle that a # 0 and F # {0} . Put

1 2a

U := -'U

+ -21 7 1 ~

Then

5.2.3.5 a (Propositio~~

+ e) and

for any x E F . Moreover, for x E Ker (1 - u ) ,

x

=

7 i ~ xE F

(Proposition 5.2.3.1).Hence

llull

=

1,

F = Ker (1 - u)

By Proposition 5.2.4.3, U7lp

= K F U = 7lp

and this implies U T p = T F U = (Yrp.

The final assertion follows from Corollary 5.2.3.12 b).

62

5. Hilbert Spaces

Remark. T h e relation

holds even if la1 # llull but it is, in general, not the case that KFU

= 071~

if la1 # llull. For instance, let E := lK2,let u have the matrix

and let cr := 1 and F := Ker ( 1

respectively.

-

u ) . Then

XF

and

KFU

have the matrices

5.2 Orthogonal Projections of Hzlbert Space

5.2.5 The F'rBchet-Riesz Theorem Proposition 5.2.5.1

a) x E E

(0)

Let E be a pre-Hilbert space.

* (.Ix) E E l , 11(.1~)11= llxll .

b ) T h e map E+E1,

x*(.~x)

is conjugate linear, injective, and continuous. a) ( . ( x )is linear. By Schwarz' Inequality, ( . J x )is contir~uousand

11(.1x)115 11x11 . It follows from

that

b) The rrlap

is conjugate linear. By a), it preserves the norm. Hencc it is contiriuous and injective. ¤ Theorem 5.2.5.2 ( 0 ) (FrCchet-Riesz, 1907) Let E be a Hilbert space and take x' E E' . Then there is a unique x E E with

The uniqueness follows from Proposition 5.2.5.1 h). To prove the existence, we rnay assuIrie that x' # 0 . Put

F := Ker x' F is a closed subspace of E with F # E . Hence FL# (0) (Corollary 3.2.3.10 b + a). Let y E F1\{O). Put

5. Halbert Spaces

Y' := (.IY) Then Ker x f = F c Ker y' Hence

for some a E IK (Lemma 1.2.6.3). Since yf

# 0 , we have that a # 0 . Put

Then

Corollary 5.2.5.3

(0)

If E zs a Hzlbert space, then the map

E - - t E', x+-+(.lx) is a n isometry of real Banach spaces The assertion follows immediately from Proposition 5.2.5.1 and Theorem 5.2.5.2. W Corollary 5.2.5.4 (Hellinger-Toeplitz, 1910) Let E be a Hilbert space and u : E + E a lznear map such that

for all x , y E E . Then v zs continuous The assertion follows immediately from Corollary 1.4.2.20 and Corollary 5.2.5.3. Corollary 5.2.5.5

(0)

All Hilbert spaces are reflexive

Let E be a Hilbert space and consider

Take x" 6 E" . Then s " is continuous and linear (Proposition 5.2.5.1). By the Frkchet-R.iesz Theorem (Theorem 5.2.5.2), there is an x E E with

5.2 Orthogonal Projections of Hilbert Space

ux = 5''

0

U

Take y' E E' and y E E with

uy = y' (Theorem 5 . 2 . 5 . 2 ) . We have ( ~ E XY, ' ) = ( X I Y')

=

= ( y ,?G) = XI'

0

- -

(XI?/)

=

( ~ 1 2= )

( Y , 215) =

u ( y )= (xf',uy) = (xl',y') .

Since y' is arbitrary,

jEx = x'' . Hence jE is surjective and E is reflexive.

Corollary 5.2.5.6 Let E be a pre-Hilbert space, F a subspace of E , and y' E F' . Then there is a unique x' E E' such that

We may assume E to be complete (Corollary 5.1.1.7) and F to be closed. By the FrCchet-Riesz Theorem, there is a y E F such that

Define

x ' : E -+lK,

x ~ ( x l y )

Then

(Proposition 5.2.5.1 a)). Take 2' E E' with

ztIF = Y f ,

l l ~ ' l l = IIY'II

.

By the FrCchet-Riesz Theorem, there is a z E E such that

66

5. Hilbert Spaces

Then, given any x E F , ( x i 2 - y ) = ( x l z ) - ( x l y ) = z l ( x )- y l ( x ) = 0 ,

that z - y E F'. By Proposition 5.2.3.1 (and Proposition 5.2.5.1 a)), we now see that

SO

Proposition 5.2.5.7 Let T be a Hausdorff space, p a positive Radon measure on T , and F a closed vector subspace of L 2 ( p ) , such that 3# is equicontznuous and the map

zs continuous for every t E T . If we consider 3 with the induced Hilbert space st.-ucture (Example 5.1.2.l ) , there there is a unique contznuous functio;.

such that k ( . , t ) E T and ( x l k ( . ,t ) ) = x ( t ) for every t E T and x E 3 . Moreover, k ( s , t ) = ( k ( . ,t ) l k ( . , s ) ) ,

and

for all s , t E T and the m a p

is continuous. If .F # (0) , then for every t E T , the functzon smallest norm i n the set { x E 3 1 Iz(t)l = 1 ) .

has the

5.2 Orthogonal Projections of Halbert Space

67

Take t E T . By the Frkchet-Riesz Theorem (Theorem 5.2.5.2), there is a unique kt E F with

for every x E 7 .Consider

We have

so that

for all s, t E T . Take t E T and E > 0 . Since hood U of t such that

for every x E

F# is equicontinuous, there is I neighbour-

F # and s E U . Thus

=

sup I(xlk, - kt)l = sup Ix(s) - x(t)l 5 ~ € 9

ZE~-#

for every s E

U (Proposition 5.2.5.1 a)). Hencc the map T

-+ 3,

is continuous. Given (so,to), (s, t) E T x T I

t

ct

k(., t)

E

5. Hilbert Spaces

by Schwarz' Inequality. Hence k is continuous. In order to prove the last assertion, let x E 3 with Ix(t)l = 1 . Then

Example 5.2.5.8 Let T be an open set of some Euclidean space, X the Lebesgue measure on T , and

3 := { x E L2(X)I x zs harmonic) be endowed with the induced Hilbert space structure (Example 5.1.2.1, Example 3.1.2.9). Then there is a unique continuous function

such that k(., t ) E 3 and ( x l k ( . ,t ) ) = x ( t ) for every t

E

T and x E 3 . Moreover, gzven s,t E T , k ( s , t ) = ( k ( . ,t ) I k(., s ) )

;

and

and the map

is continuous. If 3 # { 0 ) , then for each t E T , the function smallest norm in the set { x E F I x ( t ) = 1 ) .

~

has the

The assertion follows immediately from Example 3.1.2.9 and Proposition 5.2.5.7.

69

5.2 Orthogonal Projections of Halbert Space

Example 5.2.5.9 Take IK = C. Let T be a n open set of C . Let X denote the Lebesgue measure o n T and take

A := { x E L 2 ( X )I x is analytic) with the induced Hilbert space structure (Example 5.1.2.1, Example 3.1.2.10). T h e n there is a unique continuous junction

such that k ( . , t ) E A and

for every t E T and x E A ( k is called the Bergman kernel of T ) . Given s:t E T :

and

and the map

is continuous. If A # (0) , then for each t E T , the function smallest n o r m i n the set {x E A l x ( t ) = 1 ) .

$$

has the

The assertions follow immediately frorrr Example 3.1.2.10 and Proposition 5.2.5.7.

Example 5.2.5.10 Take K = C . Let T # C be a simply connected domain of C. Let X denote the Lebesgue measure on T and k the Bergman kernel of T . Take s E T . Let f be a bijective conformal map of T o n q ( 0 ) wzth f ( s ) = 0 . Then

and

for every t E T

5. Hilbert Spaces

Given r E ] 0 , l [, put

Tr

:= f-'

(m), 4

Let x be an analytic function on T and take T E ] O , l [ . is a rnerornorphic in s is function on T analytic on T\{s} . The residue of

4

lirn

-

T3t-3 tfr

By the Theorem of Residues

since If 1 = r on T,. By Green's formula,

Putting x := f' in the above equality,

/1

f'2dX = m 2 ,

T,

i.e. f ' E L 2 ( X ) and

11 frll2 = fi.If

x E L 2 ( X ) ,then by the above equality,

"'"' = -(I.1 f'(s)

n

f')

,

Hence

1k ( . , s ) = - f l ( s )f' , n and so

ff(s)f'(t) k ( s ,t ) = K

for every t E T , by Example 5.2.5.9.

Example 5.2.5.11

The Bergman kernel of p ( 0 ) is

5.2 Orthogonal Projections of Hilbert Space

Let s E q ( 0 ) . The map f : G ( 0 ) -+ G ( o ) ,

t-s

t ct 1 -st

is bijective, conformal, arid variishes a t s . Moreover,

for every t E q ( 0 ) . Thus

for every t E T and the assertion follows from Example 5.2.5.10.

Example 5.2.5.12

If T

:= { a E C

T x T -+C,

I im a > 0 ) , then 1 ( s , t ) ++-a ( -~

q2

is the Bergman kernel of T .

Take s E T . The map

is bijective, conformal, and vanishes a t s . Given t E T , fl(t) =

t-3-t+s (t - 3 2

-

s-s ( t - 312

Hence

for every t E T and the assertion follows from Example 5.2.5.10.

5. Hilbert Spaces

72

5.3 Adjoint Operators The notion of the adjoint of an operator, the most important single concept in the theory of Hilbert spaces, is introduced in this section. The central tool is the FrCchet-Riesz Theorem from Section 5.2. When endowed with this adjunction, the algebra of operators on a Hilbert space forms a C*-algebra. Important properties of selfadjoint and normal operators are established.

5.3.1 General Results Proposition 5.3.1.1 ( 0 ) Let E , F be normed spaces and f a continuous sesquilinear form on E x F . Then the map

is conjugate linear and continuous.

It is obvious that f ( . ,y ) E E' for every y E F and that the map

is conjuqate linear. By Proposition 1.2.9.2 a =. c, there is an

I f (x>Y)I

(Y

E

R+, with

I aIIxII llvll

for every pair (x, y ) E E x F . It follows that

Ilf (.?Y)II5 allvll for every y E F , i.e. the map

F

-+ E ' ,

y ++ f ( . , y )

is continuous. Corollary 5.3.1.2 ( 0 ) Let E , F be Hilbert spaces and f a continuous sesquilinear form on E x F . Then there is a unique map g : F + E , such that

f (x,Y ) = ( x I ~ ( Y ) ) for every (x,y) E E x F . g is linear and continuous.

5.3 Adjoint Operators

The uniqueness follows from Proposition 5.2.5.1 b). Define

u : E 4E ' ,

x-

(.1x),

(Corollary 5.2.5.3). Then

for every ( x ,y ) E E x F (Corollary 5.2.5.3), which proves the existence. Since u-' and v are conjugate linear and continuous (Proposition 5.2.5.1 b), Proposition 5.3.1.1); g is linear and continuous.

Theorem 5.3.1.3 (Lax-Milgram, 1954) Let E be a Hilbert space, f a sesquilinear form o n E , and a:P strictly positive real numbers with

If

( x ?x)I 2 PIIxIl2

for every x , y E E . T h e n there is a unique u : E

f

3E

with

( x ,Y ) = ( x l u y )

for every x , y E E . u is an isomorphism of Banach spaces and

The existence, uniqueness, continuity, and linearity of u all follow from Corollary 5.3.1.2. It follows from

I I ~ Y I I =~ (UYIUY)

=f(vy,y)

that

Ilvyll I allvll for every y E E , so that

1 1 ~ l l5 a .

I alluyll llvll

5. Hilbert Spaces

74

It follows frorn ~ 1 1 x 1I 1I~~ ( X , X = )I (IX I U X ) II I I X I I1 1 . ~ 1 ~ 1 1 that

ll~xll2 Bllxll for every x

E and D11x1I2 = 0

for every x E (Im u ) . Hence

( I mu)' = 0 and -

Imu= E

(Corollary 5.2.3.10 b + a). By Proposition 1.2.1.18 b),d), u is an isomorphism of Banach spaces and 1121-

1

11 I -D1.

Theorem 5.3.1.4 ( 0 ) Let E , F be Hilbert spaces and take u E L ( E , F ) . There is a unzque map u* : F -t E such that

for every ( x ,y) E E x F . Putting

w :F

+F ' , y c - t ( - ( y ) ,

then U* = v - ' O U ' O W

E L(E,F)

(Corollary 5.2.5.,?) and llu*ll U*

=

IIuII

>

lluf O uII

is called the adjoint (operator) of u .

=

llu112

5.3 Adjoint Operators

The map E x F +K ,

( 2 , ~C) f ( " ~ I Y )

is a continuous sesquilinear form. The existence, uniqueness, continuity, and linearity of u* all follow from Corollary 5.3.1.2. Given any (x, y) E E x F , (ux(y) = (112, wy) = (x, ulwy) = (x/v-'ulwy), so that

l l ~ * I l = l l ~ ' I l = llull (Corollary 5.2.5.3, Theorem 1.3.4.2 b)), and IIu*

41 5 l l ~ * l ll l ~ l l= 1 1 ~ 1 l ~

(Corollary 1.2.1.5). We have ( ( U X ( (= ~

(ux(ux) = (x(u*ux)5

((XI(

<

((u*ux(( (/u*0 u ( (( ( x ( ( ~

for every x E E , and so llul12 I llu* O 41 , Ilu* O ulI = llu1I2 . Proposition 5.3.1.5 a) u E C(E,F )

(0)

Let E , F be Hzlbert spaces.

+ u** = U .

b) The map C(E, F )

+ L(F, E) ,

u

+-+u*

is conjugate linear. a) Given (x, y) E E x F , - -

(U*YIX =) ( x I ~ * Y =)( U X I Y=) ( ~ l u x ) . Hence

5. Hzlbert Spaces

u8*= U

.

b) Take u, u E C ( E ,F ) and a , ,9 E IK . Then

+

( ( a u Pu)x1y) = (aux + Pux(y)= a(uxly) + P(uxly) =

+

+ P U * ~=) (xI(au*+

= a ( x ) u * y ) ,9(xlu8y)= ( x ( a ~ * y

for every ( x ,y ) E E x F . Hence

( a u + By)' = au* + pu' . Example 5.3.1.6

If u, and up are the right and left shift o n

e2,

then

u: = Ut, u; = U r (Example 5.1.2.3). Given

T ,y E

e2,

Hence

u: = up,

u; = u:* = U r

(Proposition 5.3.1.5 a))

Proposition 5.3.1.7 ( 0 ) Let E , F, G be Hzlbert spaces. Take u E C ( E ,F ) and u E L(F,G ) . T h e n

Given

( 2 ,y )

E

E x G,

( ( u0 u)xIy) = (uux1y) = ( u x ~ u *=~(xIutu*y) ) = (xl(ul0 u f ) y ) and so

(v 0 u)'

= U*

0

u* .

5.3 Adjoint Operators

77

the Proposition 5.3.1.8 ( 0 ) Let E be a real pre-flilbert space and complexification of the underlying real vector space (Lemma 2.1.5.1 c)). Then the m a p

i x i - a ,

is a scalar product. Its canonical n o r m zs the Euclidean n o r m o n the product E x E . If E is complete, then E is also complete. T h e complex pre-Hilbert 0 space E is called the complexification of E . 0

Given ( a .x ) , (6,y ) , ( c , z ) E E and a , p E Ill,

78

5. Htlbert Spaces

Example 5.3.1.9

-

Take n E IN and u E L ( R n ) . Then the map

+ 0

IRn +C n ,

( x ,Y )

(xk + ~

Y ~ ) ~ E L N ,

is an isometry of complex Hilbert spaces. If we use this isometry to identify the Hilbert spaces, then u and the map

have the same associated matrices. Example 5.3.1.10 Let ( T ,I , p ) be a measure space, L 2 ( p ,R) and L 2 ( p , C ) the corresponding Hilbert spaces i n the real and complex case, respectively, and k a real function on T x T belonging to L 2 ( p 8 p ) . Put

(Proposition 3.1.6.17 a)),

Then the map

is an isometry of complex Hilbert spaces and

for all x E L 2 ( p , C ) . Proposition 5.3.1.11

(0)

Let E , F be real Hilbert spaces, 0

0

h , and fi

the

0

complexification of E and F , respectively, and L ( E , F ) , L ( F , E ) and K ( E , F ) the complexification of the underlying real vector spaces of L ( E , F ) , L ( F , E ) and K ( E , F ) , respectzvely. For

define

5.3 Adjoint Operators

a ) If ( u ,V ) E

79

i ( F~ ) then , (G) E L ( i , i) ,

b) The maps

E

F +

6 ,)

i i, K

)

(u,21)

--( u ,v ) ,

( u ,v ) H( u ,V )

are bijective and linear.

-

a) By Lemma 2.1.5.3, ( u , v ) is linear. Choose (x,y) E E . B y Proposition 5.3.1.8,

Hence (u,li) is continuous and II(W v)11 5

IIuII + llvll

Let x E E . By Proposition 5.3.1.8,

5. Hilbert Spaces

Il.xl12

-

+ ll.x1I2

= l l ( ~ x > v x ) I 1=2 I I ( ~ > v ) ( x > o )5l l ll(X?)l1211xl12 2 >

so that sup{ll.lll Let ( a ,b) E

h and

( c ,d ) E

((=)(a,

b)

= ( u a - vb(c)

1

IIvII)

5 Il(.?

-. v)ll

% . W e have

( c ,d ) ) = ( ( u a - vb, u b

+ ( u b + vald)

-

+ v a ) I ( c ,d ) ) =

i ( u a - vbld)

+

= ( ( a ,b)l(u*c v t d , utd - v * ~ )=)

(

+ i ( u b + valc) =

I*+

( a ,b) ( u , -v ) ( c ,d )

)

,

so that

b) follon,~from a ) and Proposition 2.1.5.6.

b,i ( ~ )

Corollary 5.3.1.12 ( 0 ) Let E be a Hzlbert space and the complexijcations of E and of the underlying involutive algebra of L ( E ) (Proposition 2.3.1.40), respectzvely. For ( u , v ) E L ( E ) , define

-

0

0

( u , ~ )E: -+ E ,

( 2 , ~ ) -

(~x-u~,uY+ux).

T h e n the maps

i (4 ~ ~) ( 5 ,v ) (74

--

( u ,v ) ,

0

0

are bijectzve involutive unital algebra homomorphisn~s(hence L ( E ) and K ( E ) 0 0 m a y be identified via this map with L ( E ) and K ( E ) , respectzvely).

5.3 Adjoint Operators

81

By Proposition 5.3.1.11, the above maps are well-defined and are bijective, linear, and involutive. By Corollary 2.1.5.8, the map is a unital algebra homornorphism. 8

( 0 ) If E # {O} is a Hilbert space, then C ( E ) is a unital C*-algebra with respect t o the involution

Theorem 5.3.1.13

In the complex case, the assertion follows from Theorem 5.3.1.4, Proposition 5.3.1.5, and Proposition 5.3.1.7 (and Example 2.2.2.4). The real case follows 8 from the complex one and from Corollary 5.3.1.12. Remark. By Proposition 2.3.1.14 (and the above theorem)

for every u E C(E) . Example 5.3.1.6 shows that ap(u*) may be different from {Z I CY E U ~ ( I L )They }. nevertheless coincide if u is normal (Corollary 5.3.4.5) or compact (Corollary 5.3.2.7). Corollary 5.3.1.14 If E is a Hilbert space, then there is a unique order relation o n L(E)' which renders it a n ordered Banach space and

C(E): = {u' E ReC(E)'l u E L ( E )

u'(u*u) E IR+}

In the complex case,

The assertion follows imrricdiately from Theorem 5.3.1.13 and Proposition 2.3.4.13 b),d). 8

Let F be a closed vector subspace of the Hilbert space E the inclusion map. Then

Proposition 5.3.1.15

E and i : F

-t

for every x E E and y E F . We have

82

5. Hilbert Spaces

for every z E F (Proposition 5.2.3.1), so that i ' x = TFX

and (TFZIY)= ( ~ I Y=) (zliy) for every z E E (Proposition 5.2.3.1). Thus 7rZ-y = i y . Proposition 5.3.1.16 ( 6 ) Let u be a normal operator o n the cornplez Hilbert space E # {O) and a a n isolated point of a ( u ) (a is zsolated if e.g. u is compact and a # 0). T h e n the resolvent of u has a pole of order 1 at cu and its residue is the orthogonal projectzon of E onto Ker (a1 - u) . By Proposition 4.1.3.17 and Theorem 5.3.1.13, the resolvent of u has in a a pole of order 1 , its residue is an orthogonal projection p in E , and there is an operator v on E , such that (al-u)p=O,

v(a?-u)=p-1

If x E Ker ( a 1 - u) , then

and so Ker ( a 1 - u)

c Imp

If x E I m p , then

and so Imp

c Ker ( a 1 - u) .

If u is compact and cu # 0 , then, by Theorem 3.1.5.1 b), a is an isolated W point of a ( u ) .

5.3 Adjoint Operators

83

Proposition 5.3.1.17 If E is a Hilbert space, then K ( E ) is a n involutive closed ideal and so a hereditary C*-subalgebra of C ( E ) (Theorem 5.3.1.13). If 3 zs a Gelfand C*-subalgebra of K ( E ) , then u ( F ) is discrete. By Corollary 3.1.1.13, K(E) is a closed ideal of C(E)and so by Proposition 4.3.4.5 e), a hereditary C'-subalgebra of L ( E ) . Take u E 3 . Then

(Theorem 4.1.2.12) and so u ~ ( u ) \ { ~is) discrete (Theorem 3.1.5.1 b)). By Proposition 4.1.2.26, u ( 3 ) is discrete. Proposition 5.3.1.18 If u is a positive operator o n a Hilbert space, then Imua

c Imu

for every cr > 1 . Moreover,

for every vector x of the Hzlbert space Since

we have that

Im uQ c Im u For the last assertion remark that

so that

Proposition 5.3.1.19 If u is a self-normal operator on a Hilbert space, then every isolated point of u(u) is zn u,(u) .

5. Hzlbert Spaces

Let cu be an isolated point of a ( u ) . Define

f :a(.) 4 K , P t - t b ,

v:= f(u),

w := ( j - ~ f ) ( u ) .

Then

v2=2),

WV=O,

U=QV+W

and

1E

f ( 4 ~=)~ )(( uf) )= 4 v )

(Corollary 4.1.3.5), so that

v#O,

Imv#{O).

Given x E I m v ,

tux = IlIVX = 0 ,

so that

aE

( J P ( ~ )

Example 5.3.1.20 The right (left) shift of e2 is an extreme point of L(e2)# (Example 4.1.1.7), 1 zs its rzght (left) carrzer, and

is its left (rzght) carrier. Let x be the right shift of e 2 . Then x* is the left shift of e2 (Example 5.3.1.6) and 2'3: = 1 , xx* = u . By Theorem 4.3.3.20 b + a & e , x is an extreme point of L(e2)# and 1 and u are its right and left carrier, respectively. The assertion for the left shift of e2 follows.

5.3 Adjoint Operators

85

Proposition 5.3.1.21 ( 0 ) Let E , F be Hilbert spaces and u a bijective operator from E to F . u is a n isometry iff u-' = U* . If u is an isometry then ( x 1 , u - l ~ )= ( u x l y ) for every ( x , y )

€

E x F (Corollary 5.1.1.4). Thus u-I = u*

(Theorem 5.3.1.4). Conversely, if

then

for every x E E (Theorem 3.3.1.4). Thus u is an isonietry.

Corollary 5.3.1.22 ( 0 ) A n operator o n a Hilbert space zs a n isometry ifl it is unitary. Its spectrum is then contained zn {a E IK 1 la1 = 1). Let E be a Hilbert space and take u E L ( E ) . If u is an isometry, tlicn by Proposition 5.3.1.21

Thus

showing that u is unitary. If u is unitary, then u is invertible and u* = u-1. By Proposition 5.3.1.21, u is an isometry. The final assertion follows from Corollary 2.2.4.7.

5. Hilbert Spaces

86

5.3.2 Supplementary Results Proposition 5.3.2.1 ( 0 ) Let ( S ,(5, p ) , ( T ,I , v ) be measure spaces. Take k E L 2 ( p 8 v ) , and let u be the operator L 2 ( v ) + L 2 ( p ),

x

H

kUx

(Proposition 3.1.6.17 a)). Then, given y E L 2 ( p )

for v almost all t E T (Example 5.1.2.1)

Take x E L 2 ( v ). Then y

@I

x E L2 ( p 8 v ) and so

( y @ I x ) kE L ' ( p 8 ~ ) (Holder). B y Fubini's Theorem

which proves the assertion (Theorem 5.3.1.4). Proposition 5.3.2.2 ( 0 ) Let E , F be Hilbert spaces and take u E L ( E , F ) . Then the followzng are equivalent: a ) u is compact.

b ) u* zs compact c)

u* o

11

zs compact.

d ) u o u* is compact.

5.3 Adjoint Operators

a c

+c & d 3

and b + c & d follow from Proposition 3.1.1.11. a . Let ( x , ) , , ~ be a bounded sequence in E . Put 0. := sup

115,)).

nEN

By Proposition 3.1.1.19 a + b, there is a subsequence such that ( u * u ~ ~converges. ) ~ , ~ Given n , p E IN,

(yn)nEN

of

( X ~ ) , ~ N ,

IIuYn - uYpI12 = IIu(Yn - yp)I12 = ( u ( Y ~ Yp)Iu(Yn- Y P ) ) =

Thus ( u y , ) , , ~ is a Cauchy sequence and hence convergent. By Proposition 3.1.1.19 b + a , u is compact. W d + b follows from c + a and Proposition 5.3.1.5 a). Proposition 5.3.2.3 (. spaces and take ( u , ) , , ~E

0) .

be two families of Hilbert

Let

fl L(E,, F,)

with

LEI

Then

(Proposition 5.1.3.3 b)).

We have

Given ( x ,y) E

a) El

)x

),

OF,

and the assertion now follows (Theorem 5.3.1.4). Proposition 5.3.2.4 then

(0)

H

If E , F are Hilbert spaces and u 6 L ( E ,F ) ,

Ker u = Ker (u*o u ) = (1m u*)' ,

Im = Im ( u o v*) = ( ~ eus)' r

.

If E = F , K := G ,L := (Keru)' then T K and T I , are the left and the right carrier of u i n L ( E ), respectively (Theorem 5.3.1.13).

5. Hilbert Spaces

88

It follows immediately from

( x ,?I)E E x F + ( X I U * Y ) = ( 2 ~ x 1 ~ ) that Ker u = (1m us)' . T h e inclusion Ker u

c Ker (u*o u )

is trivial. Given x E Ker (u*o u ) ,

11~x11' = (uxIux) = (xIu'ux) = 0 , so that

Hence Ker (u* o u ) c Kcr u ,

Ker u = Ker (u'ou)

NO\!! (Ker v*)' = (Im u ) l L = (Proposition 5.3.1.5 a), Corollary 5.2.3.9) and In1 ( U o u * )= (Ker ( u o u*))' = (Ker u*)'

(Propositions 5.3.1.5 a) and 5.3.1.7). We prove now the last assertion. For every x E E , TKUX

=UX,

so that TKU

=U

Take v E L ( E ) with VU =

Then

0

5.3 Adjoint Operator.?

vux = 0 for all x E E , so that, by continuity,

u(K= O . Hence

and

KK

is the left carrier of x . Since

L = (Ker u)' =

mu',

it follows that T,. is the left carrier of u* . By Proposition 4.3.3.3, right carrier of u .

71.~

is the W

Corollary 5.3.2.5 ( 0 ) Let E , F be Hzlbert spaces and take u E C ( E , F ) Suppose that for some n > 0 ,

for. all (x, y ) E E x F . Then u is a71 isomorphism of Banach spaces and 1

llu-'11 5 - . Q

u* is obviously injective.

By Proposition 5.3.2.4,

Im = (Ker u*)'

=

(0)' = E

¤

and the assertiorl now follows from Proposition 1.2.1.18 b),d).

Proposition 5.3.2.6 ( 0 ) Let E: F be Hilbert spaces and u E F ( E >F ) . Then u* and u* o u are Fredholm operators and Dim Ker U * = Dim Coker u , Ind u* = -1nd u

Dirt1 Ker u = Dirn Coker u'

,

, Ind (u o u * ) = 0 ,

If u has index 0 , then Dim Ker u = Dim Ker U * = D i ~ nCoker u = Dim Coker u' < co In particular if E = F then ue(u*)=

la E ue(u)}.

5. Hilbert Spaces

Since Dim Ker u* = Dim Ker u' = Dim Cokcr u (Theorem 5.3.1.4, Proposition 3.1.3.4), we deduce that Dim Ker u = Dim Ker u*' = Dim Coker u* (Proposition 5.3.1.5 a)). In particular, u* is Fredholm and Ind u* = -1nd u .

By Proposition 3.1.3.7, u o u* is Fredholm and Ind(u o u') = 0 . Since I m u and Im (u o u*) are closed, Im u = Im ( u o u*) = (Ker u')' by Proposition 5.3.2.4. The assertion for Ind u = 0 follows. In order to prove the final assertion take a E IK. By Proposition 3.1.3.25 a),

Corollary 5.3.2.7 Let E be a Hilbert space. Take u E K ( E ) and a E IK . If E i s finite-dzmensional or if a # 0 , then Dim Ker ( a 1 - U) = Dim Ker (Gl - u*) =

= Dim Coker ( a 1 - u) = Dim Coker (81 - u*)

Im ( a 1 - U) = (Ker (51 - u*))' ,

< cc ,

5.3 Adjoint Operators

By the Corollaries 3.1.3.12 and 3.1.3.13,

a1 - u

E

3 ( E ) , I n d ( a 1 - u) = 0 .

By Proposition 5.3.2.6,

Dim Ker ( a 1 - u) = Dim Ker(G1 - u*) = = Dim Coker ( a 1 - u) = Dim Coker(a1 - u*) < oo ,

Im(a1 - u )

=

(Ker ( ~ - lu*))'

It follows that a E up(u) e CY E u p ( u * ) . Proposition 5.3.2.8 ( 0 ) Let u be a projection i n the Hilbert space E . Then the following are equivalent: a)

u is a n orthogonal projection (in the sense of Corollary 5.2.3.2).

b) u is selfadjoint, i.e. u is a n orthogonal projection of the C*-algebra C(E) (Theorem 5.3.1.13 and Definition 4.1.2.18). c) u is positive.

a a b . Take x , y ~ E . T h e n (uxly-uy) = o = (ux -x1uy) (Proposition 5.2.3.1 a (uxly) = (uxly - uy)

+ b), so that

+ (uxIuy) = (uxIuy) = (ux

i.e. u is selfadjoint (Theorem 5.3.1.4) b + a . Since

-

xluy)

+ (xluy) = ( ~ I U Y ) ,

92

5. Hilbert Spaces

By Proposition 5.2.3.5 e + a , u is an orthogonal projection. b 3 c follows from Proposition 4.2.1.22. c 3 d is trivial. d 3 a . Xow

i.e. u'u is a projection. Since u'u is selfadjoint, we deduce from b is an orthogonal projection. Hence

* a , that

U*U

llu1I2 =

l l ~ * ~ lIl 1

(Theorem 5.3.1.4, Proposition 5.2.3.5 a

* e), so that

.

llull 5 1 and u is an orthogonal projection (Proposition 5.2.3.5 e

+ a).

Proposition 5.3.2.9 ( 0 ) Let F be a closed vector subspace of the Hilbert space E . Take u E L ( E ) . T h e n F is u-invariant iff F L is u* invariant. In particular, F reduces u iff F is u-invarzant and u* -invariant. In this case,

for every x i F , where

If 3 is a n involutzve set of C ( H ) , then F is 3 - i n v a n a n t iff

T.D

E

3 '

The first assertion follows from x, Y E E

* (uxly) = ( x l u * ~ )

.

and F = F1' (Corollary 5.2.3.9). Then second assertion follows from

The final assertion follows from the first one and Corollary 5.2.3.12 b).

Let E be a Hzlbert space. Take u E L ( E ) . Let (F,),,I be a famzly of closed vector subspaces of E such that each F, ( L E I ) reduces u . Then both F, and the closed vector s~ibspaceof E generated by U F, Corollary 5.3.2.10

reduce u .

LEI

LEI

¤

5.3 Adjoint Operators

99

Corollary 5.3.2.11 Let E be a Hilbert space, u a normal (selfadjoint, unitary) operator on E , F a closed vector subspace of E which reduces u and

W

Then v is normal (selfadjoint, unitary).

Definition 5.3.2.12 ( 0 ) Let E be a pre-Hilbert space and F a vector space. Given ( x ,y) E E x F , define

I f F is normed, then (.lx)y E L ( E ,F ) and II(.Ix)yII = llxll llyll for every ( x ,y )

E

E x F.

Proposition 5.3.2.13

(0)

Let E , F, G be Hilbert spaces.

a ) The map

is sesquzlinear. b) ( x ,Y ) E E x F C)

* ((.lx)y)' = ( . I y ) x .

( x ,y) E E x F , v

E

C ( F ,G ) 3 v

o

( ( . l x ) y )= ( . l x ) u y .

d ) u E L ( E , F ) , ( y ,2 ) E F x G 3 ( ( . I Y ) z 0) u = (.lu*y)z e)

(X,YI)EEXF,(YL,Z)EFXG* 3

f)

If

((.l742)2)0 ( ( . I x ) Y I= ) (Y~IY~)(.Ix)z.

( X I , yl), ( x 2 ,y

then

where

~E)

E x F such that

5. Halbert Spaces

a ) is easy to see. b) Given (a, b) E E x F , (((.lx)y)alb) = ((alx)ylb) = (alx)(ylb) = (al(bly)x) = (al((.ly)x)b) . c) Given a E E ,

u

0

((.Ix)y)a = u(a1x)y = (a1x)uy = ((.[x)uy)a.

d ) Given a E E ,

e) By c) and a) ( ( ' 1 ~ 2 )O~ (('Ix)YI) ) = ('Ix)((YIIY~)z) = (YI~Y~)(.~x)z. f) We have

and

It follows

Remark. If we put

for every n E N (Example 5.1.2.3),then (x;x,),,~ (x,x;),~N is not. Indeed, by b) and e),

is sumrnable in C(e2) , but

5.3 Adjoint Operators

for every n E N . Take A E !Q,(lr\J). For

E E e2,

(Pythagoras' Theorem) and

so that

By Proposition 1.1.6.6, ( X ; X , ) , ~ ~is summable and (X,X:),~~ is not sumrnable.

Corollary 5.3.2.14

(0)

If E

zs

a Hilbert space, then

Take u E {(.lx)x 1 x E

ElC.

Define

@N if x # O f:E+K,

x-

11~112

O

if x = O .

If x E E , then by Proposition 5.3.2.13 c),d),f),

It follows

= (f(x+y))(x+y) = f ( x + y ) x + f ( x + ~ ) ~ 7

f(x) = f ( x + y ) = f

(~)

for all x, y E E . Hcncc f is constant and so u E E ~ E .

5. Halbert Spaces

96

Proposition 5.3.2.15 ( 0 ) Let E be a Hilbert space, F a Banach space, @ a71 upward directed set of closed subspaces of E the union of which is dense i n E , and 5 the upper section filter of 6 . If u E K ( E , F ) , then

Let v be the isometry of real Banach spaces

(Corollary 5.2.5.3). We have u' E K ( F t ,E ' ) (Theorem 3.1.1.22 a is compact (Proposition 3.1.1.1 I ) ,

(Corollary 1.3.4.5, Theorem 5.3.1.4, Proposition 5.3.2.8 a

+ b ) , v-'

out

+ b ) , and

for every G E 6 (Theorem 1.3.4.2 t))). By Proposition 5.2.3.3, lim IIu c,3

0

KG -

ull = lim llrGo (v-I G.3

0

u t ) - (v-I

0

ul)ll = 0 ,

Corollary 5.3.2.16 ( 0 ) Let E be a Hilbert space, 5 art upward directed set of finite-dimenszonal vector subspaces of E , the union of which is dense i n E , 6 the upper section filter of 5 , and define

Then ~ ( 6 zs) an approximate unzt of K ( E ) (Proposition 5.3.1.17). The assertion follows immediately from Propositions 5.2.3.3 arid 5.3.2.15 (and Corollary 5.2.3.2).

Corollary 5.3.2.17 ( 0 ) Let E, F be Hilbert spaces, E (resp. 5 ) an upward directed set offinite-dzmensional vector subspaces of E (resp. F ) the union of which is dense i n E (resp. i n F ) , and take u E K ( E , F ) . Then for every E > 0 , there are Eo E E and Fo E 5 such that llrHo u o KG - u I I < E

forall G E E and HE^ with Eo c G , I;bc H

5.3 Adjoint Operators

By Corollary 5.3.2.15, there are Eo E @ and Fo E

5 such t h a t

for all G E @ and H E 5 with Eo c G , F o c H . Take G E @ and H E 5 with Eo C G , Fo C H . Then (Theorem 5.3.1.4, Proposition 5.3.1.7, Corollary 5.2.3.2)

< ~ ~ ~ * - u * o n , , ~ ~ + ~< ~-2+u- =o2E n. ~ - u ~ ~ E

E

Example 5.3.2.18 If u,, up denote the right an left shift of and q denotes the quotient map

e 2 , respectively,

then qu, , qup are unit(l,ry and (qur)' = qur, but there is no v E L ( t 2 ) / K ( t 2 )with

By the last assertion of Proposition 5.3.2.2, q is a homomorphism of involutive unital algebras (Proposition 5.3.1.17) and by Example 5.3.1.6,

By Example 3.1.2.16,

Hence qu, and qup arc unitary. Assume there is a v E L ( t 2 ) / K ( e 2 )with ev = qur

Take w E C(e2) with

98

5. Hilbert Spaces

Then

Hence, by Proposition 3.1.3.21 c), Ind u ,

=

Ind ew = 0

since eW is invertible. This is a contradiction, since Indu,

=

¤

-1.

Definition 5.3.2.19 ( 0 ) Given the Hilbert space E , take x E E and 3 C L ( E ) . T h e n 3 is sazd to act irreducibly on E if 3\{0) # 0 and if 0 and E are the only 3-invariant closed vector subspaces of E . 3 zs said to act non-degenemtely on E if

x is called cyclic for 3 , if

x is called separating for 3 if

It is obvious that if there is a cyclic element for degenerately on E .

Proposition 5.3.2.20 a ) If every x E E\{O)

(0)

F ,then 3 acts non-

Let E be a Hzlbert space and take 3 C L ( E ) .

is cyclic for 3 , then 3 acts irreducibly o n E .

b) If 3 is a subalgebra of L ( E ) acting irreducibly o n E , then every

x E E \{O) is cyclic for 3 . If p is a n orthogonal projection of E belongzng to 3 such that p 3 p is one-dimensional, then there is a n x E E such that

5.3 Adjoint Operators

99

a) It is obvious that 3 \ { O ) # 0. Let F be an 3-invariant closed vector subspace of E . Assume that F # (0) and take x E F\{O). By Corollary 5.2.3.9,

E = {ux 1 u

E 3)"

c F"

=F

Hence F = E and 3 acts irreducibly on E . b) We put

Then F is an F-invariant vector subspace of E and so F is also 3-invariant. Assume F = {O). Then IKx is 3-invariant, so that IKx = E . Take u E F \ { O ) . Then ux # 0 and this contradicts the assumption F = ( 0 ) . Hence F = E . By Corollary 5.2.3.9,

x is thus cyclic for 3. Assume Im p is not one-dimensional. Then there are x,y E I m p such that

Since x is cyclic for 3 , there is a u E 3 such that

Since p 3 p is one-dimensional, there is an cu E IK such that pup = a p . Thus

which is a contradiction. Hence I m p is one-dimensional. Take x E I m p with 11x11 = 1 . Then Ker p = (Imp)' = Ker (.lx)x (Proposition 5.2.3.5 a

+ d ) and

Hence p = (.Ix)x.

5. Halbert Spaces

100

Proposition 5.3.2.21 ( 0 ) Let E be a Hilbert space and take 3 c C ( E ). Then the following are equivalent:

a ) 3 acts non-degenerately on E b ) Given x E E\{O} , there is a u E 3 with u'x # 0 . If these conditions are fulfilled, 3 is a subalgebra of L ( H ) , and 5 is an approximate unit of 3 , then limux = x u,3

for every z E E

a

+ b.

.

Assume that

u w x= 0 for every u E 3 . Then

(xluy) = (u*xIy)= 0 for every

(11,

y) E 3 x E . Hence x E {uy I ( u ,Y ) E 3 x E l i = { O ) , x=o.

b

+ a.

Take

x E

{UY

I ( u ,y) E 3 x

ElL.

Then

11u*x11' = ( ? ~ ' X ~ U * X=) (xIuu*x)= 0 for every u 6 3 ,so that x = 0 . Thus {UY

I ( u ,Y ) E 3 x El'

1

{ I L ~ (u,y )

E

T x E)"

= (01,

=E

5.3 Adjoint Operators

Hence 3 acts non~-deneratelyon E . We nowr prove the last assertion. Given

(11,

x) E 3x E ,

lirn u u x = u x u,3

Hence if F is the vector s~ibspaceof E , generated by { v x 1 ( v ,x ) E 3 x E ) , then lirn u x = x u,3

for every x E F . By a ) and Corollary 5.2.3.9, F is dense in E . Hence lirn u x = x u,3

for every x E E

Proposition 5.3.2.22

(0)

Let E be a Hilbert space and take 3 C L ( E ) .

a) If x is cyclic for 3 , then x separates 3" b) If x separates P r F and 3 is a quaszunital involutive subtr1,qebra of L ( E ) acting non-degenerately o n E , then x is cyclic for 3 . a) Take u E

FCwith

u v x = vux = 0 for every v E F .Hence if F denotes the vector subspace of E generated by { v x I u E 3 1 , then

F c Keru. By Corollary 5.2.3.9,

Hence x separates 3' b) Define

5. Halbert Spaces

102

F is the closed vector space generated by {ux 1 u E 3 ) (Corollary 5.2.3.9).It is therefore 3-invariant. By Corollary 5.3.2.9, T F E 3 ' . Let 5 be an approximate unit of 3 . By Proposition 5.3.2.21,

Hence

(1

-

rF)x = 0.

Since 1 - K F E Pr 3' and x separates Pr F C , it follows that

F=E Hence x is cyclic for 3 . Proposition 5.3.2.23 ( 0 ) Let E be a separable Hilbert space and 3 a commutative involutive subalgebra of L ( E ) . If 3 acts non-degenerately o n E , then 3 has a separatzng vector. Let '2 be the set of subsets A of E such that

for every x E

A and

for distinct x ,y E A . By Zorn's Lemma, ZI contains a maximal element A . Take y E { U X ~ ( U , XE ) 3 x A)'

Take u,,v E 3 and x E A . Then

5.3 Adjoint Operators

(ux Ivy) = (v'ux I y) = 0 Hence

Since A is maximal in U,

Hence

= U ( ~ e r u ) ' c {uxI(u,x) E 3 x

A)"

~ € 3

(Proposition 5.3.2.4),

Since E is separable, A is countable. Let cp : A put

+ IN

be an injective map and

Take v E 3 with

Since the elements of the fanlily ( V X ) , , ~are pairwisc orthogonal,

for every x E A (Pythagoras' Theorem). Hence vux = UVX = 0 for every (u, x) E 3 x A . It follows that

(Corollary 3.2.3.9). Thus y separates 3 .

5. Hilbert Spaces

104

Proposition 5.3.2.24

(0)

Let E be a Hilbert space, F a closed vector subspace o j E , and 3 a subalgebra of L ( E ) acting irreducibly or1 E . Given ~ € 3 define ,

T h e n {ii I u E 3 ) acts irreducibly o n F Take x , y E F with x # 0 . By Proposition 5.3.2.20 b), x is cyclic. Hence ~ that ~ there is a sequence ( u ~ in ) 3~such lim u,x = y

n+m

(Corollary 5.2.3.9). Thus lim ZLnx = y .

n+cc

Hence x is cyclic for {ii 1 u E 3). By Proposition 5.3.2.20 a), {ii 1 u E 3) acts irreducibly or1 F . Definition 5.3.2.25 Let E , F be Hilbert spaces. A n operator u called a partial isometry if

1121x11=

:

E

-t

F zs

I I X I I.

for all x E (Ker u)' . Proposition 5.3.2.26 Let E, F be Halbert spaces and take u E L ( E , F ) . T h e n the jollowzng are equivalent:

a ) u is a partial isometry.

b)

U*

zs a partial isometry.

c) u' o u zs a n orthogonal projection

d ) u o u* is a n orthogonal projection. If these conditzons are fulfilled, then (Ker u)' = Irn u* ,

(Ker u')'

= Im u

,

and u* 0 u (resp. u 0 v* ) is the orthogonal projection of E on Im u' (of F on Im u ).

5.3 Adjoint Operators

a

+ b & c, and the last assertion. By the hypothesis, the map

is an isometry. Hence I m u is closed and by Proposition 5.3.2.4, Im u = (Ker u')'

.

Take x, y E (Ker u)' . Then (ux I UY)= ( ~ I Y ) (Corollary 5.1.1.8 a

b), so that

Hence the map Im IL + (Ker u ) ~ , z

++ U*Z

is the inverse of u . It follows that u' is also a partial isometry and therefore Imu* = ( ~ eu)' r Moreover, u* o u (resp. u o U* ) is the orthogonal projection of E on Irn u* (of F or1 I m u ) . c + a . We have Ker u = Ker (u* o u ) , ( ~ e11)'r

= (Ker (u'ou))'

=

Im (U*OIL)

(Proposition 5.3.2.4) and = llu*112 = llu* 0 uI1

I1

(Theorem 5.3.1.4). Take x E (Ker u)' . Then 11x11 = Ilu'uxll I IluxII

I IIxII

3

so that IIuxll = llxll and u is a partial isometry. b+a&dandd+b followfroma+b&candc~a.

5. Halbert Spaces

106

Proposition 5.3.2.27 Let (H,),,, , (K,),,, be two families of Hilbert spaces ~ and E the C*-direct s u m of the family (K(H,, K L ) ) L .EPut

and for each u E E , define

Then

and the m a p

is un injective homomorphism of znvolutive vector spaces (algebras i f (HL)lE,= (K')Gl ). The first assertion follows from Proposition 3.1.2.17. It is obvious that the map

is injective. By Proposition 5.1.3.4 and Proposition 5.3.2.3, this map is a homomorphism of involrltive algebras. Proposition 5.3.2.28 E:

T h e following are equivalent for every Hilbert space

a ) E is finite-dzmensional.

c) K ( E ) is unital. a + b is trivial. b + c follows from Theorem 4.4.1.8 h). c + a . Let p be the unit of K(E) and put F := I m p . Then p E Pr K(E) so that p = T F (Proposition 5.3.2.8 b + a). Take x E F 1 . By Proposition 5.3.2.13 c),

5.3 Adjoint Operators

so that

Hence F'

= (0)

,

and

By Proposition 3.1.1.10, E is finite-dimensional.

Proposition 5.3.2.29 Let E be a n infinite-dimensional Hilbert space, 3 a closed ideal of L ( E ) , and G a C*-subalgebra of L ( E ) . T h e n 3 G is the C' -subalgebra of L ( E ) generated by 3 u G and the C * - algebras (3 G ) / F and G / ( F n G) are canonically isometric. If i n addition F n G = (0) then

+

for all ( u ,v) E 3 x

+

6

By Theorem 5.3.1.13, L ( E ) is a C'palgebra and the assertion follows from W Corollary 4.2.6.7.

5. Hilbert Spaces

108

5.3.3 Selfadjoint O p e r a t o r s P r o p o s i t i o n 5.3.3.1 ( 0 ) Let E , F be Hilbert spaces and S the involutive vector space of continuous sesqilinear forms on E (Proposition 2.3.3.3 c)). Given u E L ( E ) , define

a) ii E S for e v e y u E L ( E )

b) The map L ( E ) -+ S , u

-

-

.ii

zs an isomorphism of involutzve vector spaces. c) u* o u is positive for e v e y u E & ( E , F ) . a ) Now

and so, by Proposition 1.2.9.2c b) Consider

+ a , .ii is continuous.

It is obvious that cp is linear. By Corollary 5.3.1.2 and Theorern 5.3.1.4, cp is bijective. Take u E L ( E ) . Then

-

- u * ( x ,y ) = (u'xly) = ( x l u * * y )= ( x l u y ) = ( u y l x ) = .ii(y,x) = & * ( x ,y )

for all x , y E E (Theorem 5.3.1.4, Proposition 5.3.1.5 a ) ) . Thus

and 9 is involutive. c) Given x E E ,

-

Thus by b), u* o u is positive. Corollary 5.3.3.2 An operator u on a real Hilbert space E is selfadjoint ifl (ux~?,)=

for all x , y E E .

I -

4

((u(x

+ Y ) lz + Y ) - ( 4 5

-

Y ) Ix

-

Y))

5.3 Adjoint Operators

109

This follows immediately from Proposition 5.3.3.1 b) and Proposition 2.3.3.7. H Corollary 5.3.3.3 selfadjoint iff

(0)

A n operator u o n a cornplez Hilbert space E is

for every x E E This follows immediately from Proposition 5.3.3.1 b) arid Proposition 2.3.3.8 a ~ b . Corollary 5.3.3.4 that

for every z

E

(0)

Let u be an operator o n a Hilbert space E such

E . If u is selfadjoznt or zf IK = C , then u = 0

By Proposition 5.3.3.1 b) and Proposition 5.1.1.9,

for every x E E and so u

=0

Remark. Let u be the operator on R 2 ,defined by the matrix

Then

for every x E IR'. This example shows that the above Corollary is false without some hypothesis such as the selfadjointness of u or IK = C . P r o p o s i t i o n 5.3.3.5 space E , then

(0 )

If u is a selfadjoint operator o n the Hilbert

11u1( = sup I(uxIx)I = inf { a E r€E#

R+ ( x E E ==+ )(uxlx)l 5

all~11~) .

5.3 Adjoint Operators

Theorem 5.3.3.6 E # 10) and put

(0)

111

Let u be a selfadjoint operator o n the Hilbert space

/3 := sup{(uxlx)

IX

E E , llxll = 1 1 ,

(Corollary 5.3.3.3). T h e n c-u, P E a ( u ) C [a,Dl.

Take y

€1

-

co,a [ .Then

0 < cu - Y I (UXIZ) - Y(XIX) = ((u - Y ~ ) X I I X )I I ( U -Y~)XII for every x E E with

((XI(

= 1 . By Corollary 5.3.2.5, y

4

$ a ( u ) . Hence

~ c )[ a ,

I t follows that

arid

4.1 c [a,PI . Put y := inf a ( u ) .

Then " ( u - y l ) = U(U)- 7 C I R + ,

i.e. u - y l is positive. Thus (uxlx) = ((u - y1)xlx)

+ y = ((u

for every x E E , llxll = 1 , and

and

SO

-

yl)1'2x I (u - y1)1'2x)

a 2 y . Hencc

+y > y

112

5. Hilbert Spaces

Corollary 5.3.3.7 ( iff zt is seljadjoznt and

0)

T h e operator u o n the Hzlbert space E is positive

for every x E E . I n the complex case, we m a y ornzt the adjective "selfadjoint". In particular, (.lx)x and the map

are positive for every x E E The first assertion follows from Theorem 5.3.3.6 and the second from Corollary 5.3.3.3. The final assertion follows from the first one, Corollary 4.2.2.10, and Proposition 5.3.2.13 b). Remark. The example given in the Remark to Corollary 5.3.3.4shows that we cannot drop "selfadjoint" in the real case. Corollary 5.3.3.8 ( 0 ) If F and G are closed vector subspaces of a Hilhert xc:. In particular, if H is the closed vector space E , then F c G iff n~ subspace of E generated by F u G (by F n G ) , then x~ is the supremum (znfimum) of { n ~xG} , zn Pr L ( E ) .

<

This follows from Corollary 5.3.3.7 and Proposition 5.2.3.13 a Proposition 5.3.2.8 a e b).

c (and

Corollary 5.3.3.9 ( 0 ) Let E , F be Hzlbert spaces and take v E L ( E , F ) . Then v ' o u o v zs positzve whenever u is a positive operator o n F . In particular, v'ov is positive. v* o u o v is selfadjoint and by Corollary 5.3.3.7,

for every x E E . By Corollary 5.3.3.7 again, v* o u o v is positive. Proposition 5.3.3.10 and CY > 0 .

(0)

Let E , F be Hilbert spaces. Take

a) Ker (u*o u)" = Ker u b) u compact e (u* o u ) zs~ compact.

E L ( E ,F ) ,

5.3 Adjoant Operators

By Corollary 5.3.3.9, u' o u is positive. Thus (u'u)" is well-defined.

a)

Step1

vERe~(E),n~K~Kerv~"=Kerv

By Proposition 5.3.2.4,

Ker v2 = Ker v , and so the assertion follows by complete induction. Step 2

v

E L(E)+ ,

0 < 4 < y + ~ ev4r c Kerv*

The assertion follows frorn v* = .*-4,p.

Step 3

v E L ( E ) + + Kerv" = Ker v

Take n E N such that 2-" < a < 2" . By Steps 1 and 2,

Ker v C Ker vQ2n= Ker ( u " ) ~ '= Ker vQ c ~ ev2" r = Ker v , and so Ker v" = Kcr v Step 4

Ker ( u * ~ ) = " Ker u

By Proposition 5.3.2.4 and Step 3:

Ker (u* o u)" = Ker (u' o U) = Ker u

b)

Stcp 1

v E R e L ( E ) , n E IN

+ (v

compact H v2" compact)

By Proposition 5.3.2.2 a H c ,

v compact u v2 compact. The assertion thus follows by complete induction. Step 2

v E .C(E)+ , 0 < /3 < 7 ,va conipact

+ v*

compact

5. Halbert Spaces

114

The assertion follows from u1

= u7-4v4

and Proposition 3.1.1.11. Step 3

u E t ( E ) ++ ( v compact

@

ua compact)

Let n E IN with

2-" < ff By Steps 1 and 2, v compact ==+ uo2" compact

* ua compact

< 2".

-

(va)'" compact

*

==+ u2' compact ti u compact.

Thus

u compact eva compact. Step 4

u compact H (u*o u ) compact ~

By Propositioii 5.3.2.2 a H c and Step 3, u compact

Corollary 5.3.3.11 then, for every n E IN

u* o u compact

(0 )

(u*o u)O compact.

If u is a normal operator o n a Hilbert space,

Ker un = Ker u ,

u compact

un compact

Since

( u * u )=~ ( U * ) ~ U " = (un)*un, it follows by Proposition 5.3.3.10, that

-

~ Ker ((un)'un)= Ker un , Ker u = Ker ( u ' u ) = u compact

( u * ~compact ) ~ M (un)'un compact e

+=+

11"

compact.

5.3 Adjoint Operators

115

Remark. W e cannot drop the adjective "norrrial" in the above corollary, as the following example shows:

Then u is not compact but u2 = 0 Corollary 5.3.3.12 define

(0)

Let E be a Hilbert space. Take u E C ( E ) and

a) f is Hermitian iff u is selfadjoint.

b ) f is positive i f f u is positive. I n this case { x E El f ( x , x ) = 0 ) = Keru. c ) f zs a scalar product iff u is injective and posztive. a) follows from Proposition 5.3.3.1 a),b). b ) By a) and Corollary 5.3.3.7, f is positive iff u is positive. Take x E E . Then f ( x , x ) = (uxIx) =

(,tX

Iutx)

= luf212

Thus, by Corollary 5.3.3.11, f ( z , x ) = 0 ex

E

~erui

x E Keru

c ) follows from b ) . Proposition 5.3.3.13 ( 0 ) Let E , F,G be Hzlbert spaces. Take u E C ( E ,F ) and let p (resp. q ) be the orthogonal projection of E onto (Keru)' (of F 071 G).

a ) p (resp. q ) is the smallest orthogonal projection r in E (in F ) such that

b) Given v E L ( G , E ) with u o II = 0 , we have that p o v

=0

c ) Given v E C ( F ,G ) with v o u = 0 , we have that v o q = 0

5. Hilbert Spaces

116

a ) 1 - p (resp. 1 - q ) is the orthogonal projection of E onto Keru (of F on Im u' ) and so

uop =u,

(qou=u).

Now uo(1-r)=O,

((1-r)ou=O),

so that Kerr=Im(l-r)cKeru,

(ImucKer(1-r)=Irnr),

(Corollary 5.3.3.8) b) Since Irnv

c Keru = Kerp,

we have that

c) Since 1 r n q = ~ ~ K e r v , we have that voq=o Remark. If E = F = G , then the above proposition says that p and q are the right and left carrier of u , respectively. Theorem 5.3.3.14 ( 0 ) Let E be a Hilhert space, F a nonenpty upward directed set of L ( E ) , and 5 its upper section filter. Then the following are equivalent:

5.3 Adjoint Operators

a ) 3 is bounded above. b) 7 possesses a supremum i n C ( E ) If these conditions are fulfilled, then:

I n particular, C(E) and K ( E ) are C-order complete (Corollary 5.3.1.12).

>

a + b & c . Replacing F by {u - uo I u E 7, u uo) for some uo E 7 ,if necessary, we may assume that 7 c L ( E ) . Let v be an upper bound for F . Then +

for every x E E (Corollary 5.3.3.7). Define f :E x E

---t

IK ,

(x, y)

H

lim(ux1y) u,3

*

(Propositions 5.3.3.1 a),,), 2.3.3.7 a + b , and 2.3.3.8 a c & d). Then f is a positive sesquilincar form on E (Proposition 5.3.3.1 a),b)) and

for all z, y 6 E (Schwarz's Inequality). Hence f is continuous (Proposition 1.2.9.2 c + a). By Corollary 5.3.1.2, there is a U I E L ( E ) such that

for all x, y E E . Thus w E L ( E ) + (Corollary 5.3.3.12 b)) and

whenever u E F and x E E . Hence

for every u E F (Corollary 5.3.3.7). w is thus thc supremum of F in C ( E ) . Hence L ( E ) is C order cornpletc. By Proposition 5.3.1.17, K ( E ) is also C order complete. b + a is trivial.

5. Halbert Spaces

118

Corollary 5.3.3.15

If E, F are complex Hzlbert spaces, then

{u E L(E, F ) 1 u is an isomorphism (isometry))

is path- connected. Define

U V

:= {u

:= { UE C ( E ) I u is invertible (unitary)}

L ( E ,F ) 1 u is an isomorphism (isometry)) .

Take u, v E V . Then 11-' o v E U (Corollary 5.3.1.22). By Corollary 4.3.2.9 (Corollary 4.3.2.7). and Theorem 5.3.3.14, there is a continuous map

with

Then

[O,ll+V,

@c--1~f(~)

is a continuous path connecting u with v .

Remark. The corollary does not hold in the real case. In the finite-dimensional case, the set of isomorphisms (isometries) has two connected components. Corollary 5.3.3.16 (Atkinsorl, 1931) If E , F are cornplex Halbert spaces, then for each n E Z, {U E F ( E , F ) I Ind u = n )

is path-connected

Case 1

n=O

Take u, v E 3 ( E ,F) with Indu = Indv = 0

5.3 Adjoint Operators

119

By Corollary 3.1.3.15, there are isomorphisms uo, vo : E -t F and continuous paths (, 7 in F ( E , F ) connecting u and uo and v and vo, respectively. By Corollary 5.3.3.15, there is a continuous path in F ( E , F ) connecting uo and vo . Then v-'<( is a continuous path in F ( E , F ) connecting u and v . By Theorem 3.1.3.9, 17-1<< is a continuous path in { w E 3 ( E ,F)JIndw = 0 ) .

<

Case2 Take

U ,v

n#O

E F ( E , F ) with

Ind u = Ind v By Proposition 5.3.2.6, u' E F ( F , E ) , Ind u* = -1nd u Hence u* 0 v E F ( E ) , u o u* E F ( F ) , Ind (u* o v ) = Ind ( u o u * ) = 0 (Proposition 3.1.3.7). By Case 1, there are continuous maps

with

Then

is a continuous path in F ( E , F ) (Proposition 3.1.3.7) connecting u and v . By Theorem 3.1.3.9, this path lies in { w E F ( E , F)IIndw = n ) . Corollary 5.3.3.17 Every positive operator on a Hilbert space E is the supremum of an upward directed set in K(E)+. Let F be a closed vector subspace o f E and 6 the set o f finite-dimensional vector subspaces o f F . Then {nclGE 6 ) is an upward directed set o f K ( E ) +

5. Hilbert Spaces

120

(Corollary 5.3.3.8) and by Proposition 5.2.1.4 and Theorem 5.3.3.14 c), T F is its supremum. Now take u E L ( E ) + . By Corollary 4.3.2.11 (and Theorem 5.3.3.14), there is a sequence (pn),EN of orthogonal projections in E such that

is increasing and

Take n E IN. By the above considerations, there is an upward directed set 3, of K ( E ) + with supremum p, . Put

LEN,

Thus 3 is an upward directed set of K ( E ) + and it is easy to see that u is its suprern!~rnin L ( E ) . U Proposition 5.3.3.18

Let E be a Hilbert space and take u , v E L ( E ) . If uu*

< uu*,

then there is a w E L ( E ) such that

u = vw,

llwll

< 1.

If X E E , then

(Corollary 5.3.3.7). Define

By the above, wo is well-defined. Then wo E L(Imv*,E ) and llwoll 5 1. By Proposition 5.2.3.18, there is a utl

E

L ( E ) such that

5.3 Adjoint Operators

Put w := w ; .

Then llwll = llw;ll = llw1Il = llwoll

51

Take x , y E E . Then ( u x J y )= (xlu'y) = (xIwOv*y)= ( x I w I v * y ) = ( x ~ w ' u * = ~ )( v w x I y )

Thus ux = vwx ,

so that u=vw.

Proposition 5.3.3.19 define -

-

Let E be a real Hilbert space. Take u , v E L ( E ) and

0

0

( u ,v ) : E-+

(x,y ) +-+ ( U X - v y , uy + 21x1

E,

Then ( u ,v ) is positive zff it is selfadjoint and (uxlx) + (

+ 2(vxly) E IR+

7 4 ~ )

for all ( x ,y ) E E .

-

Assume ( u , v ) is selfadjoint. By Proposition 2.3.1.39, zu

*,

v=

-21'

Then

= (ux

-

+

11y1x) (uy

+ v x l y ) - i ( u x - v y l y ) + i ( u y + v x ( x )=

0

for all ( x ,y ) E E (Corollary 5.3.3.7) and the assertion follows from Corollary 5.3.3.7.

w

5. Hilbert Spaces

Definition 5.3.3.20 ( 0 ) (F. Riesz, 1918) Let E be a Banach space. Take u E L ( E ) and a E IK. The algebraic eigenspace of u corresponding to cr is defined to be

U Ker ( a 1

-

u)"

nEN

Its dimension is the algebraic multiplicity of a with respect to u , and its elements are the algebraic eigenvectors of u corresponding to a .

Since the eigenspace is contained in the algebraic eigenspace, the algebraic multiplicity is a t least the multiplicity. But if the multiplicity of a is 0 , then the algebraic multiplicity of a is also 0 . If u is compact and cr # 0 , then E, (1 - 6u) (Definition 3.1.3.18) is the algebraic eigenspace of u corresponding to a and the algebraic multiplicity of a with respect to u is finite (Theorem 3.1.3.17 d)). The algebraic eigenvector is also called "root vector", "null-element" , "generalized eigenvector" , "adjoint vector". Proposition 5.3.3.21

(0)

The eigenspaces and the algebraic eigenspaces

of normal operators on a Hilbert space coincide.

Let u be a normal operator on a Hilbert space and take a E IK. Then a 1 - u is normal, so that

U Ker ( a 1 nElN

by Corollary 5.3.3.1 1.

-

u)" = Ker ( a 1 - U),

5.3 Adjoint Operators

5.3.4 Normal Operators

Proposition 5.3.4.1 u is normal iff

(0)

Let u be a n operator o n the Hilbert space E .

1lux11 = llutxll for every x E E . In this case Kcr u = Ker u'

and

E=L@Keru. I n particular, any normal Fredholm operator has index 0 . Given x E E

11~x11~ - Iu*xI12 = (UX~UX) (u-xIutx) =

Since u'u-uu' is selfadjoint, it follows from Corollary 5.3.3.4 that the following are equivalent:

u is normal, u*u- uu*= 0 ,

x

E

E ==+ IIuxll = IIutxII

It follows irnmcdiately that Ker u = Ker U* . By Proposition 5.3.2.4, Ker u* = (Im u ), ~ so that

5. Halbert Spaces

124

(Proposition 5.2.3.7, Corollary 5.2.3.9).

If u is a normal Fredholm operator, then by Proposition 5.3.2.6, Ind u = Dim Ker u - Dim Coker u = = Dim Ker u - Dim Ker u* = 0 .

Corollary 5.3.4.2 operator o n E . Put

(0)

Let E

# (0) be a Hilbert space and

u a normal

cr := inf {lluxll I x E E , llxll = 1) u is an isorrlorphzsm of Banach spaces iff cr

> 0 and in this case

If u is an isomorphism of Banach spaces, then

for every x E E . Hence

Conversely, if cr > 0 , then

for every x E E (Proposition 5.3.4.1). Thus u is an isomorphism of Banach spaces and

(Corollary 5.3.2.5). By the above

Take

P E U F ( 0 ) .Then Ilu - ( u - l j l ) l l =

By Corollary 2.2.4.4, u

-

1

Iljl < lal = IIu-~II .

D l is invcrtbilc, so that

$ a(u)

5.3 Adjoint Operators

Corollary 5.3.4.3 E # (0) . Take Q E

125

(0)

Let u be a normal operator o n the Hilbert space IK . Then the following are equivalent:

a1 - u is a normal operator on E and the assertion follows from Corollary 5.3.4.2.

Corollary 5.3.4.4 E # ( 0 ) . Take a E

(0) IK

Let u be a normal operator o n the Hilbert space and let F=Ker(crl - u ) .

a) F = Kcr ( 5 1

-

u*) .

b) F reduces u . c) If u is compact and a

# 0 , then

a) a 1 - u is normal, so that the assertion follows from Proposition 5.3.4.1. b) Take x E F . By a ) ,

Hence

By Proposition 5.3.2.9, F redr~ccs11. c) a 1 - u is a Freldholrn operator (Corollary 3.1.3.12) and so Irn ( a 1 - u) is closed. The assertion now follows from Proposition 5.3.4.1. Corollary 5.3.4.5 E # (0) , then

(0 )

If u is a normal operator o n the Hilbert space

and Kcr ( a 1 - u ) l K e r (Dl - u) for distinct a, 13 E K .

5. Hilbert Spaces

126

The first assertion follows from Corollary 5.3.4.4 a). Take x E Ker ( a 1 - u) , y E Ker (Dl - u) . Then

(Corollary 5.3.4.4 a)), so that

and Ker ( a 1 - u ) l K e r (Dl - u ) Proposition 5.3.4.6 ( 0 ) . u zs unitary iff

Let u be a normal operator o n the Hilbert space E

#

for every x E E

By Corollary 5.3.1.22, u is unitary iff it is an isometry, and by Corollary 5.3.4.2, this is equivalent to

Proposition 5.3.4.7 ( 0 ) Let u be a self normal operator o n the HiLbert space E # (0) , such that a(u)\{O} is discrete. Given a E IK , let a, be the orthogonal projectzon of E onto Ker ( a 1 - u) .

a) u =

an,

(the spectral decomposition o f u ) .

oEo(u)

b) o , p E IK + x,ao = Gaga, .

then

for every rr

E

IK\{O) . In particular,

5.3 Adjoint Operators

127

e) u and {T, I a E a(u)\{O)) generate the same closed unital subalgebra of E . It is finite-dimensional iff u(x) is finite. f ) Given f E C(u(z)) , with f (0) = 0 whenever u ( x ) is infinite,

a) - f) . Let A denote the set of isolated points of u(x) . By Theorem 4.1.4.6 a + b & c, and Theorem 5.3.1.13 (and Proposition 5.3.2.8 b + a), there is a family ( u , ) , ~ ~of orthogonal projections in E such that

a ( u ) finite

==+

u, = 1 , QEA

a E u(x)\{O)

==+ u, # 0 .

Take cu E A and z E In1 u, . We get, successively, that

x E Ker ( a 1 - u) ,

Im u,

c Ker ( a 1 - u) , U,

(Proposition 5.2.3.13 a

3

c)).

= u,a,

128

5. Hilbert Spaces

Let a , p be distinct elements of A . Then, by Corollary 5.3.4.5, Ker ( a 1 - u ) l K e r (PI - u) . Thus, by Corollary 5.2.3.15 a

+b, 7raXfl = 0 .

We deduce that U,7rfl

= uo7ra7rfl = 0 .

Take a E A\{O) arid x E Ker ( a 1 - u) . We get, successively, that

X=U,X

E Imu,,

K e r ( a 1 - u ) c Imu,, Ker ( a 1 - u) = 1111 u, ,

All the assertions follow now from Theorem 4.1.4.6. g) {uIrCis a unital C'subalgebra of L ( E ) (Corollary 4.1.4.2 b)), so that

by e). It remains to prove no E {u}". Take v E {u)" and x E E and put

By b), for every B E v f ( A ) ,

C T,

a €B

is an orthogonal projection so that

5.3 Adjoint Operators

(Pythagoras' Theorem). Hence (T,x),,~ is surnrnable in E (Corollary 5.2.2.4). By a ) and b), we obtain successively,

Remark. r o does not belong in general to the unital C*-subalgebra of L ( E ) generated by u . Indeed, let

and put Pn :

@

e2,

( x ~ ) ~ E N (6nkxk)kE~

for every n E IN.Then

is the involutive unital closed subalgebra of C ( E ) generated by u , and not belong to it.

TO does

130

5. Hilbert Spaces

5.4 Representations A representation of an involutive Banach algebra E consists of a homomorphism of involutive algebras from E into the C'-algebra of operators on a Hilbert space. We show that in the case of a C'-algebra even faithful -- that is to say norm-preserving - representations can be found. We shall return to the problem of representations later in the book. 5.4.1 Cyclic Representation Definition 5.4.1.1 ( 0 ) Let E be a (unital) involutive Banach algebra. A (unital) representation of E is a pair (H, cp), where H is a Hilbert space and cp is a (unital) involutive algebra homomorphism E + L ( H ) . (H, cp) is called irreducible if cp(E) acts irreducibly on H . (H,cp) is called nondegenerate if cp(E) acts non-degenerately on H . (H, cp) is called faithful (the 0-representation) if cp is injectzue (if cp(E) = ( 0 ) ) ) . An element of H is called cyclic for (H,cp) if it is cyclic for cp(E). An element of H is called separating for (H, 9)if it is separatzng for cp(E) . Two representations ( H ,9) and (K, q9) of E are called equivalent if there is an isometry u : H + K such that

cpx = u'

0

(Qx) 0 U

for every x E E (it is an equivalence relation). In this case u is called an equivalence of the two representations ( H , cp) and (K, $)I .

If E is a C*-algebra, then a representation (H,cpj of E is faithful iff it preserves the norms (Theorem 4.2.6.6). If a C-algebra is simple then any representation of it is either faithful or the 0-representation.

Theorem 5.4.1.2 ( 0 ) (Gelfand-Naimark, 1943, Segal, 1947) T h e GNSconstruction) Let x' # 0 be a continuous positive linear fonn on an involutive (quasiunztal) normed algebra E and f the positive sesquilinear form

(Proposition 2.3.4.6 b)). Put

Let u : E

+E/F

he the quotzent map (Proposition 5.1.2.9 b)),

5.4 Representations

E/F x E/F

-+ I K ,

( X ,Y )

ct ( X I Y )

-

the associated scalar product (Proposition 5.1.2.9 d)), and E/F be completion of E / F (Corollary 5.1.1.7). E / F is called the Rilbert space associated to x1 .

-

*E/F

c ) E separable

f)

x

E

separable.

E , Y E E / F , y,,y2 E Y

* u(xy1) = u(xy2).

Put

-

g) For ?very x E E , there is a unique operator Z o n E / F such that for every Y E E / F ,

Furthermore,

l l ~ l 5l

llx*xll+ 5 llxll (and xl(x*x)

5 11~11211~'11~

h ) The map cp:E--+f(T~)~

x z ?

is a continuous homomorphism of involutive algebras with n o r m at most 1 (equal to 1). p is called the algebra homomorphism associated to X' and ( E I F ,p ) is called the representation of E associated to x' .

-

i) If E is unital, then the algebra homomorphism associated to x' is also unital. j) If x' is a n algebra homomorphzsm, then

F = Kcr x1

and E / F as one-dimensional.

5. Hilbert Spaces

132

-

k) If E zs quasiunital, then there is a unique ( E E / F such that

-

~ ' ( 5= )

(uxlt)

for all x E E . ( is cyclic for ( E / F , 9) ,

11511 = llx'llf and

for all x E E . If 5 is an approximate unit of E , then u(5) converqes weakly to ( . If E is unital then ( = u l . ( is called the cyclic vector associated to x' and the tnple ( E I F , p,() is called the GNS-triple of E associated t o x' . Two GNS-triples (HI,91, ( I ) , (HZ,pz, (2) of E are called equivalent zf there is an equivalence v of the representations ( H I ,p l ) and (H2,v2) such that vtl = t2,.In this case v zs called an equivalence of the two GNS-triples ( H I ,9 1 ,(1) , (HZ,p2, t 2 ) of E .

-

a ) We have

-

f (x) = f (x,x) = xl(x*x) b) If x E E , then, by a), 1121x11~ = xl(x*x)I

I

I I X ' I II I X * X I I

11~11~

I I X ~ I IIX * I I11x11= I I X ~ I I

so that u E C(E, E I F ) and llull 5 llxlll t . If then, by a),

5

is an approximate unit of E ,

(Proposition 2.3.4.10 a)), so

-

if E is quasiunital. c) follows from the fact that E I F is separable and dense in E I F .

5.4 Representations

d ) By a ) and Corollary 2.3.4.8,

XVI

- xy2 = X(Y,- YZ)E F .

g) Uniqueness is obvious. Define

v:E/F-+E/F, v is linear. If Y E E/F and y

E

Yc+xTY.

Y , then, by a) and d ) ,

llvy1I2 = IIxTY1I2 = Ilu(xy)l12= f"(~?l)< ? ( Y ) I ~ X= * ~1lx*x11 I I lly1I2. Hence v is continuous and Ilvll 5 llx'xllf . Let Z be the coritinuous extension of v . Then

llzll = llvll I llx*xll~I llxll. If y

E

E # , then, by b),

11: 1 1211~' 1 1

> llzll'll~1I21 ll~11211~~y112 Z II:(uy)1I2

=

= l l x ~ u y 1 1=~11~(x~)112.

Hence, if

5 is an approximate unit of E , then, by a), xl(x*x) = I ( U X ( / ~

= lirn ( ( ~ ( x y ) I ) 1I~J Z ) ) ~ ) ) ~ ' ) ~ ~93

h) It is easy to see that cp is linear. Take x, y E E , A , B E E I F , and a E A , b~ B . Then

5. Hilbert Spaces

134

Hence

-

GZY=ZY, x*=21, which shows that cp is a homomorphism of involutive algebras. By g), cp is continuous with norm a t most 1 . If 5 is an approximate unit of E , then, by a),g), and Proposition 2.3.4.10 a), llxlll = limxl(x*x) 5 llxlll lirn 11Z112. 2,3

2,3

Hence, if E is quasiunital, cp has norm 1 . i) Take X E E / F and x E X . Then

-

l E = 1-E I F

'

j) We have

T(x) = xl(x*x) = xl(x*)xl(x)= 2 ' o x 1 ( x ) = ~ x I ( x ) ~ ~ Thus

F = Ker x' and E / F is one-dirner~sional. k) Take x, y E E and let 3 be an approximate unit of E . Then

<

1x1(y*x)12 xl(y*y)xl(x*x)=

( a) and Proposition 2.3.4.6 c)). Thus

<

) l ~ll~x11~ ~ X I ( X=)lim ~ ~~ x ~ ( ~ ' x llxlll Y !3

(Proposition 2.3.4.9, Proposition 2.3.4.10 a)). Let y1 be the factorization of x1 through E / F . The11

5.4 Representations

for every x E E , so that y' is continuous and

5 E/F. lly1Il

We extend y' continlio~islyto

llxflli .

By b),

< I I Y ~ I I IIU(X*X)IIL I I Y ~ I I llull llx*xll = l l ~ ' l l llx'llf llx*xll

xf(x*5)= yt(u(x*x)) for every x E E . Hence

llxlll = limxl(x*x) 2,3

< lly'll llx'lli

(Proposition 2.3.4.10 a)),

IIY' I I By Corollary 5.2.5.3, there is a

-

= llxllli .

< E E / F , such that

Then

for all x E E . The uniqueness of ( is obvious. Take x, y E E . By f),

Since y is arbitrary,

E< = u x . Hence

-

{Z
=

E/F

i.e., ( is cyclic for ( E I F ,9). For every x E E , (uxl<) = x' (x) = lim X ' ( ~ * X=) lirri (uxluy) y.3

Y,X

Hence u ( 3 ) converges weakly to < . If E is unital, then by i),

5. Hzlbert Spaces

136

Proposition 5.4.1.3 Let E be a n involutive Banach algebra, ( H , v) a representation of E , and E H .

<

a ) The map

belongs to E; and

for every x E E . If 5 is a n approximate unit of E and the representation is non-degenerate then

for every q E H and llxlll =

11<112.

b) Assume E quasiunztal and ( cyclic for ( H , c p ) and let ( K ,$, q ) be the G N S -triple of E associated to the above x ' . Then there is a unique equivalence u of the representatzons ( H ,c p ) and ( K , $) such that

I n particular, ( H , g,<) m a y be identified with ( K , 111,171 a ) For every x E E ,

Since y is continuous, x' is also continuous. For ~

E and E ~ E H ,

Since the representation is rion-degenerate,

5.4 Representations

It follows llxtll = limxt(x*x) = lim li(cpx)
~,a

2,s

(Proposition 2.3.4.10 a)). b) By a) and Theorem 5.4.1.2 k),

for every x E E . Hence the map

<

is well-defined and it is an isometry. Since and 7 are cyclic for the represcntations (H, p) and ( K , $) , respectively, the above map may be extended to an isometry u : H -t K . Take x, y E E . Theri

Sincc

< is cyclic for

( H , p) , it follolvs

Hence

(@x)(u<- 71) = 0 . Since $ ( E ) acts non-degenerately on K , it follows 11<-7=O,

u<=v

(Proposition 5.3.2.21 a + b) . Let v be an equivalence of the represcritatior~ (H:p) and ( K , $ ) with

v < = p. For every x E E , u(~x)= ( (@x)IL< = ($x)v( = 71(cpX)<. Since ( is cyclic for (H, p) , u = v .

5. Hilbert Spaces

138

Corollary 5.4.1.4 Let E be an involutive quasiunital Banach algebra. Take x E E , x', y' E EL . Let ( H ,p, <) and ( K ,*, q ) be the GNS-triples of E associated to x' and y' , respectively.

a ) If there is a u

E

C ( H ,K ) such that

for every y E E and

then

b ) The following are equivalent: b l ) There is an isometry u : H

+ K such that

'Py = 21.0 (*Y)

0

'1L

for eve? y E E and ( $ 5 )= ~ u< ;

b2) ( $ x ) q is cyclic for ( K ,$) and x1 = x y f x *. a ) By Theorem 5.4.1.2 k ) , if y E E , then

( Y , x ' )= ((vy)
= (*(x'yx)qlq) = (x'yx, y') = ( y ,x y f x ' ) .

Hence

5.4 Representations

bl +b2. Since u< is obviously cyclic for (K,@) from a).

139

, the assertion follows

b2 ~ b , By . Theorem 5.4.1.2 k), if y E E , then

= ((~/~(x*Yx))= T ~((~cIx)*(*Y)($x)T~T) T) = ((*Y)(V~X)TI (*x)T).

bl) now follows from Proposition 5.4.1.3 b) .

Corollary 5.4.1.5 Let E be an involutive quasiunital complex Banach algebra, F the C*-hull of E , and u : E -+ F the canonical map. Then Y'

0

,

u E E:

IlY'll = l l Y ' ~ ~ l l

for every y' E F; , and the map

F:+E:,

y'ey'ou

is bijective. Take x' E E; and let ( H , p , < ) be the GNS-triple of E associated to x' . By Proposition 4.1.1.22 f), there is an involutive algebra homomorphism : F + L ( H ) , such that

+

Define

By Proposition 5.4.1.3 a) and Theorem 5.4.1.2 k), y' E F: ,

IIY' I I

=

ll<1I2

= llx'll

and

for every x E E . Herice y'

0 U

= x'

140

5. Hilbert Spaces

Proposition 5.4.1.6 ( 0 ) Let E be a quasiunital involutive Banach algebra. Take x' E E'+ and let ( H , p,() be the GNS-triple of E associated to x1 . a) If v E , C ( H ) ~n p ( E ) ' , then the map

is a positive linear form less than x' . b) If y' is a positive linear form on E less than x', then there is a unique u E ,C(H)!/ n p ( E ) ' such that ~ ' (= 4 (((PX)U
for every x E E . a) For any x E E , = (ucp(x)*FIJ) = ~ ' ( x *= ) ( ( ~ x ' ) v < l O= (~~lp(x*)FlO

= ( ~ ( P x ) < I ( P x ) OE IR+

(Corollary 5.3.3.7). Hence y1 is positive. Since (x, x1 - y') = (x, xl) - (2, 9') =

for every x E E , it follows from the above result that x1- y' is positive. Hence y' 5 x' . b) Let q : E + E/Nz, be the quotient map and define g : E x E 4I K ,

(x, y)

+ yr(y*x) -i .

g is a positive sesquilinear form (Propositiori 2.3.4.6 b)). By Proposition

2.3.4.6 c) and Theorem 5.4.1.2 a):

5.4 Representations

141

for all x , y E E .Hence there is a unique positive sesquilirlear form h on E/N,l such that

for all x, y E E . By the above inequality,

-

for all X , Y E E / F . By Proposition 1.2.9.2 c + d, we may extend h continuously to H := E/N,>. By Corollary 5.3.1.2, there is a u E L ( H ) such that

for all

X,Y

E

H . We have -

( X I u * Y )= ( u X I Y ) = ( Y I u X ) = = h ( Y ,X ) = h ( X ,Y ) = ( X I u Y ),

for all

A',

Y E H . Thus u is selfadjoint. Since

( u X I X ) = t l ( X ,X ) E [O,1] for every X E H # ,

4 u ) c I0,ll by Theorem 5.3.3.6. Hence u E L ( H ) , # . We have

for allx, y E E . Put F := p x for every x E E and take x, y, r E E . Thcn

= ( W Y ~ ~ ( X *=ZY) ~) ( Z * X= Y )( U Q ( X Y ) I Q= Z )(uFq2jlqr).

Hence

5. Halbert Spaces

142

u E (p(E)'. Take x , y E E and let

5

be a n approximate unit of E . Then

Y ' ( Y X ) = (uqxlqy*)= (u?
y l ( x )= lim yl(yx) = lim(u@
y,3

Now we show that u is unique. Take v

(p(E)' such t h a t

y ' b ) = ((lpx)v
(UQXI~Y)

for all

2,y E

= Y ' ( Y * Z ) = ( Z v < I < )= (TZv
E , we have that v =U

Proposition 5.4.1.7 ( 0 ) Let E be a n involvtzve Banach algebra , E # (0) , ( H ,p) and ( K ,@) two representations of E , and u, u two equzualences of ( H ,p) and ( K ,$) . If p ( E ) > K ( H ), then there is a n cu E K with

For every x E E ,

so that

5.4 Representations

By Corollary 5.3.2.14, there is an

Q

E IK

with

It follows

for every x E E , so that

-

la1 = 1

Proposition 5.4.1.8

( 0 ) We use the notation of

Theorem 5.4.1.2 and as-

0

sume E is a real C*-algebra. We denote by E the complexificatior~of E , 0

by E/N,, the complexijication of the pre-Hilbert space E/N,f 5.3.1.8), by

(Proposition

the quotient map (Corollary 4.3.6.2 ca + cl), and by ( H ,p, (0) and ( K ,@, 70) the GNS-trzples associated to x' and (x', 0) , respectively.

A

b) Let (<, 7) E E/IV,, and take x 1: x2,yl , y2 E E such that uxl = Ux2 = < ,

UY1 = UYz = q .

Then

Put

-

( E : 7 ) :=:

(21, Y

I ).

c ) The map

is an isomorphism of complex pre-Hilbert spaces. Hence its extension

~ : H + K is an isomorphism of Hilbert spaces.

5. Hilbert Spaces

144

= (x'x

+ y'y, x') + i ( x * y - y'x, x') = (x'x + y * y , x f )

Hence

c) The surjectivity of the map is obvious. For

( G ( x 1 , y 1 ); 1

(XO,YO)

E

i,

y l ) : ( 5 2 , y2) E E ,

( ~ 2 , ~ 2= ) )( ( x 2 , Y 2 ) * ( x l I Y l () ,~ ' 1 0 )= )

which proves the other assertions

d ) For

(XI,

5.4 Representations

Hence * ( ( x , y ) ) o P =Po(cpx,cpy). c ) BY c), for (x,Y ) E E , h -

(

( x ,Y )

I ( t o , 0 ) ) = ( ( u x ,U Y ) I (<0?0 ) ) =

+

= ( x ,z') i ( y ,2') = ( ( x ,y), (x', 0)).

Hence

5. Hilbert Spaces

146

5.4.2 General Representations Proposition 5.4.2.1 ( 0 ) Let E be a (unital) involutive Banach algebra and (H,, cp,),El a family of (unital) representations of E . Define

T h e n (H, cp) as a (unital) representatzon of E ; it is called the Hilbert sum of the family (H,, cpL)rEl.If each (H,, 9,) is non-degenerate, then (H, cp) is non-degenerate.

By Corollary 4.1.1.20, IIcpL1l 5 1 for every L E I . Thus cp is well-defined. By the Propositions 5.1.3.3 and 5.1.3.4, cp is a (unital) algebra homomorphism and by Proposition 5.3.2.3, cp is involutive. The last assertion is obvious. Corollary 5.4.2.2 ( 0 ) Let E be a n involutive quasiunital (unital) Banach algebra and (xi)IEI a nonempty family i n r ( E ) . Given L E I , let (H,, 9 , ) denote the representatzon associated to xi and put

H := a)H,

(the Hilbert space associated to (x:),~I)

LEI

cp : E

-+ L(H) ,

x

+-+

a)cp,x

(the algebra homomorphism

LEI

associated to (X:),~I ).

Then ( H , cp) is a non-degenerate (unital) representation of E (the represen(F has n o r m 1 , and

tation of E associated to (x:),~, ),

Ilcpxl12 L supxI(x'x) LEI

for every x E E . If H, is separable whenever L E I (this occurs i f E is separable) and if I is countable, then H is separable.

By Proposition 5.4.2.1, (H, cp) is a non-degenerate (unital) representation of E and by Theorem 5.4.1.2 g),h), cp has norm 1 and llcpxl12 2 supx:(x*x) LEI

for every x E E . The last assertion follows from Proposition 5.1.3.1 d ) (and Theorem 5.4.1.2 c)).

5.4 Representations

14'7

Definition 5.4.2.3 ( 0 ) Let E be a n involutive quasiunital Banach algebra. For every nonempty subset A of r ( E ) , the representation of E (the Hilbert space, the algebra homomorphism) associated to (xr)zfEAwill be called the representation of E (the Hilbert space, the algebra homomorphism) associated to A . The universal representation of E is the representation of E associated to r ( E ) . Corollary 5.4.2.4 ( 0 ) Let E be an involutive quasiunital Banach algebra and ( H , cp) its universal representation. T h e n for each x 1 E E l , there is a E H such that

<

and

for every x E E . The assertion follows immediately fro111 Theorem 5.4.1.2 k).

Corollary 5.4.2.5 ( 0 ) (Gelfand-Naimark, 1943) Let E be a C*-algebra. If A is a subset of r ( E ) such that ro(E) c 2 , then the representation of E associated to A is faithful. If E is ,separable, then we m a y take A to be countable and, i n this case, the Hilbert space associated to A is separable.

E is quasiunital (Theorem 4.2.8.2). Let ( H , cp) the representation of E associated to A . By Corollary 4.2.8.5 d) and Corollary 5.4.2.2,

for every x E E . If E is separable, then r ( E ) has a countable base (Proposition 2.3.5.9 b)), so we may take A to be countable. By Corollary 5.4.2.2, H is separable is this case.

Remark. A C* algebra E niay have a faithful representation (H, 9) with H separable even if E itself is not separable as the Example L(e2) shows (Proposition 5.5.2.11 a)). See also the Remark of Example 5.4.3.7.

148

5. Hzlbert Spaces

Proposition 5.4.2.6

(0 )

Let E be a real C*-algebra,

h

its comple-

h.

Denote by H the xification, and ( H , cp) the universal representation of underlying real vector space of H endowed wzth the scalar product

a)

& is a Hilbert space and its canonical n o r m coincides with the canonical n o r m of H .

then ( f i , cpl E ) is afaithful b) If we identify E with the subset E x { 0 ) of representation of E called the complez universal representation of E . C) Let x' E E' and let

((EL,

be a finite family i n H x H such that

0

for every (x,y ) E E . Then

for every x E E . d ) For every x' E S i there zs a

<EH

such that

for every x E E i n H x H such that e) For every x' E E' there zs a famzly ( ( < f , qk))LEIN4

for every x E E . a ) is easy to see. b) follows from Corollary 5.4.2.5 (and Proposition 4.1.1.27 c),d), Theorem 4.2.8.2). c) For every x E E ,

5.4 Representatiorts

so that

d ) Put y

:

(x,y)c,x'(x)+ixl(y). 0

By Proposition 4.1.1.27 c) (and Theorem 4.2.8.2), y' E (E); and

IIY' I I

= llx'll .

By Corollary 5.4.2.4, there is a J E H such that

II(II

=

~ l ~ ~and llt

for every x E E e) Put y :

+C ,

(x, y) exr(x)

+ izr(y) .

0

0

Then y' E ( E ) ' so that there is a family (y;)kEIN4in (E): y r = y; - Yi+ i y i

-

S I I C ~ Ithat

iy;

(Corollary 4.1.2.7 d)). By Corollary 5.4.2.4 (and Theorem 4.2.8.2), for every k IN4 there is a tk H such that Y;((x>Y)) = (P((x, ~))tkl
for every (x, y) E E . Put

Then

for every (x, y) E E . By c), 4

x'(x) = C ( ( ( ( P X ) G I T ~ ) ) k=l

for every x E E .

5. Hilbert Spaces

150

Proposition 5.4.2.7 ( 0 ) Let E be an involutive quasiunital Banach algebra, A a nonempty subset of T ( E ) , and F the closed vector subspace of E1 generated by

If EF 2s Hausdorff, then the representation of E associated to A is faithful. Let cp be the algebra homomorphism associated to A and take x 6 E\{O) . Since EF is Hausdorff, there are y, 2 E E and XI E A with

Let $ be the algebra homomorphism associated to x' and let q be the quotient map E -t EIN,!. Then

(Proposition 2.3.4.10 a),b), Theorem 5.4.1.2 a ) ) , so that

Hence cp is injective.

W

Proposition 5.4.2.8 ( 0 ) Let E be a (unital) involutive Banach algebra, ( H ,p) a (unital) representation of E , and K a p(E)-invariant closed vector subspace of H . Given x E E , define

Then K' is p ( E ) -invariant and ( K ,$) is a (unital) representation of E called the compression of ( H ,p) to K . Take

< E K L and 7 E K . Then, given x E E ,

5.4 Representations

Hence K L is cp(E)-invariant Take x, y E E . Then

for every ( E K and therefore

If X E E , then

for all

c, q E K , so that

Hence ( K , *) is a representation of E . It is easy to see that if E and cp are unital, then 1C, is also unital.

.

Proposition 5.4.2.9 Let E be a n involutive Banach algebra and ( H , p) a representation of E . The-e is a famzly (HL)LEIof cp(E)-invarzant closed vector subspaces of H such that: 1) H L I H x whenever

1, X

E I are distznct.

2) T h e compression of (H, cp) to H, is cyclic for every 3) T h e compression of (H, cp) to

U H, (&I

Given [ E H , put

r

L

E

I.

is a 0-representation.

Let M be the set of subsets A of H such that

for every

<E A

and

.

for distinct [ , q E A . Let M be ordered by inclusion and let A be a maximal element of M . The family (HF)cEAobviously has properties 1) and 2). Property 3) follouls from the maximality of A and Proposition 5.4.2.8.

5. Hilbert Spaces

152

Corollary 5.4.2.10 Let E be a n involutive Banach algebra. T h e n any representation of E is the Hilbert s u m of cyclic representations and of a Orepresentation. Proposition 5.4.2.11 Let E be a quasiunztal involutive Banach algebra, F a n involutive closed zdeal of E , and ( H ,p ) a non-degenerate representation of F . Then there is a unique representation ( H , $J) of E with $IF = p . Suppose that there is a cyclic vector

x': F +IK,

< for

X H

( H ,p ) . Define

((cpx)
By Proposition 5.4.1.3 a), x' E F; Take x E E . Then

for every y E F (Corollary 2.3.4.8). Hence there is a unique Z E C ( H ) such that

for every y E F . If x E F , then

Since

for all y, 2 E F ,

for every y E F . Take x , y E E and z E F . Then

5.4 Representations

Choose x E E and y, z E F . Then

= (El(cpy)*G(cpz)E) =((YY)EI~(Y~)O

Thus

-

%' = x 8 and the map

is an involutive algebra homomorphism extending cp. Let $J : E + L ( H ) be an involutive algebra homomorphism extending p . Then, given any ( x ,y) E E x F ,

It follows that

for every x E E , which establishes the uniqueness of @ . The general casc follows from the cyclic casc and Corollary 5.4.2.10.

Proposition 5.4.2.12 Let E be an involutive Banach algebra, ( X : ) ~ , I an absol~~tely summable family in E: , and x' its sum (by Proposition 1.7.1.5 c), x' E E ; ) . Let ( H ,c p ) be the representation of E associated to x' and q : E + E/N,c the quotient map. For every L E I , let (H,, y,) be the representation of E associated to x: and q, : E + EIN,: the quotient map. a ) The map

-

is well-defined; it factors through H = E/N,, preserves the norms.

and this factorization v

5. Hilbert Spaces

154

b) x E E

C)

+ v o (cpx) =

u ( E ) is

{

)

acp,x) o v

a W x ) I E E ) -invariant and ( H ,p) is equivalent to the comL

E

I

pression to u ( E ) of the Hilbert sum of the family ( ( H , ,p,))LEl. d ) If E is quasiunital and if, for every L E I , <, is the cyclic vector associated to x: , then v ' ( < , ) , ~is~the cyclic vector associated to x ' . a) W e have

for every x E E (Theorem 5.4.1.2 a ) ) and the assertions follow. b ) Since

for every y E E ,

c ) follows from b ) and a) d ) Since

~ cyclic vector associated for every x E E (Theorem 5.4.1.2 k ) ) , v * ( < , ) ,is~ the t o x' .

Theorem 5.4.2.13 ( 0 ) (Goodearl, 1982) Let E be a symmetric involutive unztal real Banach algebra such that

for every x E E . Then I? is a real C*--algebra.

5.4 Representations

155

Take x E Re E . Since { P ( x ) l P E IK[t]} is dense in F := E ( x , l ) , all elements of F are selfadjoint. By By Proposition 2.3.2.21 a + b, F is symmetric, so that, by Proposition 2.1.5.18 b + a,

for every y E F . By Gelfand's Theorem (Theorem 2.2.5.4) and Proposition 4.1.1.2 b), r ( x ) = lim ~ l x ~ l= l i llxll n+m

Let ( H ,cp) be the representation of E associated to r ( E ) and let z E E . By Proposition 2.3.5.15 a the above,

+ c,

v A

x l x ( r ( E ) ) is the convex hull of ~ ( x ' x ) By .

By Corollary 5.4.2.2,

Hence cp preserves the norms and therefore E is isomorphic to a closed invoW lutive subalgebra of C ( H ) . In particular, E is a real C'-algebra.

Proposition 5.4.2.14

( 0 ) Let

E be a real Cb-algebra, A a subset of . r ( E ) , 0

and ( H ,9) the representation of E associated to A . If E and H denote the complexifications of E and H , respectively, then

0

(Corollary 5.3.1.12) is equivalent to the representation of E associated t o { ( x ' ,0 ) I x' E A } (Corollary 4.3.6.2 cs ful i f l ( H ,c p ) is faithful.

+ c, ).

This representation is faith-

The assertion follows from Proposition 5.4.1.8 c),d).

W

5. Hzlbert Spaces

156

5.4.3 Example of Representations Example 5.4.3.1 Let T be a normal topological space. Take p E Mb(T)+\{O) and let S be the carrier of p . Let E denote the involutive unital Banach algebra C ( T ) (Example 2.3.2.4). Put

a ) x' E E> and N,, = F . b)

The jactonzation v of the inclusion m a p E znjective.

c ) If X , Y E E / F and x

E

-+

L 2 ( p ) through E / F is

X , y E Y , then

d ) v m a y be extended uniquely to a n isometry of Hilbert spaces

-

e) If we identify E / F with L 2 ( p ) using the isometry i n d), then

Z ( Y ) = XY

1

llZll

=

llxlSllO0

for every ( x ,y ) E E x L 2 ( p ) . f)

T h e jollowzng assertions are equivalent: fl)

S is a one-poznt set.

f2)

the representation of E associated to

f3)

the Hilbert space associated to x' is one-dimensional.

Example 5.4.3.2 M b ( T ) +with

XI

is irreducible.

Let T be a Hausdorfl topological space. Take p, u E

5.4 Representations

T h e n the following are equivalent:

a ) The representations of C ( T ) associated to x' and y' are equivalent. b) There are positive real Bore1 functions f and g on T such that

f E L2(d>

v=f2./1,

g E L 2 ( v ) , p = g2.v.

Given x E C ( T ), define 'PX : L 2 ( p ) -+

L2(p),

t 4xF

By Example 5.4.2.1, ( L 2 ( p jcp) , . and ( L 2 ( v )$) , are the representations of C ( T ) associated t o x' and y', respectively. a + b . Let u : L 2 ( v )+ L 2 ( p ) be an isometry of Hilbert spaces such that

for every x E C ( T ). Let K be a compact set of T and q E L 2 ( v ) with q=7)e~.

Then for every x E C ( T ) with

Hence

5. Halbert Spaces

158

T\K. I f K is a p-null set, then K is a v-null set as well. Hence v is pabsolutely continuous. By tho Radon-Nikodym Theorem, there is a positive real Borel function f on T such that f€L2(p), v=f2.p. It follows that there is a positive real Borel function g on T such that

b

+a.

Define

u, v are operators preserving the norms and

Hence u and v are isometries and

Let x E C ( T ) and

7)

E

L 2 ( v ) .We have

so that

'px = u'

0

(*x) 0 u

Hence the representations associated to x' and y' arc cquivalent.

Example 5.4.3.3

Let H be a Halbert space, A a nonempty set of pairwise orthogonal elements of H\{O) , such that

and E a C*-subalgebra of L ( H ) . Put

x': E - + I K ,

x++x(x
5.4 Representations

f :E x E

and for

---t

IK , ( x , y )

ct x l ( y * x ) ,

< E A,

a ) x' is a positive linear form, f is a positive sesquilinear fonn, and F ={xEE

I T ( x ) = 0) ;

-

we endow E / F with the scalar product associated to f 5.1.2.9 d)) and denote its completion by E / F .

(Proposition

b) T h e factorization of the map E

-4

a ) HC

I

FEA

5

( X ~ F E A

-

through E / F preserves the norms; we identify E / F with the corresponding closed vector subspace K of a ) HF via the extension of the above Fen map (Corollary 5.1.1.4). c) If we put

and define

for every x E E , then ( K ,cp, X ) is the GNS-triple associated to x'

d ) If A contains a unique element <, then cpx = x1K for every x E E and the above representatzon is faithful if x ( K L ) = (0) for all x E E . e)

If K ( H ) c E , then E E = H F = H f o r e v e r y ( E A , K = a ) H E , and FEA

the representation from c) is faithful

5. Halbert Spaces

160

a ) is obvious. b) Given x E E

c) Let u : E

-+ E I F

be the quotient map. Then for all x,y E

E,

Hence ( K , p ) is the representation of E associated to x'. Since H t is Einvariant for all { E A (Proposition 5.3.2.9),

for every x E E . Thus

A' is the cyclic vector associated to x' (Theorem 5.4.1.2

k)). d ) Take x E E . Then

for every 7 E K , so that

ipx = x l K ,

The assertion now follows. e) Take E A and 7 E H . Then

<

and 2<=7. Hence

Choose ( Q ) < ~EA

a) HC such that

C E .A

{< E A 1 qC# 0) is finite. Then

5.4 Representations

and

for every ( E A . Hence (qc)cEA E K and K =

OH<. SEA

Take x E E and 17 E H # . Choose an Y :=

toE A and put

1

-(.llo)v

EE

llEo1I2

Then

so that

Since

71

is arbitrary we deduce that

and the representation in c) is faithful.

Corollary 5.4.3.4 Let H be a Hilbert space, (<,),El, (qL)rE1 nonempty families i n H\{O) such that

C

ll
<

< c~

,

11~1L11~

LEI

3

LEI

E un involutive, closed subalgebra of L ( H ) containing K ( H ),

y':E+K,

x ~ ~ ( x q L l v L ) , LEI

and ( K , v) , ( L ,4 ) the representations of E associated to x' and y' , respectively. T h e n cp=$. K = L = Q H , LEI

5. Hilbert Spaces

162

Example 5.4.3.5 (Calkin, 1941) Let E be a n infinite-dimensional Hilbert space, 5 a free ultrafilter o n N , and 3 the set of bounded maps x : IN + E such that x ( 5 ) converges weakly to 0 . Consider

and

a ) 3 is a vector subspace of E"

and f a positive sesquilinear form. Let

be the quotient map and endow 3 / G with the scalar product associated to f (Proposition 5.1.2.9 d)). Let denote its completion.

3 7 ~

b ) If u

E

L ( E ) , then u

o

x E 3 whenever x E 3 and

f ( u 0 x u 0 x ) 5 Il.l12f 9

(.> .)

Let 11 denote the operator o n 37~defined by

for x ~ 3 .

c) Forevey

U E

L(E),

d ) T h e factorization of the map

through L ( E ) / X ( E ) zs a faithful unital representation of L ( E ) / K ( E ) e) If for each decreasing sequence (An)nEN i n there is a n A E 5 such that Card ( A n (A,\A,+I ))

5 with empty intersection,

<1

for every 71 E I N , then (0) and 3 / G are the only znvariant vector subspace of 3 / 4 .

{G I u

E

L(E))-

5.4 Representations

a

a) is obvious. b) u O X is bounded and u o x ( 3 ) converges weakly to 0 (Corollary 1.4.1.6 c). Thus u o x E F .Hence

= lim IIux(n)l12 I Ilul121im l l ~ ( n ) 1 1 =~llu112f (x, x) .

n,3

n,3

c) If u E K ( E ) , then by Proposition 3.1.1.20 a =. c , 12 = 0 . Assume now u $ K ( E ) . By Proposition 3.1.1.20' d =. a , there is a sequence ( X ~ ) ~ EinI N E which converges weakly to 0 such that ( U X , ) ~ ,does ~ not converge to 0 . There is an E > 0 and a subsequence ( x ~ , ) , , of ~ ~( x , ) , ~ ~such that lluxkn~l

>

for every n E IN. Define x:IN+

E,

n e x t " .

Then x E 3 and n

2

11iix11' = 11 uox 11 = f (uox, uox) = lim(ux(n) lux(n)) = n,3

1 ' > E'

= lim(uxtn luxk,) = lim lluxkn n ,3

n,3

Hence 12 # 0 . d ) It is obvious that the map is a unital algebra homomorphism. Take u E L ( E ) and x, y, E 3.Then

and the factorization referred to in d) is a unital representation. By c), it is faithful.

164

5. Halbert Spaces

e) Take x, y E that

F\g. There is a decreasing sequence (An),En in 5 such 1

Ilx(i)ll > illxll for every i E A1 and

11412

1 l ( x ( ~ ) l x ( j ) ) l3 < , l(~(i)l~(j))l<~

for every n E IN and ( i , ] )E K n x A,. By hypothesis, there is an A E 5 such that Card ( A n ( A ~ \ A ~ + II ) )1 for every n E

K .By Corollary 5.2.3.20, there is a u E C ( E ) such that

for every k E A . Thus

Since x and y are arbitrary, (0) and F/G are the only {clu invariant vector subspac:es of FIG.

C(E))-

Remark. The existence of an ultrafilter 5 with the properties formulated in e) follows from the axiom 2N0 = N, (G. Choquet, 1968). Proposition 5.4.3.6 An involutive Banach algebra having a faithful representation zs semz-szmple. The assertion follows from Theorem 5.3.1.13 and Proposition 2.3.2.26. H Example 5.4.3.7 x E C ( T ), define

Let A be a dense set of the topological space T . Given

Then Z E L ( 1 2 ( A ) )for every x E C ( T ) and the map

is a norm-preserving homomorphism of involutive unital algebras (Theorem 5.3.1.13). Gi~ienx E C ( T ), Z is normal and it is selfadjoint (unitary) i f f x is real valued (iff 1x1 = 1 ).

5.4 Representations

Remark. If T separable.

:= A := K

165

, then C(T)= em is not separable but e 2 ( A )= C2 is

Corollary 5.4.3.8 Every symmetric inuolutiue semi-simple Gelfand algebra possesses a faithful representation. Let E be a symmetric involutive semi-simple Gelfand algebra. By Corollary 2.4.1.14, Proposition 2.4.2.2 b), and Proposition 2.4.2.3 a + c , the Gelfand transform

is an injective continuous homomorphism of involutive algebras and the assertion follows from Example 5.4.3.7.

5. Hzlbert Spaces

166

5.5 Orthonormal Bases Every Hilbert space has an orthonormal basis and the cardinality of this orthonormal basis characterises the Hilbert space up to isometry. This leads to a simple classification of all Hilbert spaces: up to isometry there are precisely as many Hilbert spaces as there are cardinal numbers. Operators can also be easily defined in terms of orthonormal bases. That is how, for example, the Fourier-Plancherel operator is defined. Orthonormal bases have also proved to be very important for compact self-normal operators. In this case the operators can be "diagonalized", that is to say there is an orthonormal basis with respect to which the associated infinite matrix is diagonal.

In this section E always denotes a pre-Hilbert space.

5.5.1 G e n e r a l R e s u l t s

Definition 5.5.1.1

(0)

The family ( z , ) , ~ in , E zs sazd to be orthonormal

if

for all L , X E I . An orthonormal basis of E is an orthonormal family ( x , ) , ~ , in E such that

or, equivalently,

(Propositzon 5.2.2.2 d)). The subset A of E is called orthonormal ( a n orthonormal basis of E ) if the family ( x ) * € ~is orthonormal (is an orthonormal basis of E ).

P r o p o s i t i o n 5.5.1.2 ( 0 ) If A is an orthonormal set in E , then A is an orthonormal baszs of A l l (Proposition 5.2.2.2 a)). Observe that A' n A"

= (0)

.

5.5 Orthonormal Bases

167

Proposition 5.5.1.3 ( 0 ) If !2l is the set of orthonormal sets i n E ordered by the inclusion, then a n element of I Iis maximal iff it is a n orthonormal basis of E . Take A E II. If A is not an orthonormal basis of E , then there is an x E A'\{o}. Then A U E 2l and A is not maximal. Conversely, if A is not maximal, the11 there is a B E I Iwith

{hx]

AcB,

A#B

Hence

and A is not an orthonormal basis of E .

¤

Corollary 5.5.1.4 ( 0 ) Every orthonormal set i n E is contained i n a n orthonormal basis of E . Hence E always has a n orthonormal basis. Let A be an orthonormal set in E and I Ithe set of all orthonormal sets iri E . It is easy to see that inclusion defines an inductive order on '21. By Zorn's Icontaining A . By Propositior~ Lemma, there is a maximal element B in I rn 5.5.1.3, B is an orthonormal basis of E . Corollary 5.5.1.5 Let F be a s:ibspace of E and B a n orthonormal basis of F . T h e n there is an orthonormal basis A of E with A n F = B .

B is an orthonormal set of E . By Corollary 5.5.1.4, there is an orthonormal rn basis A of E containg B . It follows A n F = B . Corollary 5.5.1.6 If F is a subspace of E , then there is a n orthonormal basis A of E for which A n F is a n orthonormal basis of F . The assertion follows immediately from Corollaries 5.5.1.4 and 5.5.1.5. Proposition 5.5.1.7

(0)

If A is a finite orthonormal set i n E , then

YEA

Given any x E E ,

In particular, every finite orthonormal basis of E is a n algebraic basis.

rn

5. Hilbert Spaces

168

Let F be the vector subspace of E generated by A . Since F is finitedimensional, it is complete. Now F = ALL (Corollary 5.2.3.9),

YEA

and

(Proposition 5.2.2.2 d)). It follows that

YEA

(Proposition 5.2.3.1) and by Pythagoras' Theorem

Proposition 5.5.1.8

(0j

a ) If f , g E IKA and

then f = g

b) A is linearly zndependent.

a ) Take y E A . Then

Let A be an orthonormal set in E

5.5 Orthonormal Bases

Thus f = g . b) Take f E IK* with { f

# 0) finite arid

By a), f = 0 , and thus A is linearly independent. c) By Bessel's Identity,

for every finite subset B of A . Hence

Corollary 5.5.1.9 ( 0 ) Let E , F be re-Hzlbert spaces and A a n infinite orthonormal set of E . Take u E K ( E , F) and put

Then

5

converges weakly to 0 and u(3) converges to 0 .

We may assume that E and F arc complete (Corollary 3.1.1.7). Takc x' E E'. By the FrCchet Riesz Theorem (Theorem 3.2.5.2); there is an x E E such that

By Bcssel's Inequality,

170

5. Hilbert Spaces

so that

Hence to 0 .

8 converges weakly to

Corollary 5.5.1.10

(0)

0 . By Proposition 3.1.1.20 a

+ c , u(8) converges

T h e following are equivalent:

a ) E is finite-dimenszonal. b) Every orthonormal set i n E is finite. c) E has a finite orthonormal basis. d ) Every orthonormal basis of E is a n algebraic basis of E . e) E has a n orthonormal baszs which is a n algebraic busis. a + b . By Proposition 5.5.1.8 b), every orthonormal set in E is linearly independent and so finite. b + c follows from Proposition 5.5.1.4. c + a follows from Proposition 5.5.1.7. b + d . Let A be an orthonormal basis of E . By b), A must be finite and so, by Proposition 5.5.1.7, it is an algebraic basis for E . d + e follows from Corollary 5.5.1.4. e + a . Let A be an orthonormal basis of E which is also an algebraic basis of E . Take f E 12(A) and put

YEA

(Corollary 5.2.2.5). Since A is an algebraic basis of E , there is a g E I K ( ~ ) with

Then f = g (Propositior~5.5.1.8 a)). Hence {f A is finite and that E is finite-dimensional.

#

0) is finite. It follows that

5.5 Orthonormal Bases

171

Corollary 5.5.1.11 ( 0 ) If E , F are Hilbert spaces, then L f ( E ,F ) is the vector subspace of L ( E , F ) generated by {(.lx)yl(x,y) E E x F ) .

Let 3 be the vector subspace of L ( E , F ) generated by

Since (.lx)y E L f ( E ,F ) for every ( x ,y) E E x F , we have that 3 C L f ( E ,F ) . Take u E L I ( E ,F ) and let A be a finite orthonormal basis of Im u (Corollary 5.5.1.10 a =+ c). Then I m u = A" (Corollary 5.2.3.9). Hence, by Proposition 5.5.1.7 and Proposition 5.3.2.13 d),

YEA

Proposition 5.5.1.12 ( 0 ) Let E 7F be Hilbert spaces and 3 a real vector subspace of L ( E , F ) such that 3 # (0) and

for all u E 3,v E L I ( E ) ,and w E L f ( F ) . T h e n

If i n addition 3 is closed, then

Take ( x , y ) E~ x F and u ~ 3 \ { O ) . T h e r e i s a ~ Proposition 5.3.2.13 c), e),

E with E u z # O . By

5. Halbert Spaces

172

so that, by Corollary 5.5.1.11,

The last assertion follows from the first one and Corollary 5.2.3.4.

Corollary 5.5.1.13

(0)

Let E be a Hzlbert space and

F

a closed ideal

of L ( E ) (of K ( E ) ). a) If

F#

(0) , then K ( E ) c

F.

b)

T h e C*--algebra K ( E ) is simple (Proposition 5.3.1.17).

c)

If K = R ,then K ( E ) is a purely real C*-algebra. a ) follows from Proposition 5.5.1.12. b) follows from a). c) follows from b) and Proposition 4.3.5.3 a

* b.

Remark. b) is a generalization of Example 2.1.4.7 Proposition 5.5.1.14 ( 0 ) If A zs a n orthonormal set i n E and x E E , then the following are equzvalent:

a 3 b is obvious. b =+ c is obtained by putting 2 := x . c 3 a . Take E > 0 . Then, for some B E

whenever C E P , ( A ) , B

c C . By

yI(A),

the Bessel's Identity,

5.5 Orthonormal Bases

for every C E V j ( A ) with B C C . Thus

Theorem 5.5.1.15 ( 0 ) If A is a n orthonormal set i n the Hilbert space E , then the following are equivalent: a ) A is a n orthonormal basis of E . b) x E E

+

(x1y)y = x (Fourier Expansion of x ) . YEA

d) x E

1 ( ~ 1 1 / ) 1=~

3

1 1 ~ 1 1 (Porseva19s ~ Equation,

1799).

rEA

If these conditions are fulfilled then:

a

+ b.

Put

U

F=

B".

BE'P,(A)

Then F is a vector subspace of E containing .4 (Proposition 5.2.2.2 a),b)). It follows that

Hence, by Corollary 5.2.3.10 b 3 a ,

Take

E

> 0 . There is a z

E

F with

Choose B E V j ( A ) with z E B". 2 E C" , SO that

Take C E g j ( A ) with B C C . Then

1 74

5. Hilbert Spoces

(Proposition 5.5.1.7). Since

E

is arbitrary,

YEA

b

+a.

Take x E A L . Then

YEA

Hence A' = ( 0 ) . b @ c @ d follows from Proposition 5.5.1.14 e) Take y E E# . By d ) ,

ZEA

rE.4

2E.4

Hence

f) Put

By e) and Proposition 4.2.4.2 b),

Corollary 5.5.1.16 ( 0 ) Let E be a Hilbert space, A a n orthonormal set in E , and take x E E . T h e n

YEA

A is an orthonormal basis of A L L (Proposition 5.5.1.2) and

(Proposition 5.2.3.1 a +b , Proposition 5.2.2.2 d)). By Theorem 5.5.1.15 a*b,

5.5 Orthonorma[ Bases

Corollary 5.5.1.17 ( E and if B c A then

0)

175

If A is a n orthonormal basis o f t h e Hilbert space

and

We have, successively, that

Take x E B L . Then

(Fourier Expansion), so that

and

(Proposition 5.2.3.7).

W

Proposition 5.5.1.18 (Gram-Schmidt Orthonormalization, Gram, 1883, Schmidt, 1907) Take q E W U (0, oo} and let (x,),,~, be a linearly independent family i n E . Then there is a n orthonormal family (Y,),,,~, such that ~~ the same vector for eve? p E IN, the families (xn)nEKpand ( Y ~ ) , ,generate subspace of E . If E is complete, then ( Y , , ) , , , ~is~ a n orthonormal basis of E iff the vector subspace of E generated by ( x , ) , ~ ~ ,is dense. We construct the family (2/n)nEKqrecursively. Take p E N, and assume N been ~ - , constructed. Put the family ( Y ~ ) ~ ~has

5. Hilbert Spaces

176

for every n E Np-, . Since ( x , ) , ~is ~linearly ~ independent, y

# 0 . Define

It is obvious that ( x , ) , ~and ~ ~(yn)n,,p generate the same vector subspace of E . This proves the existence of ( y n ) n E N q . The final assertion follows from { x n I n E N,)"

= { y , I n E IN,)"

rn

and Corollary 5.2.3.9.

Remark. The example from the Remark to Corollary 5.2.3.9 shows that we cannot drop the requirement of completeness in Proposition 5.5.1.18. Proposition 5.5.1.19 ( 0 ) Let ( x ~ ) ~, , ~ , be orthonormal families zn the pre- Hilbert spaces E and F , respectively. T h e n

AEL LEI

LEI

for any family ( ( a , ,b,))LEl i n E x F

We have

xx

I ( ~ x ~ ~ ~ ) ( ~=~ I Y A ) 1I ( x ~ I a ~ ) ( b , l ~I x)l XEL L E I

by Bessel's Inequality.

L E I XEI,

rn

Proposition 5.5.1.20 ( 5 ) Let I be a n interval i n IR, E a Hilbert space, A an o r t h o n o n a l baszs of E , and f a map I + E such that ( f l x ) is contznuously differentzable for every x E A and that

5.5 Orthonormal Bases

177

where (f lx)' denotes the derivative of

Then f is diflerentiable and

Take to E I and t E I \ { t o ) . Given x E A , there is a t, E I between t and to such that

Then (Corollary 5.2.2.5)

=

):I(f lx)'(t-)

- ( f lz)'(to)I2

zEA

(Fourier Expansion and Parseval's Equation). Take E > 0 . There is a B E CJ?,(A) such that

Furtherrrlore, there is a 6 > 0 such that

for every t E Ui(tO).It follows that

5. Hzlbert Spaces

178

for every t E Ui(tO),i.e.

Proposition 5.5.1.21 Let F be a finite-dimensional vector subspace of the Hilbert space E . Then r F L ( E ) r F is a finite-dimensional C*-subalgebra of L ( E ) and TF 2s a unit of zt. It is obvious that T ~ L ( E ) isT a~subalgebra ~ of L ( E ) and that T ~ Fis a unit of it. By Proposition 5.3.2.8 a + b , r F L ( E ) r F is an involutive set of L ( E ) . Let A be a finite orthonormal basis of F (Corollary 5.5.1.10 a c). By Proposition 5.5.1.7,

*

By Proposition 5.3.2.13 c),e),

for every u E L ( E ) . Hence ((.ly)~)(,,,),~zis an algebraic basis of r F L ( E ) a F , which is therefore finite-dimensional.

Proposition 5.5.1.22 ( 0 ) Let ( x , ) , ~ ,be a n orthononnal basis of E , F a Hilbert space, (y,),,, a n orthonoma1 family i n F and take f E e m ( I ) . Then there zs a unique u E L(E, F) such that

for every

L

E

I . Moreoz~er, llxll

I f ( ~ )I l IIuxll

i llf l l m l l ~ l l

5.5 Orthonormal Bases

for every x E E a71d

If ( Y ' ) ~ ~is/ a n orthonormal basis of F , E is complete, and inf

LEI

I~ ( L ) I > 0

then u is a n isomorphism (isometry) of Banach spaces The first assertion follows immediately from Corollary 5.2.3.20. If ( y l ) l E is ~ an orthonormal basis of F and inf I f ( ~ ) > l 0 LEI

then

4 E ) 3 {Y'IL E I ) , so that

and u ( E ) is dense in F (Corollary 5.2.3.9). By Proposition 1.2.1.18 d), u is an isomorphism of Banach spaces. If in addition,

then by the first assertion, u preserves the norms, i.e., it is an isomctry of Banach spaces. Definition 5.5.1.23 ( 0 ) Let H be a Hzlbert space and E a n znvolutive closed subalgebra of L ( H ) . A representation of E is called a n identical representation if it is equivalent to the representation ( H , p ) , where p is the

inclusion map E

+ L ( H ).

Theorem 5.5.1.24 ( 0 ) (Naimark, 1948) If E is a Hilbert space, then any representation of K ( E ) is the Hilbert s u m of identical representations and a 0-representation.

Let ( H , p ) be a representation of K ( E ) and A an orthonormal basis of E (Corollary 5.5.1.4). Given x, y E A , put

180

5. Hilbert Spaces

Then

for all x, y , a, b E A (Proposition 5.3.2.13 b),e)). It follows that for each x A , is the orthogonal projection of H on H, (Proposition 5.3.2.8 b + a ) and that

p,

whenever x,y E A are distinct. Moreover,

l l ~ z z l= l

for all x, y

I l ~ r y ~ : ~ =l l ll~:~11~ = l l ~ z y 1 1=~ l l ~ f ~ ~ z=y lll l~ y v l l

A (Theorern 5.3.1.4). Hence either Pry = 0

for all x, y E il or

for all x, y E A . Take u E K ( E ) and B, C E ? , ( A ) . Then

(Proposition 5.5.1.7, Proposition 5.3.2.13 c),e)) and therefore

Thus

5.5 Orthono~malBases

181

whenever p,, = O for all x, y E A (Corollary 5.3.2.16), i.e. (H,cp) is a Orepresentation in this case. Suppose that (H, cp) is not a 0-representation of K(E) . Then

for all x, y E A . Take a E A and

<,

11<,ll

E Ha with

= 1. Given

x

E

A , put

and let K denote the closed vector subspace of H generated by {&(x E A } . Given x, y , z E A , pyztz = Pyzpza
c A',

p,y(k)

Iri particular, (<,)

is an orthonornial basis of K . By the above formula, p (7rB11?~7rCII) (K) C K

for all u E K ( E ) and B, C E ?,(A).

It follows that

for every u E K(E) (Corollary 5.3.2.16), i.e. K is p(K(E))-invariant. Let (K,qi) bc the compression of ( H , p ) to K (Proposition 5.4.2.8). Let v be the isometry E + K defined by VX

for x E il (Proposition 5.5.1.22). Take

= TL E

K ( E ) anti x, y E A . Then

((+7~)
182

5. Hilbert Spaces

= (uv*rZlvar,) = (v.v*tZlr,)

(Proposition 5.3.2.13 b),e)). Since x and y are arbitrary,

(Theorem 5.5.1.13 a + b) . Hence ( K , $) is an identical representation of K(E) . Let @ be the set of sets fi with the following properties: 1) The elements of fi arc p(K(E))-invariant closed vector subspaces K of H , such that the compression of (H, cp) to K is an identical representation of K ( E ) .

2) If K 1 ,K2 are distinct elements of fi , then K 1 l K2 Let @ be ordered by inclusion and let R be a maximal element of @ (Zorn). Put

By Proposition 5.4.2.8, L is p(K(E))-invariant. Assume that the compression (L, $) of ( H , p ) to L is not a O-representation of K ( E ) . By the above considerations, there is a y(IC(E))-invariant closed vector subspace K of L such that the compression of (L, $) to K is an identical representation of K ( E ) . Then R U { K } E @ and this contradicts the maxirnality of R.Hence ( L , $ J )is the O-representation of K ( E ) . It is easy to see that ( H , cp) is the Hilbert sum H of (L, $) and of the compressions of (H, cp) to the K E fi.

Proposition 5.5.1.25 Let E be a Hilbert space, 3 a C'-subalgebra of L ( E ) containzng K ( E ) , ( H , cp) a representation of 7 , and K a cp(K(E))-invariant closed vector subspace of H such that the compression of (H,cplK(E)) to K is an identical representatzon. T h e n K is cp(3)-invariant and the compression of ( H ,9) to K is an identical representation.

5.5 Orthonormal Bases

183

Let (K,$) be the compression of (H,(pllC(E)) to K (Proposition 5.4.2.8) and v : E + K an isometry such that

for every u E K ( E ) . Take u E 3 and x, y E E with llxll = 1 . It follows successively that

*(u((.Ix)x)) = *((.lx)ux)

=

(.lvx)vux

(Proposition 5.3.2.13 c),d)), $((.lx)x)vx = (vx~vx)vx= v x ,

Since x and u are arbitrary, K is (p(F)--invariant. Let (K,B) be the compression of (H, (p) to K (Proposition 5.4.2.8). Then

for all x, y E E , so that

Thus ( K , 8) is an identical representation.

rn

Proposition 5.5.1.26 ( 0 ) Let E , F be infinite-dimensional Hilbert spaces. Take u E C ( E , F) and let u be the equivalence class of u i n

Let @ be the set of o r t h o n o n a l sequences i n E and @ the set of filters o n E which contain E# and converge weakly to 0 . T h e n

5. Hilbert Spaces

184

Take a > llull and

5E

By Proposition 3.1.1.20 a

@ . There is a u E K ( E , F ) such that

+ c,

u ( 5 ) converges to 0 . Since

for every x E E # , lim sup llux)1 5 a 1,3

Since cr and

5

are arbitrary, it follows that

The inequality

follows from Corollary 5.5.1.9 and the inequality

follows from Proposition 5.2.3.21.

Proposition 5.5.1.27

(7)

Let

be the quotient map. Take x' E ( L ( E ) / K ( E ) ) ; and let vector subspaces F of E such that

5

be the set of closed

. particular, T h e n { T F I F E 5 ) is upward directed and has supremum i E In ( L ( E ) l K ( E ) ) n= (0) and L ( E ) / K ( E ) is not a W'-algebra.

5.5 Orthonormal Bases

Take F, G E by F U G . Then

185

5 and let H be the closed vector subspace of E generated

XH

(Proposition 5.2.3.13 a

- XF = Tf,nFl

< H~

+ f , Corollary 5.3.3.8),

Hence 5 and {rFIF E 5) are upward directed. Let x be a lower bound for { r E- T F I F E Proposition 4.2.5.3 d), there is a u E L ( E ) + with

5) in ( L ( E ) / K ( E ) ) +. By

Assume that u $ K ( E ) . By Proposition 5.2.3.21, there is an orthorlormal sequence (x,),,~ irl E such that c-u := inf lluxnll > 0 . n€h'

By Lenirrla 1.1.2.17, there is an uncountable set 2L of irifirlite subsets of {x, n E IN) such that A n B is finite for distinct A, B E U. Then TALLKRLL

= X ( A ~ R ) I IE K ( E )

(Corollary 5.2.3.9), .

.

T ~ L I X ~ = L L0

for distinct A, B E M . Thus

I

5. Hilbert Spaces

186

for every Q E 'J3,(2L)

(Corollary 4.2.7.5). Hence there is an A E 2l with

It follows that

(Proposition 4.2.7.1 d

+ f),

0 < a 5 inf lluxnll = inf IIunAl~xnll= 0 nEA

nEA

(Propositions 5.5.1.7 and 5.2.3.21), which is a contradiction. Hence u E K ( E ) , E 3) in L ( E ) / K ( E ) (Corollary E 5) in L ( E ) / K ( E ) . 4.2.2.22). Hence GE is the supremum of {;,IF By Corollary 4.4.4.3 a + b, L ( E ) / K ( E ) is not a We-algebra.

x = 0 and 0 is the infimum of {iE - nFIF

Proposition 5.5.1.28 ( 7 ) Let (n,),El be a finite family i n I N , E the C ' d z r e c t s u m of the family (L(IKnL)),EI,and ( u ~ ) ~ a~ summable /, family i n E+ . Then

This inequalzty is sharp.

Case 1

Card I = 1

By Theorem 5.5.1.15 f),

for every X E L . It follows

5.5 Orthonormal Bases

XEL

Case 2

XEI, k = l

k=l

XEL

I arbitrary

Put ( ~ X L ) L E I:= U X

for every X E L . Then by Case 1,

1

lluXll

= C s u p I I ~ X L I II

XEL

XEL ' E l

LEI XEL

LEI

IIuXlII

=

XEL LEI

LEI

XEL

Now we show that the inequality is sharp. For

L

XEL

E I and k E

Nn,define

:= ( ~ X L P A ~ ) A E I

'&k

Then

for every

L

E I and k E

Nn,and

Proposition 5.5.1.29 Let (H,),,, , ( K h ) X E L be families of Hilbert spaces, E and F the C*-direct products (sums) of the families ( K ( H L ) ) L Eand I ( K ( K A ) ) X E I , , respectively. Let u : E + F be a n isomorphism of C*-algebras. Then there is a bijective map cp : L -t I and for each X E L a n isomorphism of Hilbert spaces ux : I I , ( A ) + K x such that u X

for every x E E .

( u X O %,(A)

O

ui)AcL

5. Hilbert Spaces

188

By Corollary 5.5.1.13 b), K ( H , ) and K ( K x ) are simple for any X E L . F o r L E Iand X E L , p u t

F,

:= {y E

FIX' E L\{X)

L

E I and

yw = 0 ) .

By Proposition 4.3.5.5, there is a bijective map II, : I

-t

L such that

F,(,) = 4 E L ) for every

L

E I . We define cp := II,-'

and for each X E L

where we have identified K(H,(x)) canonically with E,(x) and K ( K A )with F A . Then for each X E A , v~ is an isomorphism of C*-algebras. Take X E L . By Theorem 5.5.1.24, there is an isomorphism of Hilbert spaces ux : H,(h) -t K h such that

for every x E K(H,(A)). Ct'e get

for every x E E . Proposition 5.5.1.30 Let E be a complex Hzlbert space. Denote by F the underlying real vector space of E endowed with the scalar product

and put

for eve?

u E L ( F ).

a) F zs a real Hilbert space and the canonzcul norrns of E and F coincide.

b) L ( 6 ) is a unital real C * subalgebra of L ( F ) .

5.5 Orthononr~alBases

C)

11 E C ( E ) for every u

6

L ( F ) and the map

C ( F ) +L ( F ),

u Hh

is an ir~volutzveoperator of norm 1 and a projection of C ( F ) onto L ( E ) . d ) If A is an orthonorrnal set (basis) of E , then A U i A is an orthonormal set (basis) of F . a),b), and d ) are easy t o see. c ) Take u E L ( F ) . By Lemma 1.3.1.5, 1

11cx11 I ,(lluxIl

E

L ( E ) . For every x E E# ,

1

+ Il.(ix)Il) I ,(1lu11 + 1 .1 )

=

11~11

so that

IcI

I llull .

Moreover, if u E C ( E ) then

Ex

=

1

-(ux 2

-

1

i u ( i x ) )= - ( u x 2

+ u x ) = ux

for every x E E so that

Hence our map is an operator of norrn 1 and a projection of C ( F ) orlto C ( E ). We want to show now that

-

u* = 11' for every u E C ( F ). Take x, y E E . Then

re (Gxly) = Are ( u x - iu(z.x)1 y) = 2

190

5. Hzlbert Spaces

-

1 = -re (xlu'y - i u 8 ( i y ) )= re (xlu'y) . 2

It follows irn (Gxly) = re (-i(Gx1y)) = re ( G x J i y )=

5.5 Orthonormal Bases

5.5.2 Hilbert Dimension Proposition 5.5.2.1 subset of E . If

(0)

Let A be a n orthononnal basis of E and B a

BL = {0) , then Card A 5 Card B First suppose B is finite. It follows from B1 = (0) , that B" = E , i.e. E is the vector subspace of E generated by B (Corollary 5.2.3.9). Hence E is finite-dimensional and A is an algebraic basis of E (Corollary 5.5.1.10 a + d). Thus Card A 5 Card B . Now let B be infinite. Consider

By Bessel's Inequality,

for every y E B . Thus f (y) is countable. From

it follows that

Hence Card -4 5 Card ( B x No) = Card B

Corollary 5.5.2.2 ( 0 ) All orthonormal bases of E have the same cardznality. This cardinal number (Corollary 5.5.1.4) is called the Hilbert dimension of E . It is less than the algebraic dzmension of E . If E is infinitedimensional, then its Hilbert dimension is equal to its topological cardinalzty.

192

5. Hilbert Spaces

The first assertion follows from Proposition 5.5.2.1. Let A be an algebraic basis of E and take x E A'. There is an f E IK(") with

YEA

It follows that z E A"

and z = 0 (Proposition 5.2.2.2 a),b),e)).Hence

and the second assertion follows from Proposition 5.5.2.1. In order to prove the final assertion, let A be an orthonormal basis of E . By Corollary 5.5.1.10 c + a, A is infinite. For every B 6 p i ( A ) , let B' be a countable dense set of B L L (Corollary 5.2.3.9). Put

Then Card C

< N o x Card g i ( A ) = N o x Card A = Card A

and -

C 3

U

El=

B€'PI(A)

U

B1'.

BE?)~(B)

Thus C is dense in E . Sow let D be a dense set of E . There is a map f : A

-t

D such that

1

Ilf (2) - zll < -2

for every x E A . Then, for all distinct z , y E A ,

Ilf (XI - f (y)ll 2 JZ - 1 > 0 Hence f is injective and therefore Card A 5 Card D Thus the topological cardinality of E is equal to Card A , i.e. to the IIilbert dimension of E .

5.5 Orthonormal Bases

193

Corollary 5.5.2.3 ( 0 ) Let E , F be Hilbert spaces and u E C ( E , F) . If u is injective (if I m u is dense), then the Hilbert dimension of E is smaller (greater) then the Hilbert dzmension of F . Let A, B bc orthonormal bases of E and F , respectively (Corollary 5.5.1.4). Suppose that u is injective. Take x E u*(B)' . Thcn

for every y E B . Wc deduce successively that ux E BL, ux = 0 , x = 0 , u8(B)' = (0) , Card A 5 Card u*(B) 5 Card B (Proposition 5.5.2.1). Now let Im u be dense. Take y E u(A)'. Thcn

for every x E A . We deduce successively that u'y E A'-, u'y = 0 , y E Keru* =

mu**)'

= (1mu)'- = (0)

(Proposition 5.3.2.4, Proposition 5.3.1.3 a), Corollary 3.2.3.10 a

+ b) ,

Card B 5 Card u(A) 5 Card A (Proposition 3.5.2.1).

Example 5.5.2.4 Let (El)lEI be a famzly of pre- Hilbert spaces, E its Hilbert sum, and for each L E I let A, be a n orthonormal basis of EL and u, the canonical embedding of EL into E . T h e n U u,(A,) is a n orthonornlal basis of E and

LEI

Card A, is the Hilbert dzmension of E

194

5. Hilbert Spaces

Theorem 5.5.2.5 ( 0 ) If E is a Hilbert space and A is a n orthonormal basis of E , then the map

(Theorem 5.5.1.15 a

+ d ) is a n isometry with inverse

(Corollary 5.2.2.5). Consider

By Parseval's equation,

for every x E E . Thus u preserves the norms. Since

uvf

=

f

for every f E e 2 ( A ) , u is surjective and v is its inverse.

Corollary 5.5.2.6 equivalent:

(0)

I

Let E and F be Hilbert spaces. T h e following are

a ) E , F are isometric b) the Banach spaces E , F are isomorphic c) E and F have the same Hzlbert dimension

a + b is trivial. b + c follows from Corollary 5.5.2.3. c + a follows from Corollary 5.5.1.4 and Theorem 5.5.2.5.

Remark. Identifying isometric Hilbert spaces (Corollary 5.1.1.4), the map

is a bijection fro111 the class of cardinal nurribers on the class of Hilbert spaces (Theorem 5.5.2.5, Corollary 5.5.2.6 a @ c).

5.5 Orthonormal Bases

Corollary 5.5.2.7 valent:

(0)

195

Let E be a Hilbert space. T h e following are equi-

a) E is separable. b) the Hilbert dimension of E is at most N o . c) E is finite-dimensional or it is isometric to

e2

a =+ b follows from Corollary 5.5.2.2. b c . Let A be an orthonormal basis of E . By b),

*

Card A 5 No and so the assertion follows from Theorem 5.5.2.5. c =+ a follows from Example 1.1.2.5. Corollary 5.5.2.8 ( 0 ) If E , F are infinite-dimensional Hilbert spaces, then K ( E , F ) is not a complemented subspace of L ( E , F ) .

Let A and B be orthonormal bases of E and F , respectively (Corollary 5.5.1.4). By Theorem 5.5.2.5, we may identify E and F with e2(A) and e2(B), respectively. By Corollary 3.1.2.12, K(12(A),e2(B)) is not 2 complcmented subspace of L(e2(A),e2(B)) . Corollary 5.5.2.9 If the Hilbert dimension of a n infinite-dimensional Hilbert space its N, then its algebraic dimension is N*O.

This follows imniediately from Theorem 5.5.2.5 and Proposition 1.1.2.21.

Corollary 5.5.2.10 ( 0 ) Let A be a n o r t h o n o m a l basis of the Hilbert space E . Given x E em(A) , let Z denote the unique operator o n E such that

for every a E A (Proposition 5.5.1.22). T h e n the map

is a n norm-preserving homomorphism of involutive unital algebras.

5. Hilbert Spaces

196

-

Take x , y E ew(A) arid a , P E IK. Then for a E A h -

+

cry + pya = ( a s p y ) ( a ) a = ( a x ( a ) + p y ( a ) ) a = a x ( a ) a + D ~ ( a ) = a

;:a

=a,

and

(bl$a) = (bl?t(a)n) = x(a)(bla) = x(b)(bla) = = (x(b)bla)= (Zbla) = (b15'a)

for every b E A , so that

-

x'a

= ?*a.

Hence the above map is an involutive unital algebra homomorphism. By Proposition 5.5.1.22, it preserves norms.

Proposition 5.5.2.11 ( 0 ) Let E , F be Hilbert spaces such that at least one of them is infinite-dimensional. Let a and b be the Hilbert dimensions of E and F , respectively, and put c := inf{ab,b"}

a ) I f a = b , then c - 2 " . b) If inf{a, b )

> 2,

then the topologzcal cardinality of L ( E , F ) is equal to

C.

c ) If both E and F are dzfferent from (0) then L ( E , F ) is separable z f f one of these spaces zs finzte dimenszonal and the other is separable. a) \$'e have

b) The map

5.5 Orthonormal Bases

is an isometry of real Banach spaces, so we may assume a 5 b . Then, by a), bO

< bb = 2b < ab ,

so that

By Example 1.1.2.5 and Theorem 5.5.2.5, there is a dense set B of F with Card B = b Let A be an orthonormal basis of E . Then

F := {U

= C(E, F)lu(A) c B }

is a dense set of L ( E , F ) and Card F 5 Card BA = bn = c Now Irt A and B be orthonormal bases of E ant1 F , respectively. Put

G For f E G let

UJ

:= { f E B" (f is injective}

.

be the element of L ( E , F ) with "J" = f (")

for every x E A (Proposition 5.5.1.22). Then

1 1 , -~ ~"911 > JZ for all distinct f , g E G . Hence, if 31 is a dense set of C(E, F ) , then Card 31

> Card G = bO = c .

c) If E is finite-dimensional, then CardL(E, F ) = Card Fa and so L ( E , F ) is separable iff F is separable. We use again the isometry L ( E , F) --+ C(F, E ) ,

u c,u * ,

to conclude that if F is finite-dimensional, then C(E, F ) is separable iff E is separable. If both E and F arc: infinite-dimensional, then, by b), L(E, F ) is not separable.

198

5. Hilbert Spaces

Proposition 5.5.2.12 ( 0 ) Let E , F be Hilbert spaces different from ( 0 ) . Let N ( E ) and N ( F ) be the Hilbert dimensions of E and F , respectively, and

put

a) A n y dense set of K ( E , F ) has cardinality at least N b) If N is infinite, then K ( E , F ) has topological cardinality N c) K ( E , F ) is separable iff E and F are separable. d ) The C--algebras K ( E ) and K ( F ) are isomorphic i f f the Hilbert spaces E and F are isomorphic. e) If E is infinzte-dimensional, then the topological cardinality of C ( E ) / K( E ) is 2 H ( E.) a ) Let A and B be orthonormal bases of E and F , respectively. Then for all X I , 2 2 E A , x1 # 5 2 and y l , y2 E B , yl # y2 we have

Hence there is a set of pairwise disjoint open sets of K ( E , F ) of cardinality N . Thus any dense set of K ( E , F ) has cardinality at least N . b) Let A and B be dense sets of E and F , respectively, of cardinality a t most N and let ( x ,y ) E E x F . Then for all ( a ,b) E A x B ,

5 Ila - xll llbll + llxll Ilb - Yll It follows that ( . l x ) y belongs to the closure of

{(.I.)bl(a,b) in K ( E , F) . Since

E

A

x

B)

5.5 Orthonormal Bases

Card {(.la)bl(a,b) E A x B )


it follows from Corollary 5.5.1.11 and Corollary 5.2.3.4, that K ( E , F ) has a dense set of cardinality a t most N . By a), N is the topological cardinality of K(E,F ) . c) follows from a),b), and Corollary 5.5.2.7 a e b . d) We may assume that N is infinite. By b), if the C*-algebras K ( E ) and K ( F ) are isomorphic, then

By Corollary 5.5.2.6 c + a , the Hilbert spaces E and F are isomorphic. The converse implication is trivial. e) follows from b) and Proposition 5.5.2.1 1 a),b) . Proposition 5.5.2.13 ( 0 ) Let E , F be Hilbert spaces, G a separable closed vector subspace of E and take

Then there are u E C ( E ) and w E L(F, E ) such that WOUOU =KG

=U*U

By Proposition 5.2.3.21, there is an orthonormal sequence ( x , ) , ~ ~in E such that cu := lim sup IIux,,ll

>O

n+m

By Corollary 5.5.1.9 (and Corollary 1.4.1.6 a + c) , ( U X , ) , ~converges ~ weakly ~ to 0 . Hence we can construct inductively an increasing sequence ( k n ) n E in IN such that

for every n E IN. We may assume G to be infinite-dimensional. Let (y,,),,~ be an orthonormal basis of G (Corollary 5.5.2.7 a + b). By Proposition 5.5.1.22, there is a u E C(E) such that

5. Hilbert Spaces

200

vYn = X k , for every n E W and v = 0 on G I . We have

C 32"m2 = a28 < m24 -
-

-

nEb

By Corollary 5.2.3.20, there is a w E C ( F ,E ) such that WUXk,

=

Yn

for every n E W . Thus WuvYn = Yn for every n E IS and wouov=o on GL . Hence

Let ( z , ) , ~ Ibe a n orthonormal basis of G I . Then

for every n E N (Fourier expansion) and v'v = 0 on G1 . Thus v*v = ac .

Corollary 5.5.2.14 ( 0 ) If E is a Hilbert space and u E L ( E ) \ K ( E ) , then the ideal of L ( E ) generated by 11 contains { v E L ( E ) I I m v is separable)

.

5.5 Orthonormal Bases

201

Let 3 be the ideal of C(E) generated by u and take v E C ( E ) such that Im v is separable. Put

By Proposition 5.5.2.13,

T,V

6 3.Thus

Remark. The ideal generated by u E C ( E ) \ K ( E ) need not be closed. But it does contain {v E L(E)IImv is separable) which is (by Example 2.2.2.23) a closed ideal. Corollary 5.5.2.15

(0)

If E is separable, then the C*-algebra C ( E ) / K ( E )

is simple. Let 3 be a closed ideal of C ( E ) / K ( E ) and let q be quotient map

Then -b(3) is a closed ideal of L ( E ) . By Corollary 5.5.1.13 a) and Corollary 5.5.2.14,

Hence

so that C ( E ) / K ( E ) is simple Corollary 5.5.2.16

Let E be a Hilbert space and put 3 := { u E C(E)IIm u is separable).

T h e n every representation of 3 (Example 2.2.2.23) is the Hilbert s u m of identical representatzons, of represcntatzons with kernel K(E) , and of a zero representation. Let (H, rp) be a representation of 3 . Then (H, pIK(E)) (Proposition 3.1.1.18) is a representation of K ( E ) . Thus by Theorem 5.5.1.24, it is the Hilbcrt sum of identical representations and of a 0 representation. By Proposition

202

5. Hilbert Spaces

5.5.1.25, (H, cp) is the Hilbert sum of identical representations and of representations (K,11) with $(K(E)) = 0 . Ker ll, is a closed ideal of F containing K ( E ) . By Corollary 5.5.2.14, Ker zL = K ( E ) or Ker $ = 3.

Remark. If E is separable, then 3 = &(E) and a reprensentation of L ( E ) with kernel K ( E ) is in fact a representation of the Calkin algebra &(E)/K(E) . P r o p o s i t i o n 5.5.2.17 Let E be a Hilbert space and N its Hilbert dimension. T h e n there is a faithful unital representation (H, cp) of L(E)/K(E) such that the Hilbert dzmension of H is at most NNO . We may assume E to be infinite-dimensional. Then there is a dense set A of E such that Card A = N (Corollary 5.5.2.2). Let ( H , cp) be the faithful unital representation of & ( E ) / K ( E ) from Example 5.4.3.5 d). In the sequel we use the notation of this example. Define

Then Card F05 Card"'A

= NN0

Take x E 3. There is a y E 6 such that

for every n E I N . Then x - y E 4 . Hence Card (FIG) 5 Card

5 NN0.

-

By Corollary 5.5.2.2, the Hilbert dimension of 3 / G is a t most Card(F/E) . Therefore the Hilbert dimension of H is a t most NHo. P r o p o s i t i o n 5.5.2.18 A a dense set of F .

Let F be a quasiunital znvolutzve Banach algebra and

a) If (H, p) is a cyclic representation of H is at most Card A .

F , then the HilberE dimension of

5.5 Orthonormal Bases

203

b) If ( H , y ) zs the universal representation of F , then the Hilbert dimension of H is at most 2CardA. a ) By Proposition 5.4.1.3 b), (H, p) is equivalent to the representation of E associated to an x' E EL. By Theorem 5.4.1.2 b),h), H contains a dense set of cardinality a t most Card A , so by Corollary 5.5.2.2, the Hilbert dimension of H is a t most Card A . b) The map

is injective, and so Card r ( E ) 5 Card IKA 5 2CardA. By a),

Card H 5 (Card ~ ) 2 ' " ' ~ = 2CardA .

Example 5.5.2.19 Let (T,2, p ) be a measure space and H the set of maps x : T + E such that

is measurable for every y E E and that

is p-integrable. If E is separable, then:

is p-integrable for all x,y E 31.

b) 31 is a vector subspace of ET and

is a positive sesquilinear form. Put

and endow H with the scalar product associated lo f (Proposition 5.1.2.9 d)); H is called the direct integral of E with respect t o p .

204

5. Halbert Spaces

c) If E zs complete, then H i s also complete Let A be an orthonormal basis of E (Corollary 5.5.1.4). By Corollary 5.5.2.7 a + b, A is countable. a) The map

belongs to L2(p) for every

2

E A and

for every t E T (Parseval's Equation). Given t E T ,

(Theorem 5.5.1.15 a

+ c),

and so

is p-integrable. b) follows from a). c) Let (Xn)n,K be a Cauchy sequence in H and for each n E IN, choose xn E A',. Then ( ( x , , ~ z ) ) ~is, ~a Cauchy sequence in LZ(p) for every z E E . Given z E A , put z, := lim (x,lz) E L2(p) n+m

and

which is well-defined

lip

to a set of p-measure zero. The map

is nieasurable for every y E E .

5.5 Orthonormal Bases

Given

E

> 0 , there is an n, E N , such that 1IXm -

XnII < E

for every m, n E N\INnC. Take n E N , n

c2 > lirn

m+m

205

J

> n, . By Parseval's Equation,

IJx,,,(t)- xn(t)l12dp(t)= m+cc lirn

1

I(xrn(t) - xn(t)Iz)12dp(t)=

so that

Since

E

is arbitrary, ( X n ) n Econverges ~ to the equivalence class of x in H .

206

5. Hilbert Spaces

5.5.3 Standard Examples Example 5.5.3.1 For any set T , ( e t ) t E Tis an orthonormal basis of P2(T). It is called the canonical orthonormal basis of P ( T ) . Given s, t E T ,

i.e. ( e t ) t Eis~ an orthonormal family in P 2 ( T ) .Take x E {etlt E T I L .Then

for every t E T and so x = 0 . Hence

is an orthonormal basis of e 2 ( T ) .

Example 5.5.3.2 Let IK = C and X the Haar measure on T with p ( T ) = 1 . Given n E Z , consider

Then ( x , ) , , ~ zs an orthonormal basis of L 2 ( X ) . The map

where we is an isometry which coincides with the Gelfand transform on P1(Z), here identified a(!' ( Z ) )wzth T (Example 2.4.5.6). Given n, p E Z ,

and so ( x , ) , , ~ is an orthonormal family in L 2 ( X ) . By the WeierstrassStone Theorem (Corollary 1.3.5.16), the vector subspace of C(T) generated by { x n l n Z } is dense. Since the inclusion map

is continuous, it follows that

5.5 Orthonormal Bases

(Corollary 5.2.3.9) and

(Proposition 5.2.2.2 a)). Hence (x,),,~ is an orthonormal basis of L2(X) By Theorem 5.5.2.5, the map

is an isometry. Take y E e 1 ( Z ) . Then

where the sum is takcri in C(T) (Example 2.4.5.6). Since e l ( Z ) C e"Z) the inclusion map C(T) + L2(X) is continuous, y =uy.

Example 5.5.3.3

Let K = C . Consider

T and X the Lebesyue measure

071

:= G(0)

T . Let A be the closed vector subspace

{x E L2(X)lx is analytic} of L2(X) (Example 3.1.2.10).

a) Given x E t 2 , the radius of convergence of the power series

is greater than 1 . Define

b)

is a n orthonormal basis of A and the map

is a n zsometry.

and

w

208

5. Hilbert Spaces

a) The assertion follows from

b) Given n, p E IN

Hence (E,,)nEb. is an orthonormal family in A . w

Take x E A and let

C a n t n be its associated power series. Choose p E N .

n=O Then, given any p E ] O , l [ ,

Thus

We deduce that

and hence is an orthonormal basis of A . Take x E e2. Then ( x , E ~ ) , , ~ is summable in A (Corollary 5.2.2.5). By Exa~nple3.1.2.10, given any compact set K of T , the sequence (x,%lK),,~m is summable in C ( K ). Hence E E A and

5.5 Orthonormal Bases

where the sum is taken in A . By Theorem 5.5.2.5, the map

is an isometry.

Example 5.5.3.4

Take !K = R and

T

:= u p 2 ( 0 ).

Let X be Lebesgue measure on T and 3-1 the closed vector subspace

{x E L' ( A )lx is harmonic} of L 2 ( X ) (Example 3.1.2.9). Then

is a n orthonormal basis of 3-1. Here the functions ure expressed i n terms of polar coordinates r , 8 . It is easy to see that A is orthonormal . Take u E 3-1. Since u is the real part of a cornplcx analytic function on T : u may be written in the form u ( r , 8 ) = a0

+

m

( a , cos n 8

+ b,, sin no)rn

n= l where the sum converges uniformly on the compact sets of T . Take n E IN and p E ]O,1[. Then

I

'JP(O)

u(r,8 ) r n cos n 8 dX = an

J

Uo(0)

rZncos2 n 8 dX =

5. Hilbert Spaces

21 0

J

u ( r , O)rn sin n 0 dX = bn

UP(Q)

J

r2" sin2 n 0 d~ =

UP(0)

Hence

J J

ran u ( r , O)ancos n 0 dX = 2 ( n + 1) ' .ir bn u ( r ,O)rn sin n0 dX = 2(n 1) '

+

Thus if u E A', then u = 0 . I t follows that A is an orthonormal basis of

31 Example 5.5.3.5 Take a , b E R+, a < b and let X be Lebesgue measure o n [a,b] . Given n E N u { 0 ) , define

Then

(x,),,N,,(Q)

is a n orthonormal basis of L 2 ( X )

Take m, n E IN with m < n . Using complete induction and integration by parts, we see that

5.5 Orthonormal Bases

21 1

Now given n E IN U (0) , pn is a polynomial of degree n . Thus it follows that ( ~ , ) , ~ p ~ ~is~ an o ] orthonormal family in L2(X). h/loreover, { x n J nE IN u (0))" contains all polynomials. Since the set of polynorriials is dense in L 2 ( X ) ,

Hence,

is an orthonormal basis of L2(X)

Example 5.5.3.6 ( n E IN U (0) , define

4)

Let X be Lebesgue measure on IR. Given

yn:R-+IK,

Then

( X , ) , ~ ~ J ~ ( is ~ ) an

t-e?

1 iz

dne-i2 dt" '

-

orthonormal basis of L2(X)

It is easy to see that for each n Pn E R [ t ] of degree n , such that

E

IS U {0}, we can find a polynomial

for every t E IR. It follows that

Since 3 is a dense set of L 2 ( X ) ,

Take m, n E IN with m parts, we see that

< n . Using complete induction and integration by

This arid the above considerations imply that ( basis of L ~ ( X ).

x

,

) is~ arl~orthonormal ~ ~ ~ ~

~

212

5. Hilbert Spaces

Example 5.5.3.7 Let X be Lebesgue measure o n IR+ . Given n E IN U (0) , define

is)a n orthonormal basis of L 2 ( X ) . Then (xn)nENu{O

I t is easy to see that for each n E K\'U{0} , there is a polynomial Pn E IR[t] of degree n such that

for ever t E IR, . It follows that

Since 3 is dense in L 2 ( X ) ,

{ x n l nE IN Take m, n E show that

u {o)}"

=L~(X)

IN with m < n . Complete induction and integration by parts

This and the above considerations show that (xn)nENU(Ol is an orthonormal basis of L 2 ( X ) . Proposition 5.5.3.8 ( 0 ) Let ( S ,6 ,p) , (T,2, u ) be two a- finite measure spaces and 3,6 orthonormal bases of L 2 ( p ) and L 2 ( u ) ,respectively. T h e n

is an orthonormal basis of L 2 ( p8 u )

5.5 Orthononal Bases

is an orthonormal set in L2(p 8 u) Take

Then, for any ( x , y) E 3 x 6 ,

by Fubini and Proposition 3.1.6.16. Hence

for every y E 6 . Thus k = 0 (Propositiorl 3.1.6.17 e)). Hence

and so

is an orthonormal basis of L2(p @ u) . Corollary 5.5.3.9 ( 4 ) Let ( z , ) , , ~ ~ , , ( ~be) the orthonormal basis defined i n Example 5.5.3.6. Take m E N and let X be Lebesgue measure on R m . Define

K and given k E

:= (IN

u (0))"

A', consider

T h e n ( ? / k ) k Eis~ a n orthonomal basis of L2(X).

5. Halbert Spaces

214

This follows immediately from Proposition 5.5.3.8 using complete induction.

Proposition 5.5.3.10 Let ( T ,Z, p ) be a measure space, j a positive locally p-integrable junction and v := j . p . If ( x , ) , ~ is , an orthonormal basis of L 2 ( p ) and if we define

then (gx,),,, is an orthonormal basis of L 2 ( v ) We have gx, E L 2 ( v ) for every

L

E I . Moreover,

~ Let x E { g x , l ~E I ) ' . Then for all 1 , X E I . Hence ( g ~ , ) is, ~orthonormal. Jf x E L 2 ( p ) and

for all

L

E I . hence a

Corollary 5.5.3.11

x = 0 p-a.e. and therefore x = 0 v-a.e. Take IK = C ,

T := Q ( O ) , and let X be Lebesgue measure on T . Gzven n E lh' U ( 0 ) and p E Z , define

xnp : T +C, where a,,

reze

a,pyn(r)e'pe,

is given by

J Then {xn,ln

E

N

u ( 0 ),p

1~,,(~dX= 1 .

E Z) is an orthonomal basis oj L2(X)

5.5 Orthonomal Bases

215

Let p be the measure on [O,1] defined by dp := tdt and let v be Haar measure on T with v ( T ) = I , and set

By Example 5.5.3.5 and Proposition 5.5.3.10, there is a family (Pn)nEnu(ol in R+ such that (Pnyn),,,,{,] is an orthonormal basis of L 2 ( p ) .By Example 5.5.3.2, ( e ' ~ ~ ) is ~ ,an, orthonormal basis of L 2 ( v ) .Thus by Proposition 5.5.3.8, { ( p n y n )@ (e'PB)lnE N U (0) , p E Z) is an orthonormal basis of L 2 ( @ ~ v). Since

{xnplnE IN U (0) , p E Z} is an orthor~ormalbasis of L 2 ( X ) . Proposition 5.5.3.12 Take K = C. Let T be a n open set of C, X the Lebesgue measure o n T , A the closed vector subspace

of L2(X) (Ezamplr: 3.1.2.10), and ( x , ) , , ~ a n orthononnal basis of A . Then, for each t E T , ( ~ ~ ( t ) E) e2 , , and ~ ~

is the Bergman kernel of T (Example 5.2.5.9).

Take

T

> 0 . with

Then e~ E L 2 ( X ) .By Gauss' Mean Value Theorem and Bessel's Inequality, I

Hence ( ~ , ( t ) ) , ,E~e2. Take x E A . Then

12

5. Hilbert Spaces

21 6

(Fourier expansion). Since the map

is contin~ious(Example 3.1.2.10),

(Corollary 5.2.2.25). Hence k is the Bergman kernel of T

.

Remark. a ) Since L2(X) is separable, A is also separable. Hence A has a countable orthonormal basis (Corollary 5.5.2.7 a + b). It follows from function theory, that if A # (0) , then A is infinite-dimensional. Hence, taking a basis is no real restriction of generality. of A of the form (xn)nELV b) Let k be the Bergman kernel of q ( 0 ) . By Example 5.5.3.3 and the above proposition, k ( s ,t ) =

1n-sn-'r' ~ =

nEN

71

1 ( -1~ 1 ) ~

for all s , t E q { O ) , which coincides with Example 5.2.5.11. Proposition 5.5.3.13 bers such that:

Let T be an uncountable set and Nf,N"cardznal num-

1) N' < C a r d T ; 2) Nu 5 Card T ; 3) the s u m of any countable family of cardinal numbers striclty srnaller than N" is strictly smaller than N" . Denote by 3 the set of u E L(t2(T)) for which the Hzlbert dimension of In1 u is at most equal to N' (is strictly smaller than Nu) and let G be a C*-subalgebra of L ( t 2 ( T ) ) .

a ) 3 is a closed proper ideal of L ( t 2 ( T ) ) . b) 3 + G zs the C * -subalgebra of L(t2(T)) generated by 3 U G c) If 3 n G = (0) then llvll for all ( u ;v) E 3 x G .

I IIu

-t

~ l l

5.5 Orthonormal Bases

By Example 5.5.3.1, Card T is the Hilbert dimension of t 2 ( T ) . a) follows from Example 2.2.2.23 and Theorem 5.5.2.5. b & c follows from a) and Proposition 5.3.2.29.

21 7

5. Hilbert Spaces

218

5.5.4 The Fourier-Plancherel Operator

Throughout this subsection we assume IK = C .

Theorem 5.5.4.1

(4)

orthonormal basis of L ~ ( I R " )

(Plancherel, 1910) Take m E IN and ( x ~ ) ~ E Kthe defined in Corollary 5.5.3.9. Given k E K and

P E {O, l , 2 , 3 ) , define

and

Then there is a unique operator u on L2(IRm) such that

for every k E K . u is called the Fourier-Plancherel operator. Furthermore:

a) u zs unitary. b) u4 = 1 .

d) For each p E { 0 , 1 , 2 , 3 ) , F, is the eigenspace of u corresponding to (-i)P .

By Proposition 5.5.1.22, there is a unique operator on L2(IRm) with

for every k E K and this operator is an isometry. By Corollary 5.3.1.22, u is unitary. Given x E L2(IRm),

5.5 Orthonormal Bases

(Fourier expansion and Corollary 5.5.1.16). Hence, for any p E { O , l , 2 , 3 } , up(u) = { I , -2, -1, i } ,

3, is the eigenspace of u corresponding to (-i)P , and

It follows that u(u) = { l ,-2, - l , i } (Theorem 2.1.3.4 b)). By Corollary 5.3.4.4 a), FPis the eigenspace of u* corresponding to iP . Hence, given x E L2(Rm),

since the irlnctions of x k (k E K) are real. Corollary 5.5.4.2 If p E { O , l , 2 , 3 ) , then the resolvent of the FourierPlancherel operator has a pole of order one at (-i)P and (using the notation of Theorem 5.5.4.1), its residue is 7rFD .

This follows immediately from Theorem 5.5.4.1 a),d), and Proposition 3.3.1.16.

Lemma 5.5.4.3

We have

(4)

Given n E IN U ( 0 ) and t E R ,

5. Hilbert Spaces

220

d"e;(s-")2 dt"

ds = (-i)"

-m

-

(-i)ne;,z d" ( e - t 2

J21;

dt"

7

dn dt"

lt2

- ~ e - s 2 e ~ ( s - 1 t ~= 2ds

e-ts2ds) = I-i)

-m

ett2 d"eP2 dt" '

- 00

Theorem 5.5.4.4 ( 4 ) Take m E I N . Let X be Lebesgue measure or Rm and u the Fourier-Plancherel operator on L 2 ( X ) . Take x E L 1( A )n L 2 ( X ). Then

X almost everywhere on Rm. Hence u and the Fourier transform coincide on L L ( X )n L 2 ( X ). Let ( x , ) , , ~U {0} be the orthonormal basis of L 2 ( R ) defined in Example 5.5.3.6, ( y k ) k E Xthe orthonormal basis of L ~ ( X ) defined in Corollary 5.5.3.9, . 3 is dense and 3 the vector subspace of L 2 ( X ) generated by ( y k ) k E ~Then in L 2 ( X ) (Corollary 5.2.3.10 b + a). Given k E K ; define

Then

5.5 Orthonormal Bases

for every x E F and t E Rm (Lemma 5.5.4.3). T h u s

for every x E 3 and t E Rm. Given t E Rm and E > 0 , define

Then

Thus gE E L2(X) and

for every

E

s u p Ig,(s) 1 5 ( 2 ~ ) " sE In

> 0 and

for all s,t E R m and

E

> 0 . Moreover,

22 1

222

5. Hilbert Spaces

for every s E Rm. Takc x E F ,t E IRm , and

E

> 0 . By Fubini's Thcorcm,

Since g, E L2(X), uxdX =

( 2 ~T)

1

e-'(slt)x(s)gE(s)dX(s)

whenever x E L2(X), t E R m , and E > 0 . Now take x E LL(X)n L2(X). Therc is a A-null set A C Rm, such that

for every t E IRm\A (Lebesgue point). By Lebesgue's Convergence Theorem,

-

/

2 ( 2 ~7)

e-l(st)x(s)d~(s)

for every t E R m \ A . By Theorem 5.5.4.1 e), it also follows that (u-'x)(t)

=Z

A-almost everyhwere in Rm

(t)=

( 2 ~3 )

I

e-1(~ylt)5(S)dX(~) =

5.5 Orthononnal Bases

5.5.5 Operators and Orthonormal Bases

Theorem 5.5.5.1 ( 0 ) Take u E L ( E , F ) with E , F Hilbert spaces. Then the following are equivalent:

a ) u is compact b) If 5 is a weakly convergent filter on E containing a bounded set of E , then u ( 5 ) converges.

c ) If A is a bounded set of E , then the map

is continuous. d) If ( x , ) , , ~is an orthonormal family in E and (y,),,l is a bounded family in F , then ((~xLlyL))L,lE co(I) . e) I f (x,),€N and ( y , ) , , ~ are orthonormal sequences in E and F , respectively, then

lim (uxnlyn)= 0

n+m

f ) If ( x , ) , , ~and

are orthonormal families i n E and F , respective-

ly, then

g ) If v E L(P2,E ) and w

E

L(P2,F ) then

If these conditions are fulfilled and if A and B are orthonormal bases of E and F , respectively, then

This representation of u is unique in the following sense: if f E K A x Bsuch summable ~ and that (f( x ,y ) ( . ( ~ ) y ) ( , , ,is) ~

224

5. Hilbert Spaces

then

whenever (x, y ) E A x B . We have

a eb

@

c follows from Proposition 3.1.1.20, since E is reflexive (Corollary

5.2.5.5).

a + d . We may assume I to be infinite. Let 5 be the filter of cofinite subsets of I , i.e.

By Corollary 5.5.1.9, lim ux, = 0 r.3

Thus

d + e is trivial. e + b . Let 5 be a weakly convergent filter on E containing a bounded set of E . We may assume that 5 converges weakly to 0 and that E# E 5 . Assurne that 4 5 ) does not converge to 0 . Then, for some /3 > 0 ,

Hence 5 does not converges to 0 , and so there is some a > 0 with

Let 6 be an ultrafilter on E finer than 5 such that

We construct a sequence ( z , , ) , , ~ ~in E# inductively, such that

5.5 O r t h o n o n a l Bases

for every n E IN, where En arid Fn denote the vector subspaces of E and F generated by {zklk E and {uzklk E K , - I ) , respectively. Take n E IN and suppose that the sequence has been constructed up to n - 1 . Let A and B be finite orthonormal bases of En and F n , respectively (Corollary 5.3.1.10 a + c). Then

for all x E E , y E F (Proposition 5.5.1.7). Since 6 and u ( 6 ) converge weakly to {0} (Corollary 1.4.1.6 a 3 c), limrE,x=O,

limrF,,ux=O.

T,@

2,8

Hence there is a C E 6 , with

for every z E C . Choose zn arbitrarily from C . This completes the inductive construction. Given n E IN.

(~uz,- T F , , u z ~= / / ~J J

2

>P-B - > o

U Z ~-J J)TF"UZ,JI ~ ~

n

(Proposition 5.2.3.1). For n E I N , define

Then (x,),~N and (yn)nEEare orthonormal sequences in E and F , respectively.

5. Hilbert Spaces

226

Choose n E IN. From

~F,,UZ~III lluznll 5

IIuII

IIuznI12 = (UznIuzn)

<

11~2~

we get that

P2 I

0 < llull I ( ~ x n l Y n ) +I ;lluII

P + IIUII~.

By e), we get the contradiction

Hence u ( 3 ) converges to 0 . a & e + f . Assunie j ) does not hold. Then there is an for every C E P f ( I x L ) there is an ( L , A ) E ( I x L)\C with

E

> 0 such that

By induction, we construct injective maps f : IN 4 I , g : IN -t L such that

for every n E IN. Let n E IN and assume f and g have been constructed up to n - 1 . By Corollary 5.5.1.9, there are A E P f ( I ) and B E P f ( L ) such that

for every L E I\A, X E L\B, and k E INn-,. By the hypothesis of this proof, there is an

5.5 Orthonormal Bases

with

Then

L

@ f(INn-1) and X $ g(IN,-,) , so that we may put

This finishes the inductive construction. The existence of f and g contradicts e).

f +c is trivial. a & e + g . By a ) and Proposition 3.1.1.1 1, w* o u o v E K(e2) , and so, by el? 0 = lim (w*uve,le,) = lim (uvenlwen). n-tm

n+m

g =+ e . By Proposition 5.5.1.22, there are v E L(e2, E ) and w E C(e2,F ) such that ve, = x, ,

we, = y,

for every n E Dj . Now we prove the final assertion. For C E p f ( A ) (resp. D E y f ( B ) ) , let C' (resp. D') denote the vector subspace of E (resp. of F ) generated by C (resp. D ) . Take E > 0 . By Corollary 5.3.2.17, there are Co E p f ( A ) and Do E q j ( B ) such that

for every C E yf ( A ) and D E p f ( B ) with Co c C , Do C D . By Proposition 5.5.1.7 (and Proposition 5.3.2.13 c), e)),

for every C E

p

(A) and D E U

pf(B) . Hence ( u ~ I Y ) ( . I ~ ) Y(in K(E, F ) )

= (z,y)EA x B

(Corollary 5.5.1.11). It follows that

f

E K~~~ > ( f (x, Y ) ( ' I ~ ) Y ) ( ~ , ~ ) E A ~ B

is summable in K(E, F )

5. Hilbert Spaces

rn

The uniqueness of the representation of u is easy to see.

Remark. The last assertion of the theorem allows us to work with compact operators as we do with matrices.

Theorem 5.5.5.1' T h e above assertions are further equivalent to the following one: h) if ( x , ) , , ~ is a weakly convergent sequence i n E , then ges.

conver-

Since E is reflexive (Corollary 5.2.5.5), this assertion follows from Proposition 3.1.1.20'.

Proposition 5.5.5.2 Let E be a Hilbert space and take u E C(E)+.T h e n the following are equivalent: a ) u is compact b) If 3 is a filter or' E contazning a bounded set and converging weakly t o 0 , then

a + b . By Theorem 5.5.5.1 a + b , u(5) converges to 0 and the assertion follows. b + a . Let 5 be a weakly convergent filter on E containing a bounded set. We show that u i ( 5 ) converges. Without loss of generality, we may assume that 3 converges weakly to 0 . Since

for every s , we see by b), that 0 = lim(ux/z)f = lim ll?~fxll, ~~3

x,3

i.e. u f x ( 5 ) . converges to 0 . By Theorem 5.3.5.1 b Proposition 5.3.3.10 b) , u is compact.

+a,

u i is compact. By

rn

Proposition 5.5.5.3 If E , F are ir~jnite-dimensional Hilbert spaces, then L ( E , F ) / K ( E , F ) zs not separable.

5.5 Orthonormal Bases

229

Let (x,),,, and (yh)hELbe orthonormal bases of E and F , respectively (Corollary 5.5.1.4) and let

be the quotient map. We may assume that

For A C N , let

UA

be the operator E + F such that

(Proposition 5.5.1.22). Given A, B 6 ?(IN), define .4 A

-

B :-

(A\B)

u (B\A)

-- B

by

is finite.

--

-

is an equivalence relation on p ( N ) the equivalence classes of which are countable. Hence the set !Q(N)/-- of equivalence classes with respect to is uncountable. Given A E ?(IN), let A denote its equivalence class in v(K)/--. Take A, B E !J3(K) such that

and take v E K ( E , F ) . By Theorem 5.5.5.1 a

+e,

lim (~x,ly,~)= 0

n-+a

It follows that

Hence

Thus L ( E , F ) / K ( E , F ) is not separable.

Theorem 5.5.5.4 ( 0 ) Let E , F be Hilbert spaces. Take orthonormal families (x,),,~ and ( y L ) L E I i n E and F , respectively, and f € IK' . Let J : = { L E I ( f ( ~# )0 ) .

Take u E L ( E , F ) . Then the following are equivalent:

230

5. Hzlbert Spaces

and ux, = f ( ~ ) yfor , every

a ) u vanishes o n { x , ( LE

L

EI

If these conditions are fulfilled, then:

g) ( x , ) ,J ~ is a n orthonormal basis of (Ker u)'

and

Ker u = {x,Ii E J)' . h ) ( Y , ) , ~ J is a n orthonormal basis of Irnu and

Im = { Y , ( L E

J ) ~ ~ .

i ) Given y E F , y E Irn u i f l y E { y , l ~E J)"

,

(%)

LEJ

and zn this case

j)

T h e following are equivalent:

j,)

u

2.9

compact.

k) For eoenj v

E

L ( F , E ) and

L

E I ,

E P(J)

5.5 Orthonormal Bases

a =+b. Put

Then, for any x E E ,

(Proposition 5.2.3.7, Corollary 5.5.1.16), so that

b + a is trivial. c) follows from b) and Proposition 5.5.1.14 a d) follows from c) . e) By d) and the Bessel's Inequality,

+c.

)f O ( Y IYC)XL LEI

is well-defined (Corollary 5.2.2.5) and by b),

for every x E E . Hence

f) By a) and e), U*UX'

for every

L

E I . Thus, by a

= f (L)u*yc=

+ b,

If

(1)

I2x1

5. Hilbert Spaces

232

for every x E E g) By c ) ,

Keru = { x , ) E ~ J)' , and so (Keru)' = {x,Ii E J } l l By Proposition 5.5.1.2, it follows that (x,),,~ is an orthonormal basis of (Ker u)' . h) By Proposition 5.3.2.4,

Hence, by e) and g), (y,),,, is an orthonormal basis of Im u and

Im = { ~ ' I LE J ) l L . i) First suppose that y E Im u and take x E E with y = u s . By h) ,

and so, by b) and Proposition 5.5.1.2 (using the Fourier expansion),

Hence

for every

L

E .I (Proposition 5.5.1.8 a)) and

( Now suppose that

Put

( J ) LEJ

(Bessells Inequality)

5.5 Orthonormal Bases

233

(Corollary 5.2.2.5). By a) and Proposition 5.5.1.2 (using the Fourier expansion),

j, =+ j2 follows from Theorem 5.5.5.1 a 3 d jZ j 3 . Take E > 0 . Define

Suppose L E g f ( I ) and K

c L.

By a ) ,

and

on

{X,/I.

E I } ' . It follows frorn a

Since L and

E

+ d, that

are arbitrary,

LEL

in L ( E , F) . j3 + jl follows from Proposition 3.1.1.4. k) By a ) and e),

Corollary 5.5.5.5 ( 0 ) Let E, F be Hilbert spaces. Let ( z , ) , ~ I (, Y ~ ) ~ E beI orthonormal familzes in E and F , respectzvely, and take f E IK1 . Then the family (f(~)(.lx,)y,),,~is summable iff f E c o ( I ) . The necessity is trivial (Corollary 1.1.6.7), so take f E co(I) and Put

E

> 0.

234

5. Hilbert Spaces

Take K E CQf(I\J). By Theorem 5.5.5.4 b

+ d,

Hence ( f (~)(.Ix,)y,),~,is sumrnable (Proposition 1.1.6.6).

Proposition 5.5.5.6 Let E be a Hilbert space and 3 a Gelfand C * subalgebra of K ( E ) . Then there is a n orthonormal set A of E and a normpreserving homomorphism of involutive algebras

such that:

2) For every z E A , the map

zs i n a ( 3 ) and the m a p

is surjective. If, i n addition, A zs znfinite, then the image of the filter of cofinite subsets of A converges to the Alezandroff point of a ( 3 ) .

By Proposition 5.3.1.17, a ( 3 ) is discrete. Let x' E a ( 3 ) . Since the Gelfand transform on 3 is an isometry of C*-algebras (Corollary 4.1.2.5), there is an orthogonal projection K,! of 3,such that

Then K,! is an orthogonal projection in E with a finite-dimensional image. Let A,t be an orthonormal basis of Im x' (Corollary 5.5.1.4). Given distinct

5.5 Orthonormal Bases

(Corollary 5.2.3.15 b Put

* a) A :=

U

A,,.

z1€o(3)

By the above considerations, A let pu denote the map A + IK x' E 4 3 ) . Since the Gelfand C*-algebras (Corollary 4.1.2.5),

is an orthonormal set of E . For each u E 3 , which takes the value G(x') on A,! for every transform 3 + Co(a(3)) is an isometry of cpu E @ ( A ) for every u E 3 and the map

is a norm-preserving homomorphism of involutive algebras. 1) G = G(xl)ez, = G(x1)Fz,. z'Eo(3)

z'Eo(3)

Thus, by Proposition 5.5.1.7 (and Corollary 5.5.5.5),

2) Take x E A . Then

for every u E

F .Hence Z

E u ( E ) and x E A , . The other assertions follow. W

Proposition 5.5.5.7 ( 0 ) Let A be an orthonormal basis of the Hilbert space E . Take f E IKA and u E L ( E ) such that

for every x E A . a)

f

E em(A), llull = Ilf loo, and for every x E E ,

5. Hilbert Spaces

236

-1

b) a E IK

+ Ker ( a 1 - u) = f

(a)"

.

gp(u) = f ( A ) . d ) 4 u ) = f ( A ). C)

e) u is self-normal and for every g E C ( a ( u ) ) and x E A ,

f)

For every f E tCo(A), there is a unique

7E L ( E ) such that

for every x E .4. T h e map tm(A)--+L(E);

f-7

is a homomorphism of involutive unztal algebras preservzng the norms. f " K(E) ~ ifl f E co(A).

a) follows from Theorem 5.5.5.4 a b) By Theorem 5.5.5.4 a g,

*

+b&c&d&e&f

Ker ( a 1 - u) = {x E A I f ( x ) # a)'

Thus, by Corollary 5.5.1.17, -1

Ker ( a 1 - u) =

f (a)''

c) follows from b) . d ) Take a E IK\ f ('4). By b), a1 - u is injective. Take x E E . Then

5.5 Orthonormal Bases

297

(Bcssel's Inequality), and so x E Im (crl - u) by Theorem 5.5.5.4 i). Hence 01 - u is surjective. By the Principle of Inverse Operators, crl - u is invertible, i.e. cr $ a ( u ) . Hence

The reverse inclusion follows from c ) . e) Take X E E . By a),

and if f is real, then

Hence u is self normal. By a ) , again,

for every P E IK[s, t ] . But by Weierstra5s- Stone Theorem, g can be approximated unifornily on a ( u ) by such P . Thus

f) The first two assertions were proved in Corollary 5.5.2.10. The final assertion follows from Theorem 5.5.5.4 a 3 j .

Proposition 5.5.5.8 ( 0 ) Let E be a Hzlbert space and take u E C ( E ) . For every cr E a ( u ) , let T , denote the orthogonal projection of E onto Ker ( a 1 -u) . T h e n the following are equivalent: a) u is normal and

b) o(u)\{O) is discrete and there are a n orthonormal basis A of E and a n f E IKA with

for every x E A .

238

5. Hilbert Spaces

If these equivalent condztions hold, then u is self-normal and 4u)\{O) = ~P(u)\{O)= f (A)\{O). a + b and the last assertion. For cr E a ( u ) , let A, be an orthonormal basis of Ker ( a 1 - u) (Corollary 5.5.1.4). Put

A:=

U A,. oEo(u)

A is an orthonormal set of E (Corollary 5.3.4.5). If x E A L , then

and so

Hence A is an orthonormal basis of E . We define f : A + IK by f lA, = a for a E o ( u ) . Then, given x E A , x E A!(,)

C

Ker (f ( x ) l - u) .

Hence UX

= f (x)x .

By Proposition 5.5.5.7 c),d),e), u is self-normal and

Since (Ila~all)a~o(u) E co(a(u)) and

it follows that f (A)\{O) is discrete. Hence a(u)\{O) is discrete and

b 3 a . By Proposition 5.5.5.7 e), u is self-normal, so the assertion follows from Proposition 5.3.4.7 a ) . 8

5.5 Orthonormal Bases

239

Proposition 5.5.5.9 ( 0 ) Let E , F be Hilbert spaces and take u E L ( E , F ) . Then the following are equivalent: a) u is an isometry. b) If A is an orthonormal basis of E , then u ( A ) is an orthononnal basis of F . c) There is an orthonormal basis ( x , ) , ~of~ E for which orthononnal basis of F .

If E

=

( U X , ) , ~ is ~

an

F , then the above assertions are equivalent to the following one:

d) u zs unitary a

+ b . By Corollary 5.1.1.4, (""1"~)

=

( 5 1 ~= ) 6z,

for all x , y E A . u ( A ) is thus an orthonormal set of F . By Theorem 5.5.5.4 a+h, u(A)"

= G F= .

Hence u ( A ) is an orthonormal basis of F . b c . Let ( x , ) , , ~be an orthonormal basis of E (Corollary 5.5.1.4). By b), { U X , ~ LE I ) is an orthonormal basis of F . Let L , X be distinct elements of I . Assume that

*

Define

It is easy to see that A is an orthoriormal basis of E . Since u ( A ) is not an orthonormal basis of F , this contradicts b). Hence

and

(ILX,),,I

c

is an orthonormal basis of F .

* a . Take x E E . By the Fourier expansion,

24 0

5. Hilbert Spaces

so that

By Corollary 5.2.2.4,

Since

(Theorem 5.5.5.4 a 3 h), it follows from Proposition 1.2.1.18 d), that u is an isometry. a @ d follows from Corollary 5.3.1.22. Proposition 5.5.5.10 ( 0 ) Let E, F be Hilbert spaces. Take u E L(E, F) and put v := (u*O U ) ~(Corollary 5.3.3.9). Then there is a unique w E L ( E , F) such that

Moreover:

a ) 11~x11= llvxll for every x E E .

c) llwll

5 1 , Tnl= Imw

d ) llwxll = llxll for every x E ( K e r w ) l .

f)

If (x,),,,

is a n orthononnal famzly i n E and ( Q , ) , ~ Iis a family i n IK\{O} such that

then ( W X , ) , is ~ ~an orthonormal family i n F and

5.5 Orthonorma[ Bases

Step 1

a&b

Now 2

ll~x11~ = (vxlvx) = (xlv2x) = (zlu*uz) = (UXIUX) = 11uzll

,

Iluxll = IIvxlI Hence (Ker u)' = (Ker v)' = G ,

Ker u = Ker v ,

(Propositions 5.3.2.4 and 5.2.3.7) Step 2

(Existence and Uniqueness of w ) & c & d

We define

By a), wo is well-defined and linear. Moreover,

IIwoxII = llxll for evcry x E I m v . Hence there is an isometry

I m v G which extends 7uo. By b), we may define w by extending this isometry by 0 on Ker u . Then

and

11~x11 = 11x11 for every x E (Ker Step 3 We put

.

e)

By b), w is unique.

5. Hilbert Spaces

24 2

F := (Ker u)' . Then = (wxlwy) = ( W ~ F X ~ W = ~ F(KFXIKFY) Y) = (KFXIY) (W*WXIY)

for all x, y E E (d), Corollary 5.1.1.4, Proposition 5.2.3.1 a

+ b). Thus by b),

W * O W = K F , w*OU=W*OWOV=KFV=V. Step 4 BY b)

f)

3

Kerw = Kerv = { x , ~ E L I)', so that (Kerw)' = { x , ~ EL I)" By d ) , the map (Ker w)' 4Im w ,

x

++

wx

is an isometry and so (wx,),~, is an orthonormal family (Proposition 5.5.5.9 a + b). Now

(Proposition 5.3.2.13 c ) )

5.5 Orthonormal Bases

5.5.6 Self-normal C o m p a c t O p e r a t o r s T h e o r e m 5.5.6.1 ( 0 ) Let u be a self-normal operator o n a Hilbert space E . Given CY E K , let T, be the orthogonal projections of E onto Ker (a1-u) . T h e n the following are equivalent: a) u is compact.

b) a(u)\{O) is discrete and Ker (a1 - u) is finite-dimensional for every E ap(u)\{O). C)

Ker ( a 1 - u) is finite-dimensional for every a E up(u)\{O) and

d)

There is a n orthonormal basis A of E and a n f E co(A) such that

for every x E A . e) There is a n orthonormal basis A of E and a n f E co(A) with u = ):f (x)(.lx)x (diagonalization of u) zEA

If these conditions are fulfilled, then:

f ) For each g E C(a(u)), g(u) = ):g ( f (x))(.lx)x rcA

whenever E zs finite-dzmensional and

whenever E is infinite-dimensional. g) For each

a E IK\u(u),

whenever E is finite-dimensional and

whenever E is infinite-dimensional.

24 4

5. Hilbert Spaces

h) If

CY

E

K\{O), then

and

for every x E Im (01- U)

F :=

U

Ker(a1 - u ) ,

o€o(u)\(o)

then Ker u = F 1 .

a + b follows from Theorem 3.1.5.1 a),b). b =+ c follows from Proposition 5.3.4.7 1). c + (1. By Proposition 5.5.5.8 a + b , there is an orthonormal basis A of E and an f E K A such that

for every x E A . Since, by Proposition 3.1.1.4, u is compact, we see that f E co(A), by Proposition 5.5.5.4 j, + j:, . d =+ e 3 a follows from Theorem 5.5.5.4 j2 + j3 + jl . f ) By Proposition 5.5.5.7 e),

for every x E A . The assertion now follows from a g) Define

+e

5.5 Orthonormal Bases

for every x E A and the assertion now follows from f), since

h) By Theorem 3.1.3.8, a1 - u is a Frcdholm operator, so that

by Theorem 5.5.5.4 h ) and Corollary 5.5.1.17. By Theorem 5.5.5.4 i),

i) follows from c) . j) follows from d) anti Proposition 5.5.5.7 c),d).

Corollary 5.5.6.2 Take n E CV . Let u be a self-normal operator o n IKn , u the associated matrzz of u , P the characteristic polynomial of a , i.e.

P ( t ) = Det ( t l - a ) , and take cr E a ( u ) . T h e n the mulitiplicity of the eigenvalue a with respect to u agrees with the multiplicity of the zero a of P . By Theorem 5.5.6.1 a and a family (a,),,nn in

+ d , there is an orthonormal basis

( x , ) , ~ of~ ,IKn

IK,such that

for every i E N,, . If p is the n~ultiplicityof a with rcspcct to u , then

p = Card { i E Nn ( a, = a )

24 6

5. Hilhert Spaces

(Proposition 5.5.5.7 b)). Let v be the operator on I K n , given by ve, = x,

for every i E Nn . v is obviously bijective. Put

Then

we, = a,e, for every i E Nn and

Thus

Hence the mulitplicity of the zero a of P is p . Remark. The above result does not hold if u is not self-normal. Indeed, let u be the operator associated to the matrix

Then a ( u ) = (1). 1 is a zero of the charactertistic polynomial of the above matrix with multiplicity 2 , but the multiplicity of the eigenvalue 1 with respect to u is 1 .

Corollary 5.5.6.3

Take n E N and a E IK,,, . Consider

f : IKn

x IK"

+I K ,

( x ,y)

++

):a , J x , ~ J . *,]=I

If the operator associated to a is self-normal, then there is an orthonormal basts ( z ~ ) , of ~ ~IKn, and a famzly (y,),,Kn in IK such that

5.5 Orthonormal Bases

This assertion follows from Theorem 5.5.6.1 a

24 7

+ d and Proposition 5.5.5.7 a )

Corollary 5.5.6.4 ( 0 ) Let E be a real (complex) Hilbert space and F the vector subspace of C ( E ) generated by {(.Ix)x I x E

El

Then

and -

F=ReK(E)

(7=Kj(E)).

By Proposition 3.3.2.13 b),

for every x E E , so that

Take u E RcC,(E) and v E R e K ( E ) . By Theorem 5.5.6.1 a + e , there are orthonormal bases A, B of E and f E K ( ~ g) ,E c o ( B ) such that

Thus u E 3 and v E

7 and hence

Suppose that K = C and take u E C f ( E ) . v E K ( E ) . Then

(Proposition 5.3.2.13 b), Corollary 5.5.1.11, Proposition 5.3.2.2 a

+ b), so that

24 8

5. Halbert Spaces

Corollary 5.5.6.5 ( 0 ) Let u be a self-normal compact operator o n a n infinite- dimensional Hilbert space and take f E C((T(U)). Then f (u) is compact

zff f(0) = 0 .

is a diagonalization of u (Theorem 5.5.6.1 a

f

+ e)

then

= f (011 + C ( f (g(x)) - f (O))(.lx)x ztn

by Theorem 5.5.6.1 f). By Theorem 5.5.6.1 e

+a,

f ( u ) is compact iff

Corollary 5.5.6.6 ( 0 ) Let u be a self normal operator o n a Hilbert space E . Then the following are equivalent: a)

IL

zs compact

b) Gzven f E C(a(u)) such that O$suppf, Im f (u) is finite-dimensional. a

+ b.

Let

be a diagonalization of u (Theorem 5.5.6.1 a

3

e ) . By Theorem 5.5.6.1 f ) ,

so that In1 f (u) is the vector subspace of E generated by

Sincc the above set is firiitc, Im f (u) is finite-dimensior~al. b + a . Take E > 0 . Consider

and g E C(a(u)) s~ichthat:

5.5 Orthonormal Bases

1) 0@-S ~ P P ~ . 2) .9(4u)) c [0,11.

3) g = 1 on a ( u ) \ U F (0) . We put

By b) , f (u) E L f ( E ) and by Theorem 4.1.3.1 b),

Since

E

is arbitrary,

Corollary 5.5.6.7

(0)

C f ( E ) is a hereditary set of C ( E )

Let u E C ( E ) + , u E C f ( E ) such that

Then u E X ( E ) (Proposition 3.1.1.4), so that v E X ( E ) (Corollary 3.1.1.13, Proposition 4.3.4.5 e)). By Theorern 5.5.6.1 a =. e & j , there are orthonorrnal R ( : ' such that bases A , B of E and f E c o ( A ) + , g E l

Take y E j l ( 0 ) . Then

(Corollary 5.3.3.7), so that

5. Hilbert Spaces

250

x $ j'(0)

* xly

We deduce that -1

f (]O,m[) C ?J1(0)l C ;(lo, m [ ) l L .

Since >'(lo, m [ ) is a finite set, ?J1(]0,m [ ) l L is finite-dimensional (Corollary -I

5.2.3.9) and f (10, m[) is finite. Hence u E L I ( E ) and L,(E) is a hereditary set of L ( E ) .

Proposition 5.5.6.8 ( 0 ) Let E be a Hilbert space and take u E K ( E ) + . Then the following are equivalent:

a) There is an x E E such that

b) (resp. c)) Given v E L ( E ) + (resp. v E K ( E ) + ) with v 5 u , there is an a E IR+ such that

In partacular {(.lx)x

I

x E E , llxll = 1) is the set of minimal elements of

PrL(b)\{O). a * b . By Corollary 3.1.1.13 and Proposition 4.3.4.5 e), v E K(E)+.By Theorem 5.5.6.1 a 3 e & j , there is an orthonormal basis B of E and an f E co(B)+ with

Let A be an orthonormal basis of E containing x. Put

By Corollary 5.3.3.7,

for every z E E . Hence

5.5 Orthonomal Bases

for every z E A\{x) . Thus

for every y E C and z E A\{x) . I t follows that

for every y E C . Hence C contains a t most one element and if y is this element, then

which proves b) . b + c is trivial. c 3 a . By Theorem 5.5.6.1 a E and an f E co(A)+ with

Take y E A with f ( y )

+ e & j , there is an orthonormal basis

# 0 and put

Then

so that V

By c) , there is an a E

~

U

.

R+ such that v = ffu.

Then

f (Y)= (vyly) = a(uyl?/) = af (?J)

A of

252

5. Halbert Spaces

and f ( x ) = (uxIx)

=

(vxIx)

=o

for every x E A\{y) . Hence

Now we prove the Imt assertion. By a + b and Proposition 5.5.1.7, (.lx)x is a minimal element of P r E\{O) for every x E E with 11x11 = 1 . Let p be a minimal element of PrL(E)\{O) . Take x E I m p with llxll = 1 . Then P O ((.lx)x) 0 P = ( . I P ~ ) P=~(.lx)x (Proposition 5.3.2.13c).d)), so that (.lx)x (Corollary 4.2.7.6 f

+ a ) . Since p


is minimal, it follows

w

p = (,lx)x. Proposition 5.5.6.9 ( 0 ) Let E be a complex Hzlbert space and C'subalgebra of L ( E ) actzng irreducibly on E . If

F

a

3 n K ( E ) # {O) > then K ( E ) c 3 .

Step 1

There is a finite-dimensional vector subspace

F of E with (0) # F # E and

TF

E3

There is a u E (Re3) n K(E)\Cl

By Theorem 5.5.6.1 a

3

e , there is an orthonormal basis A of E and an

f E co(-4) such that f is not constant and

Takc

tr

E f(A)\{O) anti p i t -I

B

:=

f ( a ) , F := Ker (01 - U)

5.5 Orthonormal Bases

25.7

By Theorem 5.5.6.1 a + b : F is finite-dimensional and by Theorem 5.5.6.1 a + f & j (arid Proposition 5.5.1.7),

Step 2

3 2 E E\{O), (.1x)x E 3

Let F be a minimal vector subspace of E with the properties described in Step 1 . Given u E 3 ,define

and put

Since X F E F ,it follows that ? is a subalgebra of C ( F ) . By Proposition 5.3.2.25, 3. acts irreducibly on F . By Step 1, F is one-din~ensional.Hencc

for every x E F , llxll = 1 Stcp 3

Given x, y E E , if (.lx)x E

F ,then (.lx)y E F

We may assume that

By Proposition 5.3.2.20 b) , x is cyclic for in 3 such that

F .Hence there is a sequence (u,),~Lv

lim u,x = y

n+m

We get (.lz)y = lirn (.Ix)unx = lim n+m

Step 4

n+m

K(E) c F

By Stcp 2: 3, and Corollary 5.5.1.11,

IL, o

((.lx)x) E 3

254

5. Hilbert Spaces

Hence by Corollary 5.2.3.4, r(E) c 7.

m

Remark. A similar result does not hold in the real case as the following example shows: E.=IR 2

C o r o l l a r y 5.5.6.10

,

jr:=

-~

a

,

9

Let E be a complex Hilbert space and J~ an order a -

complete C*-subalgebra of s

acting irreducibly on E . If P r ~ \ { 0 }

has a

minimal element, then c 7. Let p be a minimal element of Pr~-\{0}. By Corollary 4.3.2.12 a =, d, pjrp is one-dimel~sional. So, by Proposition 5.3.2.20 b), there is an x c E with p

<-I~)~.

Hence p 6 K~(E) and by Proposition 5.5.6.9,

r(E) P r o p o s i t i o n 5.5.6.11

7.

m

( 0 ) Let E be a complex C*-algebra and (H, tp),

(K,~p) irreducible representations of E such that Ker~ = Kerr

p(E) N K~(H) # {0}, then the two representations are equivalent. Put F := K e r ~ = K e r r -1

G := ~ (LS(H)).

5.5 Orthonormal Bases

Let cp and and let

255

4be the factorizations through G/E' of ylG and $JIG,respectively, -+ G / F

q :G

be the quotient map ( F and G are closed ideals of E ). By Proposition 5.5.6.9, K(H)

c Y(E)>

so that

cp : G / F 4K ( H ) is an isometry. Take < , q E K , < # O , a n d ~ > O . T a k ex ~ G \ F . T h e r e i s a< o € K such that (*x)
# 0.

By Proposition 5.3.2.20 b), there are y, z E E such that II(*Y)(*x)~o- 711 <

E

5,

E

I(*zX

-
< 2(llzll llyll + 1)

We get that - (@Y)(*x)
<

Since yxz E G , it follolvs that is cyclic for $(G). By Proposition 5.3.2.20 a), is an irreducible representation of G . Hence

4

is an irreducible representation of K ( H ) . By Theorem 5.5.1.24, this representation is the identical representation. Hence there is an isometry

5. Hilbert Spaces

256

for every

7)

E K ( H ) . We get

(*x)

-

0

u = (*qx)

0

u = u 0 (Lpqx) = u 0 (cpx)

for every x E G . Take

X E

E and ~ E G We . have x y ~ G , s that o

Take ( E H and g E H\{O) . Since g is cyclic for cp(G) (Proposition 5.3.2.20

b ) ) ,there is a filter

5

on G such that

I;?(cpy)o = t We get

Hence ( H , y) and ( K ,tb) are eqriivalent.

W

Corollary 5.5.6.12 ( 0 ) Let H be a cornplez Hilbert space, E a C * subalgebra of L ( H ) containing K ( H ) , and cp : E -t E and zsometry of C* algebras. T h e n there is o unitary operator u o n H such that for each x 6 E ,

cpx = uxu-I u is unique up to a factor of absolute value 1 .

If 1;. derlotes the identity map E -+ E , the11 ( H , cp) and ( H ,Q) are irreducible representatio~~s of E . By Proposition 5.5.6.11, the two representations are equivalent. The uniqueness follows from Proposition 5.4.1.7.

5.5 Orthonormal Bases

257

Corollary 5.5.6.13 ( 0 ) Let (EL),,, be a family of (complex) Hilbert spaces, E the C*d i r e c t s u m of the family (K(E,)),,, , and F a C * subalgebra of E such that: 1) for every

L

E

I , the representation

is irreducible, 2) i f

L,

X E I are distinct, then p, and px are not equzvalent representations.

Then R e E c F ( E = F ) . Take distinct L , A E I and a E K(E,) . By Proposition 4.2.4.18, it is sufficient to show that there is an x E F with

x,=a,

xx=O.

By l ) , 2), and Proposition 5.3.6.11, Ker 9,# Ker p x Since K ( E A )is simple, Ker px is a maximal closed ideal of F . Hence p,(Ker p A ) is a non-zero closed ideal of K(E,) (Theorem 4.2.6.6). Since K(E,) is simple,

Hence there is an x E F with 2,

= p,x = a , ,

xx

= (FAX = 0

5. Hilbert Spaces

258

5.5.7 Examples of Real C*-Algebras Definition 5.5.7.1 ( 0 ) A n involutive Hilbert space is a Hilbert space H endowed with a conjugate linear involution

H

+H ,


(the involution of H )

such that

(flri) = 0 for all <,7 E H . For e v e y set T , @ ( T )will be considered a n involutive Hilbert space with respect to the canonical involution

If G and H are involutive Hilbert spaces and x E L(G,H ) then (by Definition 2.3.1 . l ) z:G+H, < e x ? .

Every involutive Hilbert space is an invol~~tive Banach space with respect to its canonical norm.

a) x E L(G,H )

(0)

Let F, G , H be znvolutive Hilbert spaces. 5 E L(G, H ) , )1Z11 = llxll, Z = x , Z* = x* .

Proposition 5.5.7.2

b ) L(G,H ) endowed with the znvolution C(G,H ) + C(G,H ) , x

HZ

is a n involutive Banach space.

c) x ~ L ( F , G ) , y ~ L ( G , H ) + y o z = y o 5 . d) The map

C ( H ) + C ( H ), x

M

z

zs a conjugate znvolution of the C*-algebra C ( H )

e) Assume IK = C and let E be a n C'-subalgebra of C ( H ) such that XEE-TEE. Then the map

i +x E~ , (x,y)c-t(x+iy,~+ijj) is a n zsomorphism of C' algebras.

5.5 Orthonomal Bases

a ) Take ( < , q )E G x H . Then

- (2<11,)

= ( Z l v ) = (xZl7) = (Clx*?) =

Hence

The other relations are easy to see. b) follows from a ) . c) follows from Proposition 2.3.2.22 i). d ) follows from a ) arid c). e) follows from d) and Theorem 4.1.1.8 e) . Proposition 5.5.7.3

Let E be an involutive Banach algebra,

a conjugate involution, and x1 E EL such that -

xl(Z) = xl(x)

for every x E E . In the sequel we use the notation of Theorem 5.4.1.2. a ) Z E F for every x E F . Put

for every X E E I F and x E X . b) X, Y E E / F

+ (XIP) = (XlY) , IlX11 = llXll,

=X.

c) E / F endowed with the involution

-

is an involutive normed space. We extend this involution to E / F .

-

d) E I F is an involutive Hilbert space.

5. Hzlbert Spaces

260

a ) We have

-

f (F) = xl(Z'Z) = xl(s'z) = xl(x*x) = 0

b) For (x, y ) E

Xx Y,

c) Since the m a p is obviously a conjugate linear involutive the assertion follows from b). d) follows from b) and c) . e) Take Y' E E I F and y E 1 ' . Then

so that cpx = gJZ

Proposition 5.5.7.4 ( 0 ) Let H be an involutive complex (real) Hilbert space. Then H has an orthonormal baszs A such that

for every

<EA

Let 2L be the set of orthonormal sets A of H such that

<

E A . Order 2l by inclusion and take a maximal elemrnt A E I I for every (Zorn). Assume A is not an orthonormal basis of H . Then there is a [ E A' with ((((( = 1. Since

for every 71 E A , it follows

F

A'. Let 0 E IR such that (
and

Hence A U 17 E 2l and this contradicts the maximality of A . Remark. It follou~sfrom this proposition that every involutive complex Hilbert space is isomorphic to P2(T) for some set T . Proposition 5.5.7.5

(0)

Let H be a n involutive Hzlbert space and

a) E is a closed involutive vector subspace of L ( H ) b) If 3 is a n upward dzrected set of L ( H ) contained i n E and if x is its supremum i n L ( H ) , then x E E . c) If x is a self- normal operator on H belonging to E , then f (x) E E for every bounded Bore1 fiinctzon j on ~ ( x. ) d) If IK = C and if x is a unitary operator o n H belonging to E , then for each n E K there zs a unitary operator y o n H belonging to E such that y" = x . Moreover, there zs an orthonormal baszs A of H such that xf = for every = A .

<

<

e) If IK = C and if x, y are unitary operators o n H belonging to E , then there is a unitary operator 2 o n H such that

262

5. Hzlbert Spaces

a) E is obviously a vector subspace of & ( H ) . By Proposition 5.5.7.2 a), it is closed. b) Take <,71 E H and let 5 be the upper section filter of F .By Theorem 5.3.3.14 c),

Hence x' = 2 and x E E . c) By Theorem 5.3.3.14, L ( H ) is C-order complete, so by Corollary 4.3.2.5 b), f ( x ) is well-defined. We denote by B the set of bounded Borel functions on a ( x ) and put a , : = {f E B I f(x)*

=fo}.

If P E IK[s,t ] , then by Proposition 5.5.7.2 a),c) (and Corollary 4.1.3.8) the map a(x)

+ IK,

a-

P(a,Z)

belongs to Boo.By a) and the Weierstrass-Stone Theorem, C(a(x)) C & . If ( fn)nElh. is an increasing (decreasing) sequence in Bo with supremum (infimum) f in B , then by b), f E Bo . Hence Bo = B . d ) By Corollary 4.3.2.5 h) (and Theorem 5.3.3.14), there is a bounded Borel function f on a ( x ) such that f (x) is unitary and f (x)" = x . By c), f (x) E E . In particular, there is a unitary operator y on H belonging to E such that y2 = x . By Proposition 5.5.7.4, we may assume H = e2(T) for some set T . b). We Then (yet)ttr is an orthonormal basis of H (Proposition 5.5.5.9 d have

*

x P t = xYet = xy8et = y2y*et= yet for every t E T . e) By d ) , there are unitary operators xo, yo on H belonging to E such that

Put

Then z is a unitary operator on H and we have

5.5 Orthonormal Bases

Proposition 5.5.7.6 ( 0 ) Take operator on H := P2(T),and

IK

=

269

C . Let T be a set, x a unitary

Then the following are equivalent.

If these conditions are fulfilled and i f E is a C*-subalgebra of L ( H ) such that u ( E ) C E , then

F

:= { a E

E ( a= u ( a ) )

is a real C* subalgebra of E such that the map

is an isomorphism of complex C*-algebras. Given a E L ( H ) , we have (by Proposition 5.5.7.2 d)) u 2 ( a )= x

~

-

x =*xZae*xf .

I f a ) is fulfilled, then xf=fl,

?x'=fl,

so that u 2 = 1 . If u2 = 1 , then (xZ)a = a(xZ) for every a E L ( H ) . Hence XZ E C1 (Corollary 5.3.2.14). Putting a := Z' in the above equality, we get

Hence there is an cu E

R such that

264

5. Hilbert Spaces

Hence Z = ax'

from which it follows that cr E (-1, + I ) . BY b),

Hence the last assertion follows from Proposition 2.3.1.43 a ) (and Corollary 4.1.1.21, Proposition 4.1.1.24).

Proposition 5.5.7.7 ( 0 ) Assume IK = C .Let T be a set and x, y unitary operators on H := e2(T) such that

in H such that if we put

a ) There is a family ([,),,I

C(,,O):=EL, for every

L

E

C(L,I)

:=x<,

I , then (
b) If T zs finite then Card T 2s even. c) There zs a unitary operator z on EI such that

a ) Take < , q E H . \Z1e have (zE1q) = (E~x'v)= -(?\%V)= -(Elq) = -(x5il<), -

xx< = xz< = -xx*[ = -[. It follows that
Let 0 be the set of orthonormal sets A of H such that [lx5j for all [, 17 E A . Since f2 is inductively ordered by inclusion, it contains a maximal

5.5 Orthonormal Bases

265

element A (Zorn's Lemrna). Assume A u {xf I ( E A) is not an orthonormal basis of H . Then there is an

with 11q11 = 1 . By the above remark, A u { ~ )E 0 ,contradicting the maximality of A . Hence A U {xf 1 ( 6 A ) is an orthonormal basis of H . By the above remark, is a family in H such that (Cx),,,,~,,,) is an orthonormal basis of H . b) follows from a). c) Let (F,),,, arid ( v ~ ) ~ be ~ ,families , in H with the property desc:ribed in a ) with respect to x and y , respectively. We may assume that I = L (Corollary 5.5.2.2). Let z be the unitary operator on H , defined by

for every

L

E

I (Propositions 5.5.1.22 and 5.5.5.9 c

+ d ) . Take

L

E I . Then

Hence

Proposition 5.5.7.8 ( 0 ) Assume IK = C . Let T be a set and s,y unitary operators o n H := t 2 ( T )such that z = z* ,

ij = y*

(resp. Z = -x* ,

Let E be a C*-subalgebra of L ( H ) such that zEz' = E for any unitary operator z o n H and such that a~E==+ii€E.

= -ye)

5. Hilbert Spaces

266

Put

E, := { a E E I xtix* = a ) , EY := { a E E I yay* = a ) T h e n there is a unitary operator z o n H such that

and the m a p

E,

+E y ,

a Hzaz*

is a n isomorphism of real C *-algebras (Proposztion 5.5.7.6) By Proposition 5.5.7.5 e ) (by Proposition 5.5.7.7 c ) ) , there is a unitary

operator z on H such that z x = yz. Take a E E . Then -

yzaz* y* = yzhz'y*

=

zxiix'z* ,

so that ( a E E,)

(a = xhx*) e(zaz' = yzaz'y')

++

(zaz* E E,)

Since the map E-+E,

a-zaz*

is an isomorphism o f complex C*-algebras (Proposition 4.1.1.24), it follows from the above that the map

is an isomorphism o f real C'-algebras. Corollary 5.5.7.9 ( 7 ) Let T be a set and HK := e2(T,IK),(i.e. the Hilbert space H over IK ). Let x be a unitary operator on Hc such that Z = x* and put

Then the real C*-algebras E and E n K(Hc) (Proposition 5.5.7.6) are isomorphic to L ( H R ) and K ( H R ) , respectively.

5.5 Orthonormal Bases

By Proposition 5.5.7.8, E (resp. E n K ( H c ) ) is isomorphic t o

(resp. { a E K ( H c ) 1 a = 3) ) .

{a E L(Hc) I a = Z )

If we identify H R canonically with a subset of H c , then for a E C ( H c ) a=Z-a(HR)c

HR.

Hence E (resp. E n K ( H c ) ) is isomorphic t o L ( H I R ) (resp. K ( H R ) ). Proposition 5.5.7.10 define

(0)

Let

IK = C and let T be a set. Given t

t1 := e(t,o),

. ET ,

t" := e (t.1) .

For s, t E T and cp E IR , define

{

5:t

:= e'?(.ltl)sl

+ e-'?

Y$

:= e'?(.ltl)s"

-

(.lt")sl'

e-'?(.lt")sl

Further, let x be the unztary operator on H such that

for all t E T (Propositzons 5.5.1.22 and 5.5.5.9 c of L ( H ) such that

+ d),

E a C*-subalgebra

and put

F ( E ) := F := { a E E I XZX*= a ) . a ) F is a real C*-subalgebra of E such that the map ;---t

E,

(a,b)c-ta+ib

is an isomorphism of complex algebras. If E is simple then F is .simple and purely real.

5. Halbert Spaces

268

b) For all q, r, s, t E T and p, I) E IR,

{

x:,

= cos

yx;,

y$ = cos

y:t

(a*xLV =

7

+ sin y x l t + sin y yf, ,

(yrt)' = -YZ

xfrx!t = 6,,x::',

= -6,,x,,

Y:,Y$

xfry$ = 6rs~;'"

7

v+*

yrrx!t = brsy$+*,

1

c) For every t E T with x i E E , x& is a minimal element of P r F\{O) .

d ) For every t E T with xf, E E the map

M -+ X:~FX\, cr+pi+ y j +6k ++

ax: + p x : , +yy:, +6yp,

is an isomorphism of real C' -algebras (Example 4.1.1.31). e)

If there is a t

E T with x: E E

, then F is a purely real C*-algebra

f ) Assume there is a t E T with x i E E . If S is a set and

G := K ( e 2 ( S ) )(resp. G

:= K ( e Z ( S )nmS) )

then F zs not zsomorphzc to G . a ) We have

By Proposition 5.5.7.6, F is a real C'-subalgebra of E and the map

is an isomorphism of complex C*-algebras. By Proposition 4.3.5.3 a is simple then F is simple and purely real. b) LVe have

x:, = (cos

X P

+ i sin 2

+ b , if

( ) s t+ <:or- - i sin - (.lt")s" 2

"7

(

7

"7

=

E

269

5.5 Orthononnal Bases

"'P = cos -((.lt')sl

2

"P (z(.lt1)s' - i(.ltl')s") + (.I t")srl) + sin 2

"9 6 = (cos 1 + i sin

=

2

"'P

= cos -((. ltl)s" -

(. lt")sl)

2

"P (z(.l tl)s" + i(.l ttl)s') = + sin 2 "P + sin -y,, 2

= cos -y,,

2

I

By Proposition 5.3.2.13 b),

i

(xyt)*= e-'? (.ls1)t' + e'? (.Isl')trl = xkv , (y,:)'

I IS")^'

= e-'':

-

(.ls1)tt1= -ylP, .

By Proposition 5.3.2.13 e), x ~ ~ r=: (el?(.r')q'

+ e-'%(.rrl)q")

(e'?(.lt')sl

(sllrl)(.~tl)ql+ e-'V ( s " j T ~ ~ ) ( . ~ t= ~ ~brs)z:*q "

= el-

ygy; = (e'?(rl)q" - e-'? (.lr")ql) = -e-'-

-(*-u) 2

(.ltl)r"

- 1 -

-

(-

2

e - ' 7 (.ltll)sl) =

"(*-u)

(sl\rl)(. 1 tl')ql

~ " = 2(el?

-

,

(sl~~l)(~~t")q"-el~(sll~r'l)(~~tl)ql=-brsz~~;-w,

z ~ v ,= $ (er%(.r~)q'+ r - ' ~ ( . / T l l ) q " )(elB(.lt/)sll -

+ e-'Y(.ltl')s")

-

+ el+ *-,"I P+*)

~(!8+Ui)

= ef7 (s'Irr) (. Itl)q" - e-f&

(s"ITb) (. Itl')ql = brS *::y

-

(.I<)v = (.I?)q, 0

x* = (.(xf)x7j

(Proposition 5.3.2.13 c),d)). It follows that

,

(.jtl),Tl + e - ~ 7 ( . 1 t ~ ~ )=s ~ ~ )

For <,?1 E H ,

x0

i-17(.jt1~)s1) =

(s"~rt1) (. 1t1)q1'= brsy$-'

IT^)^')

(.lrt)qu - e-'?

-

.

5. Hilbert Spaces

2 70

c) By b), x: E Pr F . Take p E Pr F\{0} with p 5 x i . Then p = px;tp = (.lptt)pt' + (.lpt")pt" (Corollary 4.2.7.6 a

+ f, Proposition

5.3.2.13 c),d)) and

for all s E T\{t) (Corollary 4.2.7.6 a a , lj, y, 6 E C such that

+ c). By Proposition

5.3.2.4, there are

+ Pt" pt" = yt' + 6t". pt' = at'

Thus p = (./at1+ Dttl)(at' + pt")

+ (.I-$'+ bt")(yt1+ at") =

+(pa + Ty)(.lt")tl + (/Dl2 + 1612)(.lt")t". Since p E F , p

+ ly)2)(.)t1r)t'' (a3+ y8)(.lt")t1-(@ + 67)(.t')t1' + (IPI2+ 16I2)(.1t')t'

= xpx* = (1a12

-

(Proposition 5.3.2.13 c),d)),so that

+

Ia12 IyI2 = llj12

+ 1612, 5 P + 76 = -(@ + 67)

and therefore

+

p = (laI2 IYl'')z;

+ (ap + 76)(.lt1)t"

-

alj

+ 76(.ltr1)t'.

Furthermore, P = p* = (la['

+ Iyl2)x: + CYP+ y6(.lt")tt

(Proposition 5.3.2.13 b)), so that

sp+7a=o

-

( a p + 76)(.lt1)t"

5.5 Orthonomal Bases

and therefore P = (laI2+ Ir12)xPC,

1=l

l ~ l l= (laI2+lr12)11x~tll= 1012+ lrI2,

p = xPt . d ) Put

a:=xft,

b:=x:,,

c : = ~ , ' , , d:=ypt.

By b), a E P r F ,

b2 = c2 = d2 = -a

Take y E F . By Proposition 5.3.2.13 c),e), there are a : P, y , 6 E C ,such that

aya = a a + Pb + yc

+ 6d

Since aya E F , by b),

aya=xayax* = ~ i a + p b + y c + 6 d . Since a , b, c, d are linearly independent, a , P, y , 6 E R . Hcncc aFa is the vector subspace of F generated by a, b, c, d and therefore the map

is an isomorphism of involutive algebras. e ) Assume thc contrary and put

2 72

5. Hilbert Spaces

for every y E E , i.e. u2 = 1 . By Proposition 2.3.1.43 d), there is a linear map v : E + E , such that

and

v ( y z ) = ( v y ) z = y ( v z ) , vy' = -(vy)* for all y, z E E . Then for y E F ,

vy=vuy=uvy,

vyE F .

Hence v ( F ) c F . For every y E F ,

v (x:t~x:t)= x:t(vy)x::

7

so that

v (xftF X : ~ )= x:~F X ;

.

By c), we may replace X ~ ~ F by X :El. ~ Then

vi=

= (vi)*, vj=-vj* = (vj)*.

Hence there are a , 13 E R with

vi = a l ,

v j = pl

which is a contradiction. f) Let v : F -t G be an isomorphism of real C*-a1gebras.B~c), sft is a minimal element of Pr F\{O} . Then ~ ( x : is ~ a) minimal element of PrG\{O) . By Proposition 5.5.6.8, there is a E e 2 ( S ) such that ~ ( x ! = ~ )(.I<)<. Then V ( X ~ ~ FisXisomorphic ~,) to (.l<)
<

Proposition 5.5.7.11

(0)

We assume IK =C. Let T be a set. Put

H : = ~ ~ ( T )H, R : = H ~ R ~ .

5.5 Orthonormal Bases

Let E be a C*-subalgebra of C ( H ) containing K ( H ) such that zEz* = E for every unitary operator z on H and aEE==+iiEE Further put

and let F be a real C*-subalgebra of E for which there is an isomorphisnl of complex C *-algebras v : F + E such that

for every a E F . a)

If u is a conjugate involution on E , then there is a unitary operator x on H such that

for every a E E and

u is unique up to a factor of absolute value 1

b)

There zs a unitary operator x on H such that

F = { a E E I xiix* = a } , x* = Z

or x * = -Z,

and v ( ( a ,b)) = a + ib 0

for every ( a ,b) E F . x is unique up to a factor of absolute value 1 .

c ) a ( H I R )C HR for every a E G . For a E G, define

-

a : HR

Put

-

-+ H R ,

<-

a<.

G:={Z~~EG}.

5. Hilbert Spaces

d)

2. zs a C'-subalgebra

of L(HR) containing K(HR) such that zGz* = 2.

for every unitary operator z on HR . e) If E = L ( H ) (resp. E = K ( H ) ) then

G = L(HR)

(resp.

G = K(HR)).

f) If Z = x* then there is a unitary operator z on H such that z G * E

for every a E F and the map

-

F

a ~ z a z *

is an isomorphzsm of real C'-algebras. g) If x* = -Z then there is a unitary operator z on H such that

z F z W= F ( E ) (Proposition 5.5.7.10) and the map

F

+F ( E ) ,

a+ zaz'i

is an isomorphism of real C*-algebras. a ) The map E-+E,

a-iZ

is an isomorphism of complex C*-algebras. By Corollary 5.5.6.12, there is a unitary operator x on H , unique up to a factor of absolute value 1 , such that

for every a E E . By Proposition 5.5.7.6 b =. a,

b) By Proposition 2.3.1.43 b), there is a conjugate involutive u on E such that

By a), thcrc is a unitary operator x on H , unique up to a factor of absolute value 1, such that

5.5 Orthonoma1 Bases

and

~ = x *or z = - x * . By Proposition 5.5.7.6, the map

is an isomorphism of complex C*-algebras. It follows

v ( ( a ,b ) ) = a + ib

F

for all ( a ,b) E . C) For HR

<

so that a< E

HR

arid

a(HR) C HR .

d) For b E L ( H R ) , define

Take b

C ( H R ) . Then

IEL(H).

0

0

b=b,

0

b=b.

Hence if E E , then b E G . It is easy to see that G is a Ca-subalgebra of L ( H m ) . Since E K ( H ) whenever b E K ( H R ) , we deduce that K ( H m ) C G .

i

NOWtake b E

g . Then

i E G . Since

is a unitary operator on H ,

Since

we get that 0

A

zbz* E G , zbz* E

e,

zez* = G

5. Hilbert Spaces

2 76

e) is easy to see. f) By Proposition 5.5.7.8, there is a unitary operator z on H such that

and the map

is an isomorphism of real C*-algebras. For a E F ,

ZG* E G and the map

F-G.

-

a-zaz*

is an isomorphism of real C'-algebras. g) follows from Proposition 5.5.7.8

Corollary 5.5.7.12

(0)

Assume that IK = C . Let S, T be sets and let := e2(T), respectively, such that

x,y be unitary operators o n e 2 ( S ) and H

Further, let E be a C*-subalgebra of L ( H ) contaznzng K ( H ) such that

for every unitary operator z o n H and such that

Put

F := { a E E I yay' = a ) ,

G

:= { b E

K ( e 2 ( s ) 1) X ~ X *= b} .

If either S or T zs nonempty, then the real C *-algebras F and G (Proposition 5.5.7.6) and F and K ( e 2 ( S ) ) are not isomorphic. By Proposition 5.5.7.11 d),c),f),g), F is isomorphic to F ( E ) of Example 5.5.7.10 and G is isomorphic to K(Y2(S)n R").The assertion follows from Proposition 5.5.7.10 f ) .

5.5 Orthonormal Bases

277

Corollary 5.5.7.13 ( 0 ) W e adopt the hypotheses and notation of Proposition 5.5.7.11. Let CE be the class of real C *-algebras A such that the complex 0

C*-algebras A and E are isomorphic.

a ) If n := C a r d T is an odd natural number, then every element of & is isomorphic to R,,,, . b) If C a r d T is not an odd natural number, then & decomposes into precisely two zsomorphism classes: those of the first class are isomorphic to G while the elements of the second are isomorphic to F ( E ) (Proposition 5.5.7.10). If in addition E = L ( H ) (resp. E = K ( H ) ) then 5 = L(HN) (resp. = K(Hm)).

c

a & b . Let .F be the set of real C' -subalgebras B of E such that the map

is an isomorphisni of complex C * -algebras. By Proposition 2.3.1.43 c), every element of CE is isomorphic to an element of 3 . Tako B 3 . By Proposition 5.5.7.11 b), there is a unitary operator x on H such that

and

-

If x* = 3 ,then by Proposition 5.5.7.11 f), B is isomorphic to G . If x* = - 3 , then by Proposition 5.5.7.7 b), CardT is not an odd natural number and by Proposition 5.5.7.11 g), B is isomorphic to F ( E ) . Hence if n := C a r d T is an odd natural numbor, then by Proposition 5.5.7.1 1 e), each element of & is isomorphic: to b,, By Corollary 5.5.7.12, G and F ( E ) are not isomorphic. Hence if Card T is not an odd natural number, then CE decomposcs in precisely two isomorphism classes: the elements of first class are isomorphic to G while the elements of the second are isomorphic to F ( E ) . The last assertion of b) was proved in Proposition 5.5.7.11 e). Corollary 5.5.7.14 Let E be a complex C'-algebra for which there is a faithful irreducible representation (H, p) such that:

2 78

5. Halbert Spaces

Let further & be the class of real C * - algebras for which E is isomorphic to their complexifications. Put

F

-

for all F, G E & . Then

G :e F and G are isomorphic

-

is a n equzvalence relation o n & and Card &/- E { 1 , 2 ) .

Since the representation ( H ,cp) is faithful, we may identify E with cp(E) . By 1 ) and Proposition 5.5.6.9, K ( H ) c v ( E ) and the assertion follows from Corollary 5.5.7.13 a),b) .

Corollary 5.5.7.15

(0)

W e assume lK = C . Let T be a nonem.pty set,

n E Pi,and let & be the class of real C ' a l g e b r a s E such that E is isomorphic to K ( H ) " . W e set E

-

F

:-

E and F are isomorphic

for all E , F E & , E ( p , q ) := K ( H ) Px K ( H I R ) qx F ( K ( H ) ) " - ~ ~ (- ~2 +~9

5 n),

for all p,q E N U { 0 ) , and

If Card T is an odd natural number (is not a n odd natural number), then

5.5 Orthonormal Bases

and the map

is bijective. I n particular 1 Carci (el-) = - ( n 2

(card

+ 1 + cor2 "" 2 )

(el-) = -41 ( ( n + 2)'

- sin2

2

By Corollary 5.5.7.13 a), if C a r d T is an odd natural number, K ( H R ) is the only real C*-algebra that has complexification isomorphic to K ( H ) . By Corollary 5.5.7.13 b), if C a r d T is not an odd natural number then there are exactly two non-isomorphic real C*-algebras, namely K(HlR) and F(IC(H)), the complexification of which are isomorphic: with K ( H ) . Now we use Proposition 4.3.5.8. If Card T is an odd natural number then

and

1 Card (El-) = - ( n 2

+ 1 + cos2 -)nn2

If C a r d T is not an odd natural number then

and Card (t/-)= 1

n-l (n k)n + 51 x (n- X + 1 + cos2 )= -

k=O

1 = - ((n

4

+ 2 ) ~ sin2 "" 2 ) -

,

Example 5.5.7.16 W e use the notation and conventions of Proposition 5.5.7.10. Let C denote the set of all

(f,9,(P, $) E

myXTx IR:XT x IRTXTx R T x T

5. Hilbert Spaces

280

such that the family

is summable and for each ( f ,g, cp, q!~)E C let z( f , g, cp, 111) denote the sum of the above famzly. a)

If ( f , g , c p , $ ) E C , q , r , s , t ~ T and , O E R ,then

+

2 ( j 2 ( s .t ) s i n 2 ( l ( 0 y ( s , t ) )+ g2(s.t ) sin2 -(-0 2 2 X

Hence z ( f , g, cp, I,!J)= 0 implies that f b)

=g =0

+ d ( s .t ) ) )xi.

=

.

Take f , g, h, k E KtTXT such that the family

( f ( s ,t)x;t + g ( s , t)x:t + h ( s , t)y,Ot+ k ( s ,t ) ~ : t ) , , ~ ~ ~ zs summable. b l ) If the svm is 0 then: f=g=h=k=O. bS) The sum zs selfadjoint iff

h ( s ,t ) = - h ( t , s ) ,

k ( s ,t ) = - k ( t , s )

for all s , t E T . c ) If E = K ( H ) , then: cl)

F is the closed real vector subspace of E generate by {xrt I

8,t

E T ,cp E

IR) u {Y:

I s, t E T ,

E

w

and F =

{ ~ ( fg,, cp, $1 I ( f , s,(P,q ~E) z).

5.5 Orthononnal Bases

c2) F is simple and purely real.

( f , g, cp, $1

~ 3 )If

-

E C then z ( f ,g, cp, $) is selfadjoint i f f

f ( s ,t ) # 0 * cp(s,t ) + cp(t,5 )

( s , t ) = f ( t ,s ) ,

f

281

g ( s , t ) = g ( t , s ) , g(s, t ) # 0

0

* cp(s,t ) - ~ ( ts ), .= 2

(mod 4 ) , (mod 4 )

for all s, t E T . d ) If T is finite, then Dim F = (2Card T)'

Cxi

and

, Dim Re F

=

2(Card T)' - Card T

is a unit of L ( H ) .

LET

a) Put z := z ( f , g , cp,+). B y Proposition 5.5.7.10 b),

8+v(s,t) -@+tl(s,t) ~ : , ~ z ~ t )( ,~ =t)xqr 7 f + g ( ~t)yqT ,

=-f

y:,z?/;,

8-lp(s,t)

( s ,t)xpt-

x:~zYP' = f ( s ,t)yqr-8-v(s,t)

-

-

g ( s , t )y, B ;Q(", L )

g ( s ,t ) ~ : ; ~ ( ~ ' ~ )

= f ( 3 , t)y,rQ+ds,O - g ( s , t)x,;B+Q("")

Y:.?~X;~

8 .

Y:~zYP,

x ~ . +~ ~ x =~ ~ =

0

2f ( s , t ) sin 5 ( 0

Y : ~ Z " : ~ - X:~ZY:,

+ cp(s,t ) ) x &+ 2g(s,t ) sin :(-0 + $ ( s , t))$, =

= -2g(s, t ) sin :(-0

2 ( f 2 ( s :t ) sin2 5 ( 0 = f ( s ,t ) sin

+ $ ( s , t))x:, + 2f ( s ,t ) sin :(O + p(s, t ) ) y i ,

+ p ( s , t ) )+ g2(s,t ) sin2 :(-0 + $ ( s , t ) ) ) ~ : =,

;(e + 4%~ ) ) ( X : , ~ Z X : , + Y : ~ Z Y ~ ) , ) -

- g ( s ,t ) sin ;(-0 +*(st t ) ) ( ~ : ~ z-x ~,8,~2Y:,). :~

Assume that z = 0 . Taking R such that K

sin - ( R 2 we see that

+ p ( s , t ) )# 0 ,

sin ?(-R 2

+ + ( s ,t ) ) # 0 ,

282

5. Hilbert Spaces

f ( s ,t ) = g(s, 1 ) = 0 . Since s and t are arbitrary, f = g = 0 . b l ) For s, t E I R , define

Hence there are cp, $ E IRTxT such that

i

f = Fcos;cp

h = Gcos%$

g = Fsin%cp,

k=Gsin;$.

By Proposition 5.5.7.10 b ) , ~ ( st ), ~ ~ =' ~ F ("r ,t ) (cos m

2

x

:

+ sin 2

= h ( s , t)ysot+ k ( s ,t ) ) ~ :. ,

Hence ( F ,G , cp, $) E C and z ( F , G , cp,$)

=0.

By a ) ,

F=G=O. Hence

b2) By Proposition 5.5.7.10 b ) ,

C ( f ( s ,t)x9t + g ( s , t)x:, + h ( s ,t1y:t + k ( s ,t)y:t)* = = C ( f ( s ,t)xp, - g(s, t)x:, - h ( s , t ) ~ : ,- k ( s , t ) ~ : , )= s,tEl'

=

C ( f ( t 7s)xPt - g ( t , s)xjt - h ( t ,~ ) y : t- k ( s 7t1y.L) .

s,lET

By b l ) ,the sum is sclfadjoint i f f

5.5 Orthononnal Bases

for all s , t E T . cl) Let G be the closed real vector subspace of E generated by {xrt ~ s , ~ E T , ~ ~ E R ) u { Y Z I ~ , ~ E T , ~ ~ E I R )

Then { z ( f , 9,cp, $1

I (f,g , P. *)

E

El C G C F

(Proposition 5.5.7.10 b)). Take z E F . By the last assertion of Theorem 5.5.5.1,

Hence by Propositiori 5.3.2.13 c),d),

Thus ( z t l J s l )= (ztl'ls")

,

(ztrlls") = (ztllsl)

(zt1(sI1)= - ( ~ t ~ ~ l s, ~ ()z t r l J s l )= -(ztllsl')

for all s, t E T , so that

Given s , t E T , define ( f , g , cp, $) E

x 1~;~''

*TxT

IRTXT'

by

284

5. Hilbert Spaces

f ( s , t ) e ' : ~ ( := ~ ~(ztfls') ~) g ( s , t)e1T*(3J):= ( z t fIsff). Then

( f , g, cp, @) E C and

Hence

c2) By Corollary 5.5.1.13 b ) , E is simple so that the assertion follows from Proposition 5.5.7.10 a). c3) By Proposition 5.5.7.10 b),

-

g ( t , r) (cos % ( t , s ) d 2

K

+ sin -+(t. 2

s ) ~ . : , ).

By b l ) and Proposition 5.5.7.10 b), z( f , g , cp, @) is selfadjoint iff for all s , t E T ,

f ( s , t ) cos ;cp(s, t ) = f ( t , s ) cos ; p ( t , s )

f ( s ,t ) sin :cp(s, t ) = -f ( t , s ) sin ;cp(t, s ) g ( s , t ) cos ;dJ(s, t ) = - g ( t , s ) cos ;dJ(t, S ) g ( s , t ) sin :@(s,t ) = - g ( t , s ) sin ;dJ(t, 3) ,

+

f ( s , t ) 2 = ( f ( s ,t ) cos :cp(s, t ) ) 2 ( f ( s ,t ) sin ; p ( s , t ) ) 2=

+

= ( f ( t ,S ) cos ; v ( t , s ) ) ~ ( f ( t , S ) sin :cp(t. s))' = f ( t , s ) ~ ,

g ( s , t)"

( g ( s ,t ) cos ;$(s, t ) ) 2+ ( g ( s ,t ) sin ; $ ( s , t ) ) 2=

= g ( t , s ) cos :@(t,s))'

+ ( g ( t ,s ) sir1 ;$(t, s ) ) =~ g ( t , s

) ,~

5.5 Orthonormal Bases

i.e.

f

( s ,t ) = f ( t ,s) ,

f

(s, t ) # 0 ===+ p ( s , t )

g(.%t ) = g ( t , s ) , g(s, t ) # 0

+~

-

0

(mod 4) ,

**(s, t ) - *(t,s) = 2

(mod 4) .

( ts ),

d ) By Proposition 5.5.7.10 a ) , Dim F = 4(Card T)' . P u t n := Card T . By b 2 ) ,

By c l ) and Proposition 5.5.7.10 a),b),

C XI: LET

is a unit o f C ( H ) .

5. Hzlbert Spaces

286

5.6 Hilbert right C*-Modules Hilbert C* -modules are generalizations of Hilbert spaces in the sense that the scalar product takes its values in a C*-algebra rather than in the ground field. In this section, E is always a C*-algebra. 5.6.1 Some General Results

Definition 5.6.1.1 ( 0 ) A weak semi-inner-product right E-module is a vector space H endowed with a bzlinear map

H xE

-+

H,

(<, x ) c,(x

(right multiplication)

such that

(e.1~ = F(xY) for all

<E H

and x, y E E and wzth a sesquihnear m a p

H x H

+ E , (<, q) * (<1q)

(inner-product)

such that

for all < , q E H and x E E . H is called a semi-inner-product product) right E-module if i n uddition

(and ((I<) = O for all (,

E

==+ < = 0)

H , where E denotes the complexzfication of E .

(inner-

5.6 Hzlbert right C*-Modules

287

The above notions were introduced by Kaplansky (1953) for E unital, complex, and commutative and by Paschke (1973) and Rieffel (1974), for the general case. If E is not unital and E is the unital C*-algebra associated to E then, by setting f 1 = f for every E H , H becomes a weak semi-inner-product right E-module. If E is unital then under a very weak hypothesis, which will be always fulfilled in this book the axiom

<

is automatically fulfilled (Proposition 5.6.1.2 f)). Let E be a complex C'-algebra. If Eo denotes the underlying real C*algebra of E (Theorem 4.1.1.8 a)) then restricting the scalars of H to R we get a weak semi-inner-product right Eo-rnodlile. The axiom

is exactly the assertion that the above sesquilinear map is Hermitian. Proposition 5.6.1.2 ( 0 ) Let H be a weak semi-inner-product right E-module. Then for all x , y E E and f , 7, ( E H :

f) The map

is a semz norm. If it is a norm then for every mate unit 5 of E ,

<E H

and every approxi-

5. Hilbert Spaces

288

g) If K = C then H is a semz-inner-product right E -module. h) For e v e y of E .

< E H , {x E El<x = [x* = 0 )

i) {x E El< E H j)

+ [x

is a hereditary C'--subalgebra

= 0 ) is a closed ideal of

E.

The vector subspace of E generated by {((Iq) IC, 0 E H ) is an ideal of E.

= x'(SI0x

+ 2re(~*(Flv)x)+ Y*(V\V)Y.

It follows (F + VIE

+ 0) I (F + v1F + 0) + (F - vIF - 71) = 2((FIF) + (017))) .

c) By b) and Corollary 4.2.2.3, 0I ((5 - v1Fx - 0) = xl(
Ill(~l~)l l 2re((~l71)~) ~*~ + (~10) If II(
:= (q(<),we scc

=

-(
that

+ (~llrl),

5.6 Hilbert right C ' M o d u l e s

289

d ) follows from c) and Corollary 4.2.1.18. e) follows from b). f) By b) and d),

= X*(
(
for every x E E so that lim((x

-

2,3

(I(x - <) = 0

(Proposition 2.3.4.9), lim(x = ( . z,3

g) Take

[,TI

E H . By b) and Proposition 4.2.2.15 b),

(([I<)

+ (vlv)

= ((<

?

(vl<) - ((17)) =

+ ivI< + iv), O) E +;

.

h) Put

G := {x E El(x

= <x* = 0 ) .

I t is obvious that G is ax1 involutivc subalgebra of E . By e), it is closed, i.e. it is a C*--subalgebraof E . Take x E E and y, z E G . Then

5. Halbert Spaces

290

so that yxz E G . By Proposition 4.3.4.6 b subalgebra of E . i) follows from e) . j) follows from a).

3

a , G is a hereditary C*-

Proposition 5.6.1.3 Every weak semi-inner-product right M-module (Example 4.1.1.31) is a semi -znner-product right M -module. Let H be a weak semi-inner-product right M-module and let <,7 E H . Put

Then

(Proposition 5.6.1.2 d ) ) . We identify M with a unital real involutive subalgebra of C2,2as in Example 2.3.1.46. Then

By the above,

=

(m-

2 0.

0

C2.2

and M with

5.6 Hilbert right C*-Modules

By Example 4.2.3.5,

Definition 5.6.1.4 ( 0 ) A Hilbert right E-module is a n inner-product right E -module for which the n o r m

(Proposition 5.6.1.2f ) ) is complete. If E is not speczfied, then we simply speak of a Hilbert right C*-module. A (unital) Hilbert E-module is a Hilbert right- E-module H endowed with a bilinear map E x H --+ H ,

(x,I) c,XI

(left multiplication)

such that

for all x,y E E and

I,17 E H

By Corollary 4.2.1.18,

so that by Proposition 5.6.1.2 e), a Hilbert E-module is an E-module with respect to its right and left multiplications.

292

5. Hzlbert Spaces

Let E be a complex C*-algebra, F its underlying real C*-algebra (Theorem 4.1.1.8 a)), and H a Hilbert right F-module. If E is unital, then H endowed with the scalar multiplication

becomes a complex vector space and so a Hilbert right E-module. But if E is not unital the above extension of the structure of H is not always possible. Take

Then H endowed with the right multiplication

H x F ---,H ,

( ( 2 ,a ) ,Y

) -I+

(XY

+ cry, 0)

and with the inner product

is a Hilbert right F-module. Assume it is possible to extend the structure of H to a Hilbert right E -module and put ( x ,a ) := i ( 0 , l ). Then ( 0 ,-1) = i ( z ( 0 , l ) )= z(x,a ) = ( i x ,0 ) + ( a x ,a') and we get the contradiction

Example 5.6.1.5 ( 0 ) Let F be a nght zdeal (closed ideal) of E . Then F endowed with the maps

-

FxE-+F,

(<,z)++<x,

F x F -4 E ,

(I,7l)

ExF+F,

(x,O++x<,

(and with the maps

F+F,

v*<

<++<*,

(Corollary 4.2.6.2)) is an inner-product right E-module (and an involutive unital E-module).

293

5.6 Halbert right C' -Modules

The first and the third map are bilinear, the second one is sesquilinear, and the last one is an involution such that F is an involr~tiveE-module whenever F is a closed ideal of E . Take <,7 E F and x E E . Then

(x
5

11x112<*<= 11~11~(<1<)

(Corollary 4.2.2.3)).

Proposition 5.6.1.6

(0) 0

Assume IK = R . Let H be a (semi ) inner0

product right E-module and E , H the complexifications of E and of the underlying real vector space of H , respectively. Then H endowed wzth the maps

0

is a (semi-) inner product right E-module called the complezification of H . 0

If H is a Hilbert right E-module then H is a Hilbert nght E-module such that s ~ P { I I < I1I1~71?12} 5 11(<>7)112 = ll((
for all ( 5 , ~ 6 ) H and the canonical map H x H real Banach spaces.

+H

5

(ll
is an isomorphism of

5. Hilbert Spaces

294

By Lemma 2.1.5.1 a),b), the map

is bilinear and

for all (<, v) E

fi

and (x, y), ( u ,v ) E

For all (<, v), (<, A) E

fi

i

0

Take (<, A ) E H . Then (.I<) and -(.IA) are linear. By Lemma 2.1.5.3, the map

is linear. It follows, by the above result, that the map ((C, A)I.) is conjugate linear. Hence the map

is sesquilinear. We have

for all <, 7 E H and x E E . By Lemma 2.1.5.1 b),

h.

is a and (x, y) E By Proposition 5.6.1.2 g), for all (5, v), (C, A) E (semi-) inner-product Hilbert right E-module. Now suppose that H is a Hilbert right E-module. Then, by the above relation,

5.6 Hilbert right C'-Modules

(Proposition 5.6.1.2 d)). By Proposition 4.3.6.1 c) , s ~ P { I I < Illvl12} I ~ ~ 5 Il(cl0 + (vlv)ll 5 ll(<7v)l12 0

It follows that the canonical map H x H -t H is an isomorphism of real Banach 0 W spaces and H is a Hilbert right E-module. Definition 5.6.1.7 ( 0 ) Let G, H be Hilbert right E modules. W e define LE(G, H ) t o be the set of maps u : G + H for which there zs a map u* : H + G such that (u
G and H are called isomorphic if there is a bzjective u E LE(G, H ) such that

for all (, 7 E G

Let F be a C*-subalgebra of E such that

for all <,7 E G and <,q E H . Then G, H are Hilbert right F-modules and

Proposition 5.6.1.8

(0)

Let F, G , H be Hilbert right E-modules.

a ) If u E LE(G, H ) , then there is a unique map u* : H

+G

such that

for all (<, 7) E G x H . u* is called the adjoint of u . It belongs t o LE(H, G ) and u** = u .

5. Hilbert Spaces

296

b ) LE(G,H ) zs a closed vector subspace of L(G,H ) and llull = IIu'II for every u E LE(G,H ) . c ) The map LE(G,H ) + C E ( H G , ) , u w U* is conjugate linear

g) Assume that F and G are (unital) Hilbert E-modules and for x E E

and u E L E ( F ,G ) put xu : F

+G ,

ux : F

+G ,

< H x(u<) < * u(x<)

Then xu, ux E L E ( F ,G ) and ( x u ) *= u*x*,

(ux)*= xfu*

for every ( x ,u ) E E x L E ( F ,G ) , and L E ( F ,G ) endowed with the maps

{

E x L E ( F ,G ) + L E ( F ,G ) , ( x ,U ) eX U L E ( F ,G ) x E

+ L E ( F ,G ) ,

( u ,X )

ct

ux

is a (unital) E-module. In particular L E ( F ) is an involutive (unital) E-algebra with respect to the involution L E ( F )+ C E ( F ) , u HU* . h ) Assume that H is a Hzlbert (right) E-module and put

( Z :H

--+H ,

[c--t<x)

for every x E E . Then

for every x E E (z E E C ) and the map E (resp. E c )

+ L E ( H ) , x e%

is an involutive algebra homomorphism (c) and e)).

5.6 Hilbert right C*-Modules

a) Take v : H

+G

with (utlo) = (FIvv)

for all ( t , q) E G x H . Then for all q E H , 11u'q

SO

-

vq11' = (u'q

-

vq1u'q

-

vq) =

u* = 21. For all (E, q) E G x H ,

Hence u* E L E ( H ,G) and u** = u . b) Take U E L ~ ( G , H )a , , P ~ K , t , q ~ G , a < n ~d H . T h e n

at + 1817)IC) = ( a t + PqIu'C) = 4uEIC)

Since

<

= a(EIu*C) + P(vb*C) =

+ P(.LL~IC)b u t + PuqIC) .-I

is arbitrary, we get u(aC + P7) = o u t + P w ,

i.e. u is linear. For q E H # , put

v, is linear and contiriuous and

for all F E G (Proposition 5.6.1.2 d)). By the Theorem of Banach-Steinhaus, a := sup ((v,J(< cm VEII#

5. Hilbert Spaces

298

ll(u<117)11= Ilvtl
IIuFII 5 aI1tII for all

< E G , and so u E L(G,H ) . From r

I I ( F I U * V ) I I= I I ( U ~ I V ) 1 I I I I U ~ III IV I I

II
IIUII

for all (<,7) G x H (Proposition 5.6.1.2 d ) ) ,we get

for all 17 E H , so that JJu8(l 1 llull. By a), IIu'II = llull. L It is easy to see that E ( G ,H ) is a vector subspace of L(G,H ) . Take u E L(G,H ) and let (un)nEn.be a sequence in LE(G,H ) converging to u . Then ( u ; ) , ~is~a Cauchy sequence in L ( H ,G ) , so it converges to a v E L ( H ,G ) . We get

for all (<,q) E G x H (Proposition 5.6.1.2 d)). Hence u E L E ( G ,H ) and L E ( G ,H ) is a closed vector subspace of L(G,H ) . c) Take a , /3 E M and u , v E LE(G,H ) . Then for all (<, 7 ) E G x H ,

so that

d) For all

EH

,

5.6 Hilbert right C*-Modules

.u(<x)= (u<)x. e) For all (<, 7) E F x H , ((v o ~ ) < I v= ) (v(uO1v) = (utlvfv) = (
0

u*

.

f ) We have llu
cEG

l l ~ * ~ lltl12? ~ll

(Proposition 5.6.1.2 d)), so llu1I2 I IIu'ouII .

Since

IIu' o uII

I IIu*II IIull r

we get by b), Ilu1l2 = lIu*ouIl . It follows by a) and b), lluou*ll = IIu**ou*II= IIu*Il2= Ilu112 g) For

( F , 77)

E

F x G,

( ( x ~ ) E I= v )( X ( U O I=~ (U
E

(ux)* = xtu* .

E and u E LE(F,G) . Then for any

< € F,

((xY)u)<= (xY)(u<)= x(Y(u<))= X((YU)<) = (X(YU))<,

299

300

5. Halbert Spaces

((lu)< = l(uF) = u t ,

( u l ) < = u(1<) = 4

,

so that (XY)U= ~ ( Y u ) ,~ ( X Y=) (ux)y,

(xu)y = ~ ( u Y ) ,

Since the above maps are bilinear, t E ( F , G) is a (unital) E -module Similarly it can be proved that

for all x E E and u, v E L s ( F ) h) For < , v E H ,

((:
= (<xIv) = (t1v)x = x(t1v) = (
so that

-

I E C E ( H ) , ? = x' Take x , y E E (resp. EC). For every

.

< EH,

Gj<= (xy)< = x(y<) = z(g<) = %g<

(G<= E(xY) = <(Yx)= ( E Y )=~ I(?<) = z g < ) , so that

5.6 Hilbert right C*-Modules

Hence the map

E (resp. EC)

+ L E ( H ),

x

H

Z

is an algebra homomorphism. By the above, this map is involutive.

Remark. The inclusion map co + 5.6.1.5). Proposition 5.6.1.9 0

0

(. 0 )

em does not belong to Lem(co,em) (Example

Assume IK

,

=

R . Let G , H be Hilbert right

E-modules, G , H the complexification of G and H , respectively (Proposition 5.6.1.6) and ;(G, H ) , ;(H, G ) the complezification of the underlying real vec0 tor spaces L ( G ,H ) and L ( H , G ) , respectively. Given ( u ,v ) E L ( G , H ) (resp. ( u ,V ) E & H , G )), define

-

0

(resp. ( u ,v ) : H +

6,

(<, 7 ) c--f (u<- ~ 7217: + ~

Let p be the isomorphism of complex vector spaces c~ : ;(G, H )

---t

~ ( 2i f ,) , ( u ,V )

--

0. )

(21,~)

(Proposition 2.1.5.6). Then

and

-.

(11,

~ ~ i(f ) 8 .

-

v ( i E ( ~H. ) ) =

v ) = (u*,-v*)

for all ( u ,v ) t ;,(G, H ) (hence ~ ~ h ( )2may , be identified with the complexification of L E ( G ,H ) ) . Take ( u ,v ) E i ( G ,H ) 0

Step 1

( u ,v ) E L E ( G ,H ) +

-

(z) ~~ ( 2 , t

-*

if),

( u ,V ) = ( u f ,-v*) Take (<, 7) E

6 and

(C, A)

E

i f . We have

5. Hilbert Spaces

302

= ((f, v)I(u*C

-

Hence ( u , u ) t

L;(;, i ( )

step 2

Put

For

(<,v) E

Gx H.

+ u*X,u*X - u * ( ) ) =

and

(G)~ ~ i()* 2 (.u , u ) t J?E(G, H ) )E

5.6 Hilbert right C*-Modules

303

Definition 5.6.1.10 ( 0 ) A (unital) E X * - a l g e b r a is a (unital) involutive E algebra which is at the same time a C*-algebra. A n isomorphism o f E C'-algebras is an isomorphism of C'algebras which is at the same time a71 isometry of involutive E-modules. Theorem 5.6.1.11

(0)

Let H be a Hilbert right E-module.

a ) L E ( H ) endowed with the Banach algebra structure induced by L ( H ) and with the involution

(Proposition 5.6.1.8 a), b)) is a C*-algebra. b) Let IK = C and u E L E ( H ) (let

for every

IK = R and u E R ~ L E ( H ) )If.

< E H , then u E Re L E ( H )

(U

=0)

c ) If IK = C ( I K = IR) then the following are equivalent for every L E ( H ) ( u E R ~ L E ( H ): )

u E

If these conditions are fulfilled then

and (u
for every

ll~ll(
<E H .

d ) If H is a Hilbert E-module then L E ( H ) is an E-C*-algebra (Proposition 5.6.1.8 g)). e ) The following are equivalent for all p, q E Pr L E ( H ) el) p < q .

e2) I m p C I m q .

304

5. Hzlbert Spaces

f)

The followzng are equivalent for every p E L E ( H ) with pZ = p .

g)

The following are equivalent for all p, q E Pr L E ( H ) :

a ) In the complex case, the assertion follows from Proposition 5.6.1.8 a),b),c),e),f). In the real case, it follows from Proposition 5.6.1.9 that the com0

plexification of ,Cti(H) is the C*-algebra L & ( H ) , so L E ( H ) is a real C*algebra. b) The map

is sesquilinear. By Proposition 2.3.3.8 b

+a

(by Proposition 2.3.3.7 a

which implies

for every cl c2

< = H , i.e. u = 0 ) .

+ c p . (u
c 1 . If IK = C then

By the Polarization Identity (Proposition 2.3.3.2 d)),

+ b),

5.6 Halbert right C*-Modules

t, q E H , so that u = u* , i.e. u E Re C E ( H ) . It follows (for IK E {IR,C) ),

for all

( a ) and Theorem 4.2.2.9 a)), (u-[It)

I (u+tlt).

Hence ( ( ~ - ) ~ t l= t ) ( ( ~ - ) ~ r l u - tI) (ufu-tIu-<) = 0

BY cl

=+ c2,

((u-)3t10 = 0 so that by b),

Now assume that u is positive. Then

= sup{(luf
From c2) and u

H#} =

IIU?II'=

llull.

I I(u)(l(Corollary 4.2.1.17 a + b) it follows II~II( ~(utlr) I ~ ) = ( ( I I ~-I uI )~t ~ < E ) E+,

d ) follows frorn a ) and Proposition 5.6.1.8 g) el =+ e2. For every E H ,

<

305

5. Halbert Spaces

306

PE (Corollary 4.2.7.6 a

= qpE

+ b), so that imp^ Imq.

e2

+ el.

For every

< E H, pt E Imp C Imq

so that

Hence

(Corollary 1.2.7.6 d + a ) . f, + f2. We have

f2

+ fl.

For all

< , vE

H,

so that

Hence p = p* and p E Pr C E ( H ) . g, +g2. Since

5.6 Halbert right C*-Modules

92 3

61. For every

< E H,

b q f l p q f )= (qllp'pql) = (qflpqf)= 0

so that

Hence

( 0 ) Let G ,H be Hilbert right E-modules. Then for every u E LE(G,H) and v E L E ( H ) +,

Corollary 5.6.1.12

In particular,

(uFIuO 5 ll~112(f10 for every

<E G.

For every f E G ,

so that

by Theorem 5.6.1.11, c2

3

c , (and Proposition 5.6.1.8 a),e)). It follows

(Proposition 3.6.1.8 e)). By Theorem 5.6.1.11 c) and Proposition 5.6.1.8 f),

Proposition 5.6.1.13 Let H be a Hilbert right E-module. For every x E E , define

a)

The following are equivalent for all x,y E E :

5. Hilbert Spaces

308

all

J,v E H =+ (t17)x = ~(t117)

-

a?) Z E L E ( H ) , z' = y * . b ) The followzng are equivalent for every x E E :

a) For all J , 77 E H ,

(ZJIv) = ( t x l v ) = (J1v)x

1

(Proposition 5.6.1.2 a ) ) .al a a2 follows. b ) follows from a ) .

Example 5.6.1.14 Let I be a finite set, a E (E\{O})', and b E ( E # ) ' . Then E' endowed wzth the right and left multiplications

E x E' -+E' ,

( x , J )++(x<,),cI,

and with the inner-product

is a Hilbert E -module. First we remark that

for every J , 17 E E1 . By Corollary 4.2.2.3, a:b:b,a, for every

< E E'

and

L

I ala, ,

J:a:b:b,a,E,

< <:ala,J,

t I . Put

cL := (a:aL- a:b:bLaL)

5.6 Hilbert right C*-Modules

for every L E I . By the above, for [ E E l ,

Take <, E E l . Then

( ( F I E ) + ( v l v ) , (710 - ((17)) =

We have

and for

L

E

I, O5

(CLCL?

cL~]L)*

=

(cL
(E;C;S~

cLvL)

= (sL*~L

-v:cL)

(cLcL?

cLvL)

+ 17:cTv~,<:cTa- v:cTFL) .

Hence

((
+ ( V I ~ , ) (VIO ,

The other axioms are easy t o verify.

-

((171)) E

i+.

=

5. Hilbert Spaces

310

5.6.2 Self-duality Proposition 5.6.2.1 every E H put

<

(8 )

Let H be a Hilbert right E-module and for

< E H * (.I<) E L E ( H ,E), (.I<)* = F,II(.I<)II

a)

IItII

( E z a m ~ l e5.6.1.5).

b) If E is unital then the map

is conjugate linear and an isometry of real Banach spaces with inverse m aP

If H is a Hilbert E-module then for every x E E and ( E H ,

C)

x((.IO)

=

(.lCx*)

1

((.l<))x = (.lx*<)

(Proposition 5.6.1.8 g), Ezample 5.6.1.5). a) By Proposition 5.6.1.2

d),

ll((.l<))vll = 1l(v1C)11 5 lltll llall for cvcry 9 E H and therefore

(.I<)

E C ( H ,E) and

II(.IOII 5 llrll. From

ll(.lOIl IICII 2 ll((.l<))
ll(~l<)ll2 ll
3

ll(.l<)ll = ll
For (9,x) E H x E , (((.1<))171.)

=

((17101.)

= .*(VIE)

=

5.6 Hilbert right C*-Modules

= (v1Fx) = (v1Fx)

(Example 5.6.1.5, Proposition 5.6.1.2 a)) so that

(.I<)

E CE(H, E ) 1

b) For u E CE(H, E) and

(.IS)* =

F.

<E H,

(
The assertion now follows from a ) c) By Proposition 5.6.1.2 a ) ,

(((.lS))x)v = (xvlF) = (vIx.0 for every 17 E H so that x((.lE)) = (.I<%*):

((.IS)). = (.lx*E).

Remark. If E is not unital then b) may fail. Indeed, if H := E := co then CE(H, E) is isomorphic to em.

Definition 5.6.2.2

&(G, H )

(8)

-

For G , H znnner-product right E modules define

:= {U E L(G, H)I(<,X) E

GxE

u(<x) = ( u E ) ~ ) ,

i? := . Z E ( ~E, ) . H is called self-dual (Paschke, 1973) if I(.I<)IE E H I = fi (Proposition 5.6.2.1) (this zmplies that H is complete). By Proposition 5.6.1.8 d), .CE(G,H ) C .C^E(G,H) . If F, G , H are Hilbert right E-modules then u E &(F, G) arid v E &(G, H ) implies vou E E

E ( ~H ,) .

By the Theorem of Frgchet-Riesz, every Hilbert space is self-dual.

5. Hilbert Spaces

312

Proposition 5.6.2.3 ( 8 ) Let F be a closed rzght ideal of E . The innerproduct right E -module associated to F (Example 5.6.1.5) is self-dual iff there i s a p E P r E such that F = p E .

First assume F is self--dual.Denote by u the inclusion map F there is a p E F such that u=

+ E . Then

(. I P)

W e deduce successively p = u p = @Ip) = p 8 p € R e E ,

p~ P r E , F =u(F)=pE. Now assume there is a p E Pr E with F = p E . Take u E y

:= p(up)* E

Z(F,E ) and put

pE = F

Then for every x E F , (xly)= ( ~ I P ( u P ) * )

=

(UP)PX

= ~ ( P x=)

so that u = (.IY). Hence F is self-dual. Proposition 5.6.2.4 self-dual then

(8)

Let G , H be Hilbert right E-modules. If G is

Let u ~ z ~ ( G , ~ ) . Tqa~ kHe. T h e n f o r a l (l t , x ) ~ G x E ,

5.6 Hilbert right C*-Modules

Since G is self-dual, there is an element vq E G such that

It follows

<

for every E G , i.e. u E L E ( G ,H ) . Hence J ? ~ ( GH, ) C L E ( G ,H ) . The reverse inclusion follows from Proposition 5.6.1.8 d). Remark. If G is not self-dual then the above equality may fail. Indeed, the (cO,em) but not to Lc- (co,em) . inclusion map co 3 em belongs to Proposition 5.6.2.5 ( 8 ) Let F be a C*-subalgebra of E and cp : E + F an involutive continuous lznear map such that:

a ) If H is a Hilbert rzght E-module, then H endowed with the right multiplication

and with the inner product

-

is a Hilbert right F-module, which will be denoted i n the sequel by H . The norms of H and fi are equivalent so that fi is reflexive iff H is reflexive (Proposition 1.3.8.8). b) If G and H are Hilbert right E-modules then

cE(c, H) = .cF(G,H ) n &(G, H ) and the adjoint defined in LE(G,H ) coincides with the adjoint defined in

L,(C, H ) .

5. Hilbert Spaces

314

c ) L E ( H ) is a unital C'-subalgebra of d ) If

H

LF(H)

is self-dual then H is also self-dual.

e ) If E is the real C'-algebra C or IH (Example 4.1.1.31) and F := R then

has the above properties 1-4. T h e same holds for the m a p

f ) If E is the real C'-algebra C or M then every Hilbert right E-module H is self-dual and H and fi have the same norm. g)

Let G be a C ' a l g e b r a and n E

IN . If E 2s the C*-direct product of the

family ( G ) ] E N " and

then

&,asthe above properties 1 - 4 , where we identified F and G i n a nati~ral way. h) Let (E,),,I be a finite family of C*-algebras and for each L E I let F, be a C'-subalgebra of E L and ip, : EL-t F, a n involutive continuous linear map with the above propertzes 1 ) 4 ) . If E and F are the C* -direct , and i f product of the famzlies ( E l ) r E Iand ( F L ) l E Irespectively,

then 1)

-

ip

is a n involutive continuous linear map with the above properties

4).

i) If G , H are Hilbert right E-modules and u E L E ( G , H ) then the norm of u i n L E ( G ,H ) coincides with the n o r m of u i n cF(2;, H). j)

For every x E E put

Then

(1.11,

is a norm o n E equivalent to its initial norm.

5.6 Halbert right C*-Modules

a ) is an easy verification. b) Take u E c F ( G , H) n Z E ( ~H, ) . Then for (<, 7, x) E G x H x E ,

If we put x := ((u
then by 4), ( ( u ~ I v-) ( < I u * ~ ) ) ( ( u
The inverse incl~isionfollows from Proposition 5.6.1.8 d). Now take u E LE(G, H ) and (<, 7) E G x H . Then

so that

Hence the adjoint of u in LE(G,H ) and L ~ ( GH, ) coincide. c) follows frorri a) and b) (and Theorem 5.6.1.11 d ) , Corollary 4.1.1.21). d ) Take u E Z(H, E) . Then cpou E L(H, F) and for (<, y) E H x F ,

Since

fi

is self-dual, there is an q E H such that

31 6

For

5. Hdbert Spaces

(t,x ) E H

x E

, ( ~ ( ( u-t ( t 1 q ) ) x ) = v ( ( u t ) x - ( t l d x ) =

Putting

it follows by 4 ) ,

Hence H is self-dual. e) Only 3) needs a proof. First assume E = C . Take ( x ,y ) E & + . Let x 1 , x 2 ,y l , y2 E

IR

with

By Proposition 4.2.2.24,

Hence

so that ( P X , ' P Y )= ( 2 1 Y, I ) E a + .

Kow assume E = M . We identify M with a real subalgebra of C2,2 and M 0

with C2,2 as in Example 2.3.1.46. Takc (x, y ) E M+ and put

5.6 Hilbert right C*-Modules

Then x E M + = IR+ (Proposition 4.2.2.13 a)) and

By Example 4.2.3.5,

Hence

in the case F = IR and

in the case F = C . f) Let H be a Hilbert right E-module. By e) and a), space and so it is self-dual. By d ) , H is self-dual.

I?

is a real Hilbert

g) 1) and 2) are trivial. 4) follows from Corollary 4.2.1.18. In order to prove 3) we may assume IK = C . Take (x, y) E E+ . By Proposition 4.2.2.24, 0

x, f iy, E G + for every j E IN,. By Proposition 4.2.2.24, again, (x,, y,) E G+ for every j E IN, so that

h) is easy to see. , ). i) Denote by 11ull (by I I I u I I I ) the norm of u i n LE(G, H ) (in L ~ ( Gfi) By c) and Proposition 5.6.1.8 f),

j) )I.)), is obviously a norm on E with

5. Hilbert Spaces

91 8

ll.llG

i llvll 11. 1 1

Takr x E E\{O} and put

Then y E E# and so

where

a := inf{llpzII I z E E+,llzll = 1 ) > 0 .

¤

Hence Il.llG and 11.11 are equivalent.

Proposition 5.6.2.6 ( 8 ) (W. Paschke, 1973) Let E be a C*-algebra, G, H inner-product right E-modules, and u E Z E ( ~H, ) Then (uSIu0 I llu112(
< E G . In particular, if

u is an isometry then

(.
=

(El<)

< E G.

We may assume E unital. For every n E IN put 1 x := ( ( I ) + 1 , <,, :=

on,

Take n E I N . Then (
i1

(Proposition 5.6.1.2 b)), so 1l
(Corollary 4.2.1.17 b

I1

+ a). It follows

~ l( ~ F n l ~ t= n )( u ( < ~ n ) I ~ ( < x n=) ) 11~11212 l l ~ < n I I L = ( ( ~ < ) x n I ( ~ O x= n )xi(~
(Corollary 4.2.1.17 a

+ t) , Proposition 5.6.1.2 b)),

5.6 Hzlbert right C*-Modules

Proposition 5.6.2.7 XI E E: , and

(8 )

Let H be a n inner-product right E-module,

a) G,! is a vector subspace of H ; denote by qzl : H

+H/G,l

the quotient map.

b)

t E G i , v E H * x1((Flo))= xl((vl<))= 0 .

c ) Take S,Y E H/G,! and

tl,t2E X , v1,q2 E Y . T h e n

5'((<1l71))= x1t(<21v2)). Define

(XIY),e

:= x1((
dl HIG,. endowed with the m a p

( H I G i ) x ( H I G i ) + IK :

(X, Y)

(XIY),'

is a pre-Hilbert space and q,, is continuous with

ll9z~ll5 llxlll~. Denote by Hz, the completion of the pre-Hilbert space H/G,l

e) For every u E H there is a unique u,, E Hz, such that

xl(u<)= ( q , r t I ~ ~ ~ ) ~ ~ for every ( E H . Moreover,

lluz,ll I llull llxlll~. f)

< E H * (.I<)=!= q i < . a) Take 1 , E~G,c . By Proposition 5.6.1.2 b), (< + v1< + v) I 2((<10+ (vlv))

so that

31 9

5. Hilbert Spaces

320

Hence G!, is a vector subspace of H . b) By Proposition 5.6.1.2 c) ,

so that

x1((vlO(
By Proposition 2.3.4.10 b ) , 1x'((E1v))I2 5 11x'11x'((vI<)(Elv)) = 0 ,

= xl((tl

-

<21v1)) + x1((<21v1- 772)) = 0 .

d) It is easy to see that HIG,,

is a pre-Hilbert space. For every

lIqiEll2 = (qr~EIqiE) xl((EIE))

so that q,

is continuous and llqr~ll5

e) Take

IIX'II' .

< E H . By Proposition 5.6.2.6 (and Example 5.6.1.5), (~<)'(.t)

so that

I lI~'11lIEII2

= (u
I lI~1I2(tl<)

< E H,

5.6 Hzlbert right C*-Modules

5 llxtll llul12xr((cl<))5

ll~tl1211~l1211
(Proposition 2.3.4.10 b)). It follows that xrou factorizes through H/G,t to a contiriuous linear form. By the Friichet-Riesz Theorem (Theorem 5.2.5.2), there is a u , ~E Hz! such that xr(uc) = ( q z ~ < I ~ z ~ ) z ~ for every

< E H . By the above, IIUZ~II

I<

= ~u~{I(qz~11qz~<115 1 ) =

= s.p{lxt(uf)ll(

E H , xr((
5

I I X ' I I ; I I U I.I

The uniqueness follows from the fact that q,t(H) is dense in Hz,. f ) By Proposition 5.6.2.1 a) (and Proposition 5.6.1.8 d)), (.I<) E

fi. For

BEH, = xr((.1<)~) = (q=1~I(.I<)z1)=1

Since q,,(H) is dense in Hz!, (.IE)zl

Proposition 5.6.2.8

(8)

= qi<.

W e assume IK = IR, use the notation of Pro0

positzon 5.6.2.7, and extend it i n a natural way t o the complexification H of H.

b)

T h e algebmic isomorphism

associated to the quotient map 0

q(zl,o): H + H/G(z~,0) is a n isomorphism of complex pre-Hilbert spaces. W e identify the above 0

A

pre-Hilbert spaces as well as their completions Hz# and (H)(,J,o) using this isomorphism.

5. Hilbert Spaces

322

C)

d)

fi =+ ( u , = (u,,, 0) u , v € fi 3 ( ( u ,O ) ( , ~ , o ) l ( ~o,) ( z ~ , o ) ) ( ~ t=, o() u ~ f l v ~ ' ) r ' U €

o)(rl,O)

By Proposition 4.3.6.2 c3

* cl ,

0

(XI,

0) E (El;

so that

-

1

= (9(x1,0)(<> V ) 9(rr,0)(<, A))(rt,O).

Hence the map

0

r

+

0

I

,

(qrj5,qzlv)

is an isomorphism of complex pre-Hilbert spaces. .) For

( 5 ,v ) E A , ((sz,S,

q,v)l(.zr,

0))Zl =

q(r',o)(C> 7)

5.6 Hilbert right C*-Modules

= (q(zfl,o)(t, 17)

1 (21, O)(z~,o))(z~,o) .

Since (<,g) is arbitry (u, 0)(2',0) = (1111,O) . d ) For

<EH, (qz~
Since H/G,, is dense in Hz,, there is a sequence By b) and c) : ( g z ~ < n ) nconverges E~ to ~

~

1

in H such that

.

(u, O)(zl,o)= ( u ~0), = nlim + m (gz~Fn, 0) = nlim + m q(,#,o)(tn,0) It follows ( U ~ ~ I=Ulim ~ ~(q,t<,lv,~),, ) ~ ~ = n-+CC

Proposition 5.6.2.9 ( 8 ) We use the notation of Proposition 5.6.2.7 and take x', y' E E'+ with x' 5 y' .

b)

There is a unique

c) For every u E

cp,r,,t

E .L(Hyc,HZ))' such that

fi , cpz',,'Uy'

= u,,

.

324

5. Hilbert Spaces

d) For every u E

fi

there is a filter

5 o n H such that

lim q,l< = u,! F-3

for every x' E EI, and 1im q ~ z ~ , y ~0))(= < ,( u 7O)(xl,yq C,3

for every ( x ' , y') E (E): a ) is obvious. b) follows from a). c) Since H/G,l is dense in Hy, , there is a sequence that (qyc
Take

H such

)itc ~ z ~ , y ~ q=~ ) < nit qz' En

< E H and n E N . By Proposition 2.3.4.10

=

(
I l ~ ' l l ~ ' ( ( < I < n ) ( < n lO

b),

(uO(
By Proposition 5.6.2.7 e),

= (qy'(<(uO8)Iuy'- qy'
and so lim y1((u<)(u<)*- (
n+w

Again by Proposition 5.6.2.7 e),

5.6 Hilbert right C*-Modules

325

so that

= llqyf
11<11211~y~
(Proposition 5.6.2.7 e), Proposition 3.6.1.2 b),c), Corollary 4.2.2.3). It follows )) 0 lirn yl((
n+w

0 = lirn x'((
= nlirn ((qzl
Since

< is arbitrary, pz,,ytu

1

- lim - n+w

qztFn = u,, .

1

5. Halbert Spaces

326

d ) For every A E y f ( E i ) and

E

> 0 put

B ( A , E ):= {< E Hlx' E A

* Ilq,~t - u,rII

<E)

B ( A ,E ) is nonernpty. Indeed, put

Then y' E E; and x' 5 y' for every x' E A'. Since H/Gyl is dense in H d , there is a E H such that

<

ll9Y't - ~ Y l l l< E By b) and c),

<

for every x' E A . Hence E B ( A ,E ) and B ( A ,E ) is nonempty. Denote by 5 the filter on H generated by the filter basis

It is obvious that lirn q,,( = ur, t,$

for every x'

EL.

Now take (x',y') E (E)'+. By Corollary 4.3.6.2 cl

(x',Y ' ) 5 ( 2 x f 0, ) . By b),c), and Proposition 5.6.2.8 b),c),

Il9(z~.y~)(J, 0 ) - ( u >O ) ( z ~ . y ~ ) = II

By the above, lim q(r,,yl)(t> 0) = ( es

~

O)(rl.vt) 1

*

c4,

5.6 Hzlbert right C*-Modules

Proposition 5.6.2.10

(8)

327

Let H be an inner-product right E-module

0

0

and H its complexification (Proposition 5.6.1.6). W e identify L ( H , E ) and 0

0

-

L ( H , E ) using the isomorphism of complex vector spaces

(

HE)

-

( h)

( u ,v ) +-+ ( u ,v )

defined in Proposition 2.1.5.6 (Proposition 4.3.6.1 f), Proposition 5.6.1.6)

0

b) H is self-dual 2ff H is self-dual. 0

a) Take ( u ,v ) E

i?.For every

(<, 77) E

2 and

( x ,y) E

k,

Hence

and

0

Take w 6 H . By the above identification, there is a ( u ,v ) E C ( H ,E ) with

( u , v )= W For every

.

(I,x ) E H x E , ( u ( f x ) v, ( < x ) )= ~ ( ( ( 50 ), ) = w ( ( f ,O)(x:0 ) ) =

328

5. Hilbert Spaces

...

b) First assume H is self-dual. Let w E such that

Since H is self-dual, there are

to,go E H

k . By a), there is a

with

0

For every (<, g) E H ,

= ((C>v)I(CO> -70))

Hence

0

and H is self-dual. is self-dual. Take u E Now assume

k

0

fi

-

and put

w := (u, 0) . 0

By a), w E H . Since H is self dual, there is a (<, g) E H with

0

(u,v) E

fi

5.6 Hilbert right C*-Modules

For every

<E H,

=

(( -(<,7)).

Hence

u = (.It) and H is self-dual.

Theorem 5.6.2.11 ( 8 ) (Paschke, 1973) Let E be a We-algebra and H a n inner-product right E -module. For eve y E H put

<

?:= (.It): H 4E ,

7 ++

(7710

and identify H with a subset of f i uszng the znjective map

H-fi,

6-?

(Propositton 5.6.2.1 a)). For every ( u ,x ) E f i x E put

ux:H

-+ E ,

<-x*ut.

W e shall use the notation of Proposition 5.6.2.7. a ) u ~ f i : x , Ey ~~ U X E (~u x?) y, = u ( x y ) . b)

(C,X)E

H X E*?X=G.

c) For K = C there is a unique map

fixj?+E,

(u,u)e(uIv)

such that

((ulv),a ) = (uaIva)a for all u , u E f i and a € E,. d ) Assume K = IR and take u, v E f i . T h e n

((uto)l(vl0 ) ) E E x ( 0 ) and we define (ulv) E E by

5. Halbert Spaces

330

((.I.)>

0) = ((% O)I(v, 0))

Then ((21, v), a ) = (.uolva),

for every a E E+ . e)

t>v.cH

(?I$

=+

f ) ( t , U) E H x g)

E?

= ((I?).

E? + (Flu) = uC.

endowed with the nght multiplzcation E?x~--ifi,

(U,X)HUX

(whzch contains also a scalar multiplication) and with the inner-product

zs a self-dual Halbert right E-module. h)

The norm of the Hilbert right E-module fi defined in g) coincides with the initial norm of E? (Definition 5.6.2.2). a ) For every ( E H , (UX)(
so that ux E

E?

and

b) For every 17 E H ,

(Proposition 5.6.1.2 a)), so that

c) CVc use the notation of Proposition 5.6.2.9. Takc u, v E

fi. Define

5.6 Hilbert right C*-Modules

Step 1

cp may be extended linearly to E

Let (a,),El be a finite family in C and (a,),,I a family in E+ such that

Put

Then a E E+ and a, 5 a for every

L

E

I . For (, q E H ,

(Proposition 5.6.2.9 b)). It follows

(Proposition 5.6.2.9 b)). Hence cp may be extended linearly to E . We denote this extension of cp also by cp. Step 2

cp is continuous

Take a E E . By Theorem 4.4.3.9 (and Corollary 4.4.2.10), there is a family (a,),,N, in E+ such that

332

5. Hilbert Spaces

We get

(Proposition 5.6.2.7 e)). Hence cp is continuous Define

Uniqueness follows from Theorem 4.4.3.9 (and Corollary 4.4.2.10). d ) By Proposition 5.6.2.9 d ) , there is a filter 5 on H such that

for every a E E+ and lim q(a.6)(tr0) = Cj3

O)(a,b)

for every ( a ,b) E E+ . Put

,.( Y) :=

(

(

~

O)l(v, 0))

9

and take ( a ,b) E E+ . By c) (and Proposition 5.6.2.7 e)),

((x,Y), ( a , b)) = ( ( ( wO)l(v,O)), ( a ,b)) =

5.6 Hzlbert right C*-Modules

= lim ((v, O)(<,O), (a, b)) = lim ( ( v l ,O), (a, b)) F>3

333

.

( 3 3

By Proposition 4.4.4.23, y = 0 . By c) and Proposition 5.6.2.8 d),

for every a E E+ . e) First assume

IK = C . For every a

E

E+ , by c) and Proposition 5.6.2.7

c),f),

so that

If IK = IR then, by d ) and the above,

--

( ( ? 1 3 > 0= ) ((?>0)1(?,0))= ((6,0)1(770)) =

and so

f) First assume IK = C . For every a E E + , by c) and Proposition 5.6.2.7 e),f),

so that (Flu) = u s . If

IK = IR then, by d) and the above, 0) = ((?, O)l(u,0))

=

(u, O)(<,0)

=

(uE, 0)

5. Hilbert Spaces

334

and so

Step 1

u, u E

fi,x E E + (uslv) = (ulu)x

Take a E E+ . Since xa E E (Corollary 4.4.2.10), there is a family (a,)JEn, in E+ and a family (a,),EN, in C such that

(Theorem 4.4.3.9). Put

Then b is an upper bound of a and 5.6.2.9 c),

in E,. By c ) and Proposition

By a)$), and Propositions 5.6.2.7 e) and 5.6.2.9 b),c), for every

Since qb(H) is dense in Hb , it follows

<E H ,

5.6 Hilbert right C*-Modules

Hence

A

Step 2

u,v E H

+ (ulv) = (V[U)*

For every a E E+ ,

((UIV),~)= (uaIva)a = (valua), =

so that

Skp3

u~fi=+(uIu)~E+

For every a E E+ ,

((.I.)>

a)

=

(uaIua)a E JR+

so that

(ulu) E E+ by Corollary 4.4.1.6 a). Step 4

u E H , ( U ~ U=) 0 + u = 0

For every a E E+ ,

(ualua)a = ( ( u I u ) , ~ = ) 0 so that

5. Halbert Spaces

336

By Proposition 5.6.2.7 e), aou = 0 so that by Corollary 4.4.1.6 b), u Step 5

=

0.

fi is self-dual

, with Let U E ~ ( j ?E) ( v , ~E )

5x

E

U(vx) = ( U V ) ~ .

Put

<

iri for every E H . Take v E 5 and a E E+ . There is a sequence converges t o v, . For every n E I N , by c), Proposition H such that 5.6.2.6, and Proposition 5.6.2.7 f ) ,

= IIVI12(va - QoInIva - qa
It follows successively

((Vv)*(Vv),a)

=0,

(Vv)*(Vv)= 0 (Corollary 4.4.1.6 b)), vv=o,

5.6 Halbert right C*-Modules

IK=R

Case2

(u, x) E fi x E + (u, O)(x, 0) = (UX,0)

Step 6 For (E, 17) E

b,

= (xtuE1x*u17) = ( ( u x ) ~(ux)17) , = (ux)(I,17)

so that (u, O)(x, 0) = (ux, 0)

Step 7

u, v E

+

fi, x E E + (uxlv) = (u1u)x

By d) arid Steps 1 and 6 ,

((.XI.)>

0) = ((ux, O)l(v,0)) = ((u, O)(x,O)I(v, 0)) =

= ((u10)1(u1O))(x,0) = ((ulv),O)(xr0) = ((uIu)x,0)

and so (uxlv) = (uIv)x. Step 8

u, v E fi + (ulv) = (V[U)*

By d) and Step 2,

(.I) Step9

= ( ( 7 4 0)1(v,0)) = ((v, O)I(u; 0)); = (uIu)*

u~fi=+(u(u)~E+

By d) and Step 3,

((ulu),0) = ((u,O)l(u,0)) E so that

i+

5. Hilbert Spaces

338

(ulu) E E+ by Proposition 4.2.2.15 b). Step 10

..

u E H, ( U ~ U=) 0

3

u =0

BY 4, ((u, O)I(u, 0)) = ((uIu),O) = 0 and so, by Step 4,

step 11

+

u.u E fi + ( ( u ~ u ) (VIV), (v/u) - (ulu)) E

i+

By d) and Case 1 (and Proposition 5.6.1.2 g ) ) , (((ulu),0) + ((vlv), 0) ((vlu): 0) - ((ulu), 0)) = 7

so that ( ( 4 4 + ( ~ I v ) (2114 - (uIv)) E 7

i+

by Proposition 4.2.2.15 b). Step 12

fi

is self-dual ...

k.

A

2

By Proposition 5.6.2.10 a), fi may be identified with By Step 5, is self-dual, so by Proposition 5.6.2.10 b), fi is self-dual. h) Take u E fi and denote by 1 1 ~ 1 1 the norm of u in fi in the sense of Definition 5.6.2.2. For every a E E + , by c),d), and Proposition 5.6.2.7 e),

By Corollary 4.4.3.16 c),

5.6 Hilbert right C*-Modules

Let a €10, J J u J JThere [. is a

5 E H# ff

with

< Ilutll . --

#

By Corollary 4.4.3.16 c), there is an a E E+ such that

a

< (utla).

By c),d), and Proposition 5.6.2.7 d),e), ff2

< (q,clua): i llqat1l211u,ll 2 I

Hence

Proposition 5.6.2.12 Let F be a closed ideal of E and H a self-dual Hilbert right F -module. a)

For every ( t , x ) E H x E there is a unique element of Ex, such that

/or e u e y r/ E H

b) H endowed with the right multiplication

and with the inner-product

H xH

---t

E,

((1

7)

-

is a Hilbert right E-module. a) P u t u : H 4F ,

9

I-+

x*(ql[)

u is linear and continuous and for (7, y) E H x F ,

((17)

N , denoted

by

34 0

5. Hilbert Spaces

417~) = x'(17yIO = x'(171J)y = ~ ( 1 7 ) ~ . Hence u E ~ , c ( HF, ) . Since H is self-dual, there is a J s E H with u =(.I~X).

For every 17 E H (JzIv) = ( ~ l J x )= * (117). = (1117)~. The uniqueness is easy to see. b) is easy to see.

5.6 Hilbert right C*-Modules

5.6.3 Von Neumann right W*-modules Proposition 5.6.3.1

(8)

Let H , K be Hilbert right E modules. For all

(x',1,7 ) E E' x H x K , put

Take (x1,71,5)E E' x H x K .

f) If A' is an involutive set of E' then the closed vector subspace of ( L E ( H ) ) 'generated by

is an involutive unital LE(H)-submodule of (L,q(H))' Take u E L E ( H ,K ) . a ) By Proposition 5.6.1.2 d ) ,

-

Hence (XI,1,7 ) E ( L E ( H K , ) ) ' and

b) By Proposition 5.6.1.8 a ) , -

I

(stl'!, 9 )

-

(4= ( X I , 5, V ) ( U * ) = x 1 ( ( u * f I q )=)

5. Hilbert Spaces

34 2

so that

c) By Theorem 5.6.1.11 cl

+ cz ,

for every u E C E ( H ) + . Hence by b) (and Corollary 4.2.2.10), h -

(x1751 e ) e (CE(H)): and by Corollary 2.3.4.7,

d ) We have

(Proposition 5.6.1.2 a ) , Proposition 5.6.1.8. d)), so that

so that

f) follows from a),b),e)

5.6 Hilbert right C*-Modules

34 3

Definition 5.6.3.2 ( 8 ) Let E be a W * algebra and H , K Hilbert (right) E-modules. For every a 6 E and ((, 7) E H x K define

-

(a,<): H

--+

M, C

c--t

((ClO,a ) ,

and denote by H (by H ) the closed vector subspace of H' (of L E ( H x K)') generated by

(Proposition 5.6.3.1 a)). H is called a von Neumann (right) E-module if it is self dual. If E is not specified then we simply speak of von Neumann (right) W *-module. If E 6 { R , C ,M) then every Hilbert (right) E module is a von iY eumann (right) E-module (Theorem of Frbchet-Riesz for 5.6.2.5 f) for the real C*-algebras C and M ).

IR and C and Proposition

Proposition 5.6.3.3 ( 8 ) (Paschke, 1973) Let E be a W*-algebra and H a Hilbert right E--module. T h e following are equivalent: a) H is self-dual.

b)

HH#

is compact.

If these condztions are fulfilled then the map

is a n isometry of Banach spaces. a

+ b.

Let

5

be an ultrafilter on H # . Put

for every ( E H . For ( 6 H and a E E ,

5. Hilbert Spaces

344

(Proposition 5.6.1.2 d)). Hence Put

EE (E)' = E and (:I

5 ((J((for every f

EH.

Then u E C(H, E ) # . Moreover, for (J, x) E H x E and a E E ,

so that

Since H is self-dual, there is a (a,<) E

<

E

H # such that u = (.I<). For every

ExH,

=

Hence

(F,a*) = lim (((17)) a*) = lirn ((TI<), a) = v93 v.3

5 converges to

*

< in

HE

and H# is compact. H

b a and the final assertion. Identify H with a subset of injective map

fi

using the

(Proposition 5.6.2.1 a ) ) and denote by G the closed vector subspace of generated by

-

2 ~ )E

{(=)((a,

where (a, u ) is defined with respect to

E x H) , (Theorem 5.6.2.11 g)).

+ b (and Theorem 5.6.2.11 g)), fi# H gG is Hausdorff (Theorem 5.6.2.11 f)), # - fi# H I ? - ..

By a

since

fi

A

(6)'

is compact. Since G C

fi

and

5.6 Hilbert right C*-Modules

Assume IK = C and let ((a,,

34 5

be a finite family in E+ x H . Put

Take u E fi# and E > 0 . In the sequel we use the notation of Proposition 5.6.2.7. Since qaH is dense in Ha there is an 17 E H# such that IIua-9

II

E

< (1 + Card I ) ( l + sup IILII)(l + a l ~ i ) '

LEI

By Proposition 5.6.2.9 b), c), for every

L

EI,

By Theorem 5.6.2.11 c),e) and Proposition 5.6.2.7 d ) (and Corollary 4.2.1.18),

It follows

5. Halbert Spaces

34 6

Since

E

and u are arbitrary,

The reverse inequality being obvious, we have

Hence the above Banach spaces H and G may be identified. By complexification, this assertion holds also for IK = IR . By Proposition 1.3.6.27, the maps

are isomorphisms of Banach spaces. By the above identification of H and G , we get H = fi, i.e. H is self-dual. Proposition 5.6.3.4 Let E be a W'-algebra , G , H Hzlbert right E modules, and u E C E ( G ,H ) . -Y

a) ( a ,q ) E E x H

h -

+ u r ( a ,7) = ( a ,u e ~ )

c) The m a p

zs continuous.

-

d ) If G and H are self-dual then the map

H

+G , yr

ury'

is the pretranspose of u (Proposition 5.6.3.3).

-

5.6 Hzlbert right C' -Modules

<E G,

a) For every

so that

-

h -

u' ( a , 7 ) = ( a , u*q) . b) follows from a ) . c) follows from b) . d) By Proposition 5.6.3.3, G and H may be identified with the duals of G and H , respectively, and the assertion follows from c) and Corollary 1.3.4.9 a + b.

Theorem 5.6.3.5 ( 8 ) (Paschke, 1973) Let E be a \I/'-algebra and H , K von Neumann right E -modules. a) ( L E ( H ,K)):

is compact and the m a p (

HK)

+ ( H ) u ct (u, .)I H

is a n isometry of Banach spaces. b) L E ( H ) is a We-algebra wzth H as predual. c)

-

<

h -

Take a E Re E and E H and let (u, I(a,<, <)I) be the polar representatzon of ( a , t ,5 ) . T h e n

-+

-

( a ,O F ) = ( a ,u+F, u + e ) ,

--

-

,-----

( a ,<,C) = ( a ,u-<, u - 0 ,

5. Hilbert Spaces

34 8

d ) Let F be a W*-algebra,

-

an involutive algebra homomorphism,

L := { ( a ,t, 77) I ( a ,t,q ) E E x H x H I , and

a map such that

for every y E F and (a,<,q) E E x H x H . Then cp is a W * homomorphism and

where cp denotes the pretranspose of cp (Corollary 4.4.4.8 a) ( L E ( HK , ) ) : is obviously Hausdorff.Let

c)).

5 be an ultrafilter on ( L E ( H K , ))#

For a € E , ( ( , v ) E H x K , a n d u 6 L E ( H , K ) ,

so that the map (LE(HK , )); is continuous for every

---t

/ I t l l ~ K # , 2~ ++ 14

< E H . By Proposition 5.6.3.3 a + b , the maps

where the limits are taken in K , and H, , respectively, are well defined and belong t o L ( K , H ) # and L ( H , K ) # , respectively. For all a E E and (<,q)E H x K ,

-

-

( ( v t l v ) ,a ) = (a, v)(vO = h $( a ,r/)(uO =

5.6 Hilbert right C*-Modules

-

h -

= lim ((u*qlJ), a*) = lim (a*,<)(u*q) = (a*,<) (wq) = u.3

u,5

It follows

-

By the above, for a E E and (e, q) E H x K , h -

(a, E , 01 (v) = ((vEIv), a) =: :1

((~Elq),a) = l i y (a, E, II) (u)

Hence 5 converges to v in (LE(H,K)): By Proposition 1.3.6.27, the map LE(H, K )

-+( H ) ' ,

and

u

SO

(LE(H.K)):

is compact.

+-+(u, . ) ( H

is an isometry of Bariach spaces. b) By Proposition 5.6.3.1 f ) , H is an involutive unital LE(H)-submodule of (LE(H))' . By a) , L E ( H ) is a W*-algebra with H as predual. c) By Proposition 5.6.3.1 b), (a, E, e ) E Re H . By Theorem 4.4.3.9,

-

and the assertion follows from Proposition 5.6.3.1 e). d) For every y E F and (a, <, q) E E x H x H ,

5. Hilbert Spaces

350

so that

( ~ ) Since c p ' ~ j ~is~continuous,

Hence cp is a W'-homomorphism (Proposition 4.4.4.6) and

Definition 5.6.3.6 ( 8 ) Let F be a norrned space and A a subset of F' such that Fa is Hausdorff. A family (
for every a E A . [ is called the sum of the family (tL)rEl in FA and is denoted by

Proposition 5.6.3.7 ( 8 ) Let E be a W*-algebra and (
{ Ct, I J E p I ( I ) )is upper bounded. L EJ

b)

The famzly (J,),,,

is summable zn EE .

If these condztions are fulfilled then

where the limzt is taken i n E, The proposition follows from Theorem 4.4.1.8 b)

5.6 Hzlbert right C*-Modules

Proposition 5.6.3.8 right E-module, and

for all distinct

1,

(8 ) (
351

Let E be a W*-algebra, H a von Neumann

a family i n H such that

X E I and such that

is upper bounded i n E a)

(
is summable i n H I , . Denote by

<

its s u m zn H , .

For every J E P , ( I ) ,

so that

is a bounded set in H . Let

5 , 6 be ultrafilters on g l ( I ) finer than 5, and

Put

where the limits are taken in H , (Proposition 5.6.3.3 a Take X E I and a E E . Then

so that

It fc)llows

+ b).

5. Hilbert Spaces

352

and so (Proposition 5.6.3.7)

We deduce

C=v. Hence (
< is its sum

H,

Proposition 5.6.3.9 ( 8 ) Let E be a C*-algebra and H a n inner-product right E-module. Then the following are equivalent for every E H :

<

a

+ b.

By Proposition 5.6.1.2 a),b),

= ((10

-

( [ I c ) ~- ( ( I < ) ~+ ( < l o 3 = 0

so that <(
b

3

c . By Proposition 5.6.1.2 a ) , (
c 3 a . Putting 77 :=

< , we get (<1<)(<10 = (
so that (
5.6 Hilbert right C*-Modules

359

Proposition 5.6.3.10 ( 8 ) Let E be a C order a-complete C*-algebra, H a n inner-product right E-module, E H , and A a Bore1 set of IR+ with 0 $ A . Put

<

d ) If we put

5n

:= fn((
for every n E K then lim <(<xnI<xn)= < .

n+co

e ) If E is finite-dimensional and

<+0

then

(
<(EY~
for some y E E . a) By Propositiori 5.6.1.2 b) and Corollary 4.3.2.5 c),

(<xI<x) = x(
5. Hilberl Spaces

354

b) By a ) and Proposition 5.6.1.2 b),

(< - <(<xl<x)l<- <(<xl<x)) =

c) follows from b) d ) BY b) >

for every n E DJ so that

e) By Proposition 2.2.1.17, o((
for every

<EA

and

for all dzstznct <,7 E A . A maxzmal Founer set is called a Fourier basis of H. By Zorn's Lemma, every inner-product right E-module has a Fourier basis. Proposition 5.6.3.12 ( 8 ) Let H be a n inner-product right E module, A a finzte Fourier set of H , and 7 , ( E H .

5.6 Hilbert right C*-Modules

a) By Proposition 5.6.1.2 b) and Proposition 5.6.3.9 a

* C,

b) By a ) and Proposition 5.6.1.2 a),

(

17 - C<(171<)

Theorem 5.6.3.13 ( set of H , and I ) , C E H

SEA

8)

1

I)

)

- ~ < ( I ) I < )= SEA

Let H be a Hilbert right E-module, A a Fourier

.

n

~ ~ is such that ( X ; X ~ is) ~summable ~ ~ i n E then a ) If ( x ~ E ) ~(<(<)E FE.1

( < X < ) F ~isAsummable i n H . Moreover,

SEA

implies xF = 0 for every dimensional.

<EA

and so A is finite if H is finite-

b) If E is C-order a-complete, A a Fourier basis of H , and

for every

<EA

then

71 = 0

.

c) If E is a LV*-algebra then

d ) If E is finite-dimensional then ( < ( 1 ) 1 < ) ) is ~ ~summable ~ in H .

5. Halbert Spaces

356

e) If E zs a W * algebra and H zs self-dual then ( < ( v ( < ) ) is ~ ~summable ~ zn H H . f) If E is a W'-algebra, A is a Fourier basis of H , and H is self--dual then

a) Take s > 0 . There is a B E ?-?,(A) such that

for every C E P,(A\B) (Proposition 1.1.6.6). It follows

for every C E P,(A\B) (Proposition 5.6.1.2 b)). Since H is complete, the family ( < X ( ) < ~isAsurnmable (Proposition 1.1.6.6). From

it follows

for every 71 E A . b) Assume 17 # 0 . By Proposition 5.6.3.10 a),d), there is an z E E with

5.6 Hilbert right C*-Modules

Since

for every J 6 A , A U {qx) is a Fourier set of H , contradicting the maximality of A . c) By Proposition 5.6.3.12 b), the set

C (J17)(71C) €

I

B E (Pf(A)

I

is upper bounded by (717) so that by Proposition 5.6.3.7,

d ) By c) and Minkowski's Theorem, the family

is summable. By a) and Proposition 5.6.3.9 a e) By Proposition 5.6.3.12 b), the set

3

c, (J(71J))et~ is summable

is upper bounded so that by Proposition 5.6.3.8 a), the family (<(V(J))~EA is summable in H , . f) P u t (by el)

For every Jo E A and a E E ,

5. Hilbert Spaces

358

(Proposition 5.6.3.9 a

+ c). Hence

for every J E A . By b), { = 0 , i.e.

For every a E E ,

Hence

Proposition 5.6.3.14 ( 8 ) Let E be a W*-algebra , G and H von Neumann right E-modules, and A a Fourier set of G . For each J E A let be a n element of H such that

for all J , q E A . For every x E em(A) , put

(Theorem 5.6.3.13a), Proposition 5.6.3.8 a), Proposition 5.6.3.9 a a ) For 17 E G , { E H , and x E em(A) ,

+ b).

5.6 Hilbert right C*-Modules

b) If x E em(A) then -t

E LE(G,H ) ,

-+*

+

x =5,

11211 = 1 1 ~ 1 1 ~

a) For a E E , - +

((2171C)? a) = ( ( a , I),x v ) =

so that

b) By a), for all (7,< ) E G x H ,

( 7 1 5 ~=) ( $ C I ~ ) *=

so that +

x E

+

y

L E ( G ,H ) , x = x

359

5. Halbert Spaces

360

For every

<E A, 1;

2

1<;1

=

Idol

so that + 11x11 2 11x1100 ,

For 7 E G ,

tEA

==$ (&IF)

= x(t)(vIE)

so that by Theorem 5.6.3.13 e),f)

1 1 ~ ~ 51 llxll~lla1I2 1 ~ (Corollary 4.2.1.18),

Proposition 5.6.3.15 ( 8 ) Let E be a W*-algebra, H a von Neumann right E-module, and A a Fourzer basis of H . For every x E em(A) put

(Theorem 5.6.3.13 e), Proposttion 5.6.3.8 a), Proposition 5.6.3.9 a + b)) a)

The map

is an znjective unital W *-homomorphism (Theorem 5.6.3.5 b)) with

for every (a,q,C) E A x H x H .

5.6 Hilbert right C*-Modules

b)

If x

E lm(A) and B := {x

# 0) then

&

is the carrier of

a) By Proposition 5.6.3.14 b), cp is well-defined, injective and involutive. Take x,y E ["(A) . For 7 , E H ,

<

(Proposition 5.6.3.14 a)), so that +

++

xy = x y . Hence cp is an algebra homomorphism and by Theorem 5.6.3.13 f ) , it is unital. For x E ["(A) , by a),

-

so that, by Theorem 5.6.3.5 d ) , cp is a W*-homomorphism and @((al7 , O ) = ( ( ( J I O ( ~ l 0~, ) ) C EE A['(A). b) By a) ,

E

Pr L E ( H ) and + -+

+

egx = x. Let u E L E ( H ) with

Then for ( E B ,

5. Halbert Spaces

562

and so

for every 7 E H (Proposition 5.6.3.4 c)). Hence

and

&

is the right carrier of

4

2 . By Corollary 4.3.3.9, e e

+ is the carrier of z .

rn Proposition 5.6.3.16 ( 8 ) Let E be a finite-dimensional C'-algebra, H a Hilbert right E-module, and B the set of Fourier sets A of H such that (((1)zs a mznimal element of Pr E\{O} for every ( E A . Then every maximal element of M zs a Fourier baszs of E . In particular, H has a Fourier basis belonging to B . Let A be a maximal element of B and assume A is not a Fourier basis of H . Then there is a E H such that

<

By Proposition 4.2.7.22, there is a finite set B of minimal elements of P r E\{O) such that

Take p E B . Then

(Corollary 4.2.7.6 a +e, Proposition 5.6.1.2 b)) and

for every 7 6 A . Hence A A.

U {
B and this contradicts the maximality of

5.6 Hzlbert right C*-Modules

Proposition 5.6.3.17 in it W*-algebra E :

The following are equzvalent for every family ( x , ) , ~ ~

a ) ( x , ) , , ~is summable zn EE b) For every a E E and

for every K E

y f (I\J)

E

> 0 there is a J

E

qf( I ) such that

.

If these conditions are fulfilled then:

e ) ( x , ) , , ~zs summable zn E , for every J , the map

is continuous, where p ( I ) is identified with (0, I ) ' , and

is a compact set of E E and a bounded set of E .

a

+ b . There is a

J E p f ( I ) such that

for every K E p f ( I ) , J c K . T h e n for every K E

so that

363

p f (1\J),

364

5. Hilbert Spaces

b

+a.

Take a E E . By b), there is a J E p f ( I ) such that

for every K E p J ( I \ J ) . Put

Then for K E Q f ( I ) ,

Since a is arbitrary, it follows from the theorem of Banach-Steinhaus (Theorem 1.4.1.2) that

is a bounded set of E . Hence, by the Alaoglu-Bourbaki Theorem, it is a relatively compact set of E, . Let x be a point of adherence in E, of the image of the filter 5, with respect t o the m a p

Let a E E and

E

> 0 . By b) , there is a JoE pf( I ) such that

for every K E Y I ( I ) , K that

Then for K E Y j ( I ) , J

c I \ J o . Further

c K,

there is a J E p , ( I ) , Jo c J , such

5.6 Halbert right C ' M o d u l e s

Hence ( x , ) , is ~~ summable in EE and x is its sum. c) follows from a) and Proposition 4.4.1.3. d ) For every a E E ,

Since a is arbitrary,

Similarly,

and

e) By a H b , ( x , ) , , ~is summable in EE for every J C I . Take a E E E > 0 . By b), there is a J E p f ( I ) such that

for every K E

for every K

p f (I\J)

c I\J.

. By continuity,

Let K l ,K Z E p ( I ) with

5. Hzlbert Spaces

366

Then

Hence the map

is continuous. Since a is arbitrary, the map

is continuous. It follows that

is a compact set of E E . By Banach-Steinhaus Theorem (Theorem 1.4.1.2), it is a bounded set of E .

Corollary 5.6.3.18

Let E be a W*-algebra, (x,),,,

a summable family in

E E , and

(Proposition 5.6.3.17 e)). For every a E E and J C I put - ( a , J ) := sup

{

(gX.,

.)I

K

c I\.J)

5 ~llall.

5.6 Hilbert right C*-Modules

a ) If a E E , J

cI,

K E q f ( I \ J ) , and a E l m ( I ) then

b) For every a E E and

E

> 0 , there i s a J

for every a E e m ( I ) and K E C)

For every

CY

E

pf(I)such

that

(I\J) .

E e m ( I ) , ( a , ~ , ) ,i ,s ~summable i n E, and

d) T h e maps

are continuous and

as a compact set of E,.

a ) follows from Proposition 1.1.1.5. b) By Proposition 5.6.3.17 a b , there is a J E y j ( I ) such that

5. Hilbert Spaces

368

for every (Y E em(I) arid K E p f ( I \ J ) . c) By b) and Proposition 5.6.3.17 b =. a , ( ~ , x , ) , ~isIsummable in EE for every a E !"(I). The inequality follows from a ) . d) By c) , the map em(I) + E , is continuous. Let a E E and

E

a

-

E

a,xL LEI

> 0 . By b) and c ) , there is a J E p f ( I ) such that

for every a E (IK#)' and K

c I\J.

Let a ,0 E (IK')' E

1% - PtI < 3 ( 1 + 87) for every

L

E J . By c) ,

Hence the map

is continuous. Since a is arbitrary, the map

is continuous. It follows that

is a compact set of E,

such that

5.6 Hzlbert right C*-Modules

36.9

Let E be a W*-al.qebra and let (x,),,~, ( Y X ) A E L be sum~nablefamilies i n E E .

Proposition 5.6.3.19

5

a)

XEL

( ( eL XE IL ) Y X )

(..

= E (5 ")) (5") ( f Y X ) . LEI

=

XEL

XEL

LEI

b) T h e map

(Proposition 5.6.3.17 e)) is continuou.~,where p ( I ) and p ( L ) are identified with { O , l ) ' and {0,1)', , respectzvely. a ) For a E E ,

( j (eYA), (yA.atxL) ((ezL) = ( eX yE LX , a = xL ELI )

=

LEI

A €1,

XEL

LEI

= L E I (z.,

(5

X E L .A)

= XEL

a) =

XEL

LEI ( 2 .

=

yX,a)

(2

,A€ t, y X )

,

.a)

and t,he assertion follows from the fact that a is arbitrary. b) By Proposition 5.6.3.17 e), ( X ~ ) ~ ~and . J ( Y , ) , , ~ are summable in E, for all .I C I and K C L . The assertion follows from a ) and C. Constantinescu, Spaces of Measures, Theorem 3.5.2 c). Proposition 5.6.3.20

Let ( x , ) , ~be~ a fa~nily of the W * algebra E such

that

for all distinct

L,

X E I . Then the following are equivalent:

5. Hzlbert Spaces

3 70

a) (x,),,~ zs summable zn E, b ) (x:x,),,I

.

zs summable zn E,

If these condztzons are fulfilled then c)

LEI

LEI

a

LEI

* b & c . By Propositions 5.6.3.17 c),d) and 5.6.3.19 a),

( ) ( ) (5.:)(5 5 5 LEI

LEI

=

LEI

E

= ):

LEI

E

xX) =

€ 1

(x:

AEI

xA) =

E

x:xA

=

X~X,

LEI

rE1 XEI

*

b a . Let a E E and let (y, l a [ ) be its polar decomposition (Theorem 4.4.3.5 a)). For every J E T J ( I ) , b y Proposition 2.3.4.6 c ) ,

5 (Y*

(p)(5

XL)

By Proposition 5.6.3.17 a =+ b , for every that

E

Y? l a , ) ( l , , a l ) =

> 0 , there is a J

E T J ( I ) such

5.6 Hilbert right C ' M o d u l e s

for every K E g f ( I \ J ) . By the above,

for every K E p , ( I \ J ) . By Proposition 5.6.3.17 b in E,.

+a,

( x , ) , ~ Iis summable

Remark. The above conditions imply that ( ~ ~ , l is ~ sumrnable ) ~ , ~ in E, for every cr 2 2 , since

, ~ ~summable in is a cofinite subset of I . But it may happen that ( I X , ~ ~is) not EE for every (Y E ]0,2[. A counterexample in this sense is given by the sequence (x,),,~ in L(e2),where

for every n

N

Corollary 5.6.3.21 such that

for all distznct

L, X

Let E be a W*-algebra and (p,),El a family in Pr E

E I . Then (p,),,,

is summable i n E, and

The set

is upward directed and bounded (by 1) so that (P,),,~ is summable in E, (Theorem 4.4.1.8 b)). By Proposition 5.6.3.17 c) and Proposition 5.6.3.20 a =+ b&c,

5. Halbert Spaces

372

Proposition 5.6.3.22 such that

for all distinct

L,

Let E be a W*-algebra and ( P . ) [ ~ a~ famzly zn Pr E

X E I and

Then the map

LEI

(Corollary 5.6.3.18 d)) is a unztal injective W * - h o m o m o r p h i s m with pretranspose

For a , B E e r n ( [ ) ,by Propositions 5.6.3.19 a ) and 5.6.3.17 c),

Hcnce the map is an ir~volutivealgebra homomorphism. It is obviously unital and injective. Moreover, for a E t m ( I ) and a E E ,

so that the map is a W-homomorphism with pretranspose

5.6 Hilbert right G' -Modules

5.6.4 Examples Proposition 5.6.4.1 ( 0 ) Let ( H l ) , , I be a family of (weak) semi-innerproduct right E-modules and

If

:=

@ H,

:= {

fE

LEI

n

the family ((<,Ifl)),,l is summable in E ) .

LEI

a ) H is a vector subspace of

fl H,

and H endowed with the maps

LEI

as a (weak) semi-inner-product right E-module. b ) If H, is an inner-product right E-module for each inner-product right E --module and

for every f E H . If <,71 E

n H , such that

i

E I , then H is an

E H and

g

LEI

for every

L

E

I , then f E H and

< 11q11. H is a Hilbert

c ) If H , is a Hilbert right E-module for each right E -module.

L

E I , then

d) If H, is a (unital) Hilbert E-module for each E-module H endowed with the map

L

E I , then the Hilbert right

is a (unital) Hilbert E -module. Moreover if ( x , ) , , ~zs a bounded family in E then

for everg

<E H

3 74

5. Halbert Spaces

e) Take

<EH

and for every

L

E I put

is summable i n H and

Then

f ) If (IX)XEL is a parlition of I then a)

a) H ,

isomorphic to a) H, i n

X€L LEI*

LEI

a natural way. a ) It is easy to see that H is a weak semi inner-product right E-module whenever I is finite. . Proposition 5.6.1.2 d),e) Take <,71 E H and x E E . Let J E V f ( I ) By applied to the above construction with respect to J ,

By Proposition 1.1.6.6, it follows that the families ((<'I~,)),EIand (((Lxl~,x)),E, are summable. In particular (
for every

<+

71

L

E I . By the above, the family ((5,

E H and H is a vector subspace of

product right F module for each

+ 7/,1<, + qL))lE1is summable, so

n H,

. Assume

H, is a semi-inner-

LEI

L

E

I . Then for <,17 E H ,

= (1((CLlCL)+ ( a l e ) ) . x ( ( a l I , ) - (61%)) LEI

LEI

Hence H is a semi-inner product right E-module.

5.6 Hilbert right C*-Modules

11) By Corollary 4.2.1.18,

IlrLI12 = II(E~I<~)II I 11 C(E~IE~)II = II(SIE)II XEI

for every

L

E

I . Hence H is an inner-product right E-module and SUP

IlELll

~ € 1

I IIEII .

The other inequality follows from

The final assertion follows from Corollary 4.2.1.19. c) Let (<(n))n,N be a Cauchy sequence in H . For every m, E IN such that

For all n,p E N 4.2.1.18,

E

> 0 , there is an

, T L 2 m, , p 2 m, . Let J be a finite subset of I . By Corollary

for every

< E H . It follows that

for every

E

(
is a Cauchy sequence for every

> 0 and every n,p E I N , with n 2 m, , p 2 m, . In particular,

tL:= lirn

n+m

i

E I . For each

L

E I , put


Then for every finite subset J of I ,

for all

E

> 0 2nd every n E IN with n > m, . Hence

<(")

-

< E H and

5. Hilbert Spaces

376

for every

E

> 0 and n

E I N , n 2 m, . Thus f E H and ( c ( ~ ) ) , , ,converges N to

f in H . In particular, H is a right Hilbert E-module.

d) The map E x H Then

+H

is obviously bilinear. Take x , y E E and f , 7 E H .

LEI

LEI

By Corollary 4.2.1.19 and Proposition 5.6.1.2 e), ( x , J , ) , ~ E IH and

e) Let

E

> 0 . By the definition of H , there is a .I

for every L C I \ J . Hence, for K E

E

P f ( I ), such

that

P , ( I ) ,J c K ,

LEI\K

Thus ( f { , l ) , , l is sun~mablein H and

X

((1)= C

LEI

f ) follows from a).

Remark. Take

< E fl HL. If LEI

then the family ((f,lf,)),,I is absolutely summable, so it is summablc. The converse is not true as thc following example shows: E := co , I := N , HL:= E for every L E I , En := &en for every 71 E N .

5.6 Hilbert right C*-Modules

Example 5.6.4.2

(0)

377

Let I be a set, F a closed right ideal of E , and is summable} .

e 2 ( I ,F ) := { J E F'lthe family

a ) e2(1,F ) is a vector subspace of F1 and e 2 ( I ,F ) endowed with the maps

t 2 ( 1 ,F ) x e 2 v ,F )

4E

,

( I ,7 )

-

7;E' iEl

is a Hilbert right E module.

C)

If F is an ideal of E and ( x , ) , , ~is a bounded family in E then for every E E e 2 ( I ,F ) ,

In partzcular, t 2 ( 1 ,F ) endowed with the map E x e2(1,F )

('(1,F ) ,

--t

( x ,E )

(xEL)LEI

is a (unital) Hilbert E-module. d) If F is an ideal of E and

I* :=

(<:)LEI

E e 2 ( I ,F )

<

whenever E e 2 ( I ,F ) (Corollary 4.2.6.2) (this is the case when F is finitedimensional or I is finzte) then there is an a > 0 such that

IIE'II r ~ I I I I I for every J E e2(1,F ) and (x<)*= <*x*,

((x)' = XI'$*

for all ( x , J )E E x e2(1,F ) . e ) Take

(1,

E2,53, I 4

E

:

E

IR' and put

+,

L

* I I ( I , )+~G ( L +) ~h ( ~ )+j ( q ( ~ ) k .

5. Hilbert Spaces

378

Then the following are equivalent:

b,(4

E e2(I, R),

el)


e2)

< E P2(I,M) ,

e3) <* E e2(I,M) ,

e2u,R).

e4) ( I I < ( L ) I I E ),~I

If these conditions are fulfilled then

a), b), and c) follow from Example 5.6.1.5 and Proposition 5.6.4.1. d ) We may assume that I is infinite. Suppose that the map

is not continuous. Then there is a sequence (
for every n E IN. By continuity, for every n E I N , there is an InE p J ( I ) such that

We may assume that (In)nENis a disjoint sequence. Put

( <,, if L E In for some n

By Corollary 4.2.1.18

E IN

5.6 Hilbert right G*-Modules

i.e.

< E 12(1,F ) . On the other hand

for every n E IN, so that t*$! 12(1,F ) , which contradicts the hypothesis. Thus the above map is continuo~isand there is an a > 0 with

<

for every E 12(1,F ) . The other relations are easy to see. If F is finite-dimensional, then by Corollary 4.2.1.20, never t E t 2 ( I ,F ) . e) By the Examples 2.3.1.46 and 4.1.1.31,

for every

L

t* E

12(1,F) whe-

E I.

Remark. The hypothesis of d ) may fail (sez Remark of Proposition 5.3.2.13). Even if it holds, a may be different from 1 . Indeed, put

Then

so that

5. Hilbert Spaces

380

, ) be the Proposition 5.6.4.3 ( 0 ) Take n E IN and let En := t 2 ( N n E Hilbert E-module defined in Example 5.6.4.2 c). For <'E E l n , put

Then

F E (En)' and

(g

l

2

)

5

ill 5

2 llc:ll 1=1

for every <' E Eln and the map

is a homomorphism of E-modules and an involutive isomorphism of Banach spaces (Proposition 2.2.7.2, Example 2.2.7.21). We have

for every

< E En

(Example 5.6.4.2 b)),so that

Take 6' E ] O , l [ .For every z E E n , there is a

Given i E IN,, put

Then

and so

F E (En)' and

<, E

E\{O) such that

5.6 Hzlbert right C*-Modules

Since 0 is arbitrary, we see that

By Example 5.6.4.2 b), the norm of En is equivalent to the Euclidean norm. By Proposition 1.2.2.13, the map

is an isornorphism of Banach spaces. Take I' E Eln. Then

for every J E E n , so

and the above map is involutive. Moreover, for every x E E ,

for every

I E E , so that

Proposition 5.6.4.4

Let R, S, T be sets and let

be a conjugate involution. For each u E LE(e2(S, E ) , e2(T,E ) ) define

Ti : t 2 ( S E , ) 4.t2(T,E ) ,

--

ct vJ

.

5. Hilbert Spaces

382

a)

(1

17 E P2(T,E ) =+

(?I$

=

G)

b) T h e Banach space P2(T,E ) endowed with the znvolution

is a n involutzve Banach space. c) If u E LE(P2(S,E ) , P2(T,E ) ) and x E E then

- -- i z = C5

xu = xu.

(Example 5.6.4.2 c), Proposition 5.6.1.8 g)). d ) The Banach space LE(e2(S,E ) ,e2(T,E ) ) endowed with the involution

LE(P2(s,E ) , P 2 ( ~ E, ) ) 4LE(P2(s,E ) ,['(T, E ) ) , u ct ii zs a n involutive Banach space.

e ) u E L E ( e 2 ( S , E ) , P 2 ( T , E )v) E, L E ( P 2 ( R , E )P, 2 ( S , E ) )3 ~02)=COG

f ) The map

is a conjugate znvolutzon of the C*-algebra LE(P2(T,E ) ) (Theorem 5.6.1.11 a)). g)

Assume IK = C . Let F be a C-subalgebra of L E ( P ~ (E I ,) ) such that 6 E F whenever u E F and let F denote the complexification of the underlying real C ' a l g e b r a of F . T h e n the map 0

is a n isomorphism of complex C*-algebras.

a ) (?I$ =

C ;i;G LET

=

b) follows from a ) . c) h is linear and

c 2E c te'r =

LET

cv ; ~ h -

=

-V

= (117)

t ET

5.6 Halbert right C*-Modules

so that

Take ( E e2(S,E ) and 17 E E2(T,E ) . By a) ,

so that

-

ii E C E ( ! ~ ( E S ,) , t 2 ( ~ , ~ 2 ) )=, u * . Moreover,

so that

iz = iiz, (1) The involution is conjugate linear, so that the assertion follows from c). c ) For all ( E e2(R,E ) ,

f ) follows from c),d), and e). g) follows from f ) and Theorem 4.1.1.8 e) Proposition 5.6.4.5

(0)

Let E be a finite-dimensional C*-algebra and

(e,),N, an algebraic basis of E consisting of invertible elements of E . For ever9 x E E define ( x , ) , ~ N , E IK" such that

384

5. Hilbert Spaces

and put

For S , T sets and

*

:= (

a

)

E

( L ( P ( T )@ . (s)))~,

:EN.

where for

E

e2(S,E ) and j

E

N,, I1E e 2 ( S ) is defined by

Then €

-

LE(e2(s,E ) , ~ ' ( TE, ) ) , ii* = u*

for every u E ( L ( e 2 ( S )e,2 ( T ) ) ) n(Definition 5.6.1.7) and the map

zs bzjective and linear. For every z E N, L

so that

For

5 E t 2 ( S ,E )

and

71 E

t 2 ( T ,E ) ,

5.6 Hilbert right C*-Modules

Hence

ii E ,c,(e2(s, E ) , e 2 ( ~E :) ) , ii*

-

= ,u*.

It is obvious that the m a p is linear. Take

u E ( , c ( e 2 ( s )e, 2 ( T ) ) ) " and assume ii = 0 . Fix k E ISnrtake

and we obtain successively

< E e 2 ( S ) ,and define 7 E e2(S,E ) by

5. Halbert Spaces

386

the map is injective. Take

v E L E ( e 2 ( sE, ) , e2(T,E ) ). Since E is finite-dimensional, it is unital. Denote by 1 the unit of E . For every i E IN,, put

IL :=

For every

(zL,),E ~ ~( L, ( e 2 ( s )e ,2 ( ~ ) ) .) ,

< E e2(S.E ) ,

(Proposition 5.6.1.8 d)), so that ii = v and the map is surjective.

Proposition 5.6.4.6 ( 8 ) Suppose E is a We-algebra and let ( H l ) r Ebe~ a family of Hilbert right E-modules. Put

and

for every

< E n H, LEI

and J

cI.

38 7

5.6 Hilbert right G*-Modules

a ) If J , 17 E H then J EE . Put

+ 1) E H

and the family

((JLI~L))LE~

is summable i n

b) H endowed with the right multiplication

H xE

*H ,

((,I)

:= ( < L X ) L E /

and with the inner-product

is a Hilbert right E-module c)

For every a

E

E , J E H , and

E

> 0, there is a J

E

P I ( I ) such that

( ( ( ~ I < I \ K ) , ~ ) (< E

for every 17 E H # and K E P 1 ( l ) ,J

c K . In particular,

for every J E H , where for every H i n a natural way.

I we identijed H , with a subset of

L

E

d) If H , is a (unital) Hilbert E-module for each L E I then the Hilbert right E-module H endowed with the left multiplication

is a (unital) Hilbert E-module. e ) If E zs finite-dimenszonal then

f ) Assume there is a famzly ( P ~ ) i~n ~PrI E\{O)

for every

L

E I . Put ft

for every

L

such that

:= ( ~ L X P L ) X E I

I T h e n ( f L ) r E I is a Fourier basis of H .

5. Hilbert Spaces

388

a) We may assume IK = C (Theorem 4.4.1.2, Proposition 5.6.1.6). By Proposition 5.6.1.2 b),

<

for each L E I SO that + 7 E H (Proposition 5.6.3.7 a + b). The family ((
4). b) Take <,7 E H and x E E . Then

for every J E q f ( I ) (Proposition 5.6.1.2 b)) so that <x E H (Proposition 5.6.3.7 a 3 b). Moreover,

(Proposition 4.4.1.3),

(Proposition 4.4.1.3),

389

5.6 Hilbert right C*-Modules

be a Cauchy sequence in H . For every

Let

E

> 0 , there is a

p, E K such that

for all m, n E I N , m 2 p, , n 2 p, . Since for every

L

E I , the map

is linear and continuous, (
For every

E

> 0 , m E N , m > p, , and J E y f ( I )

l <$m) Hence

<(rn) -

- fJll = Iim n+m

<E H

l <$m)

-

<,y)ll5

Iirn

n+m

Il<("') -

5E

and from

it follows

(Proposition 5.6.3.7 and Corollary 4.2.1.16 a e b ) . By a ) , f E H and lim f(") = 6

m+w

Thus H is a Hilbert right E module. c) Assume the contrary. By recursion, we construct an increasing sequence (Jn)nEn in v f ( I ) and a sequence (77(n))nEN in H# such that

for every n E N . Take n E IX and assume that the sequences were constructed up to n - 1 . By the above assumption, there is a J2n-1E Q f ( I ) and an E H # such that J 2 n - 2 c J2n-L (JO:= 0) and

5. Hilbert Spaces

390

By a ) , there is a J2, E

P f ( I ), J2n-1c JZn such that

This finishes the recursive construction. P u t

for every n E N Define

i V ( " ) if

7/:I+E,

0 By Corollary 4.2.1.16 a

for every J E

P I( I )

L

E Kn for some n E N

1-

SO

otherwise.

=+ b ,

that 7 E H . For every n

Hence

which is a contradiction. d) Take <,17 E H and z, y E E . Then

N,

5.6 Hilbert right C*-Modules

for every

L

E

I

SO

that

for every J E p j ( I ) . It follows x< E H and

( x t I x 0 5 llxll2(tIO. T h e relations

are easy to verify. e ) follows from Minkowski's Theorem. f) For L , X E I ,

Let

<E H

for every

L

srich that

E

I . Then for every

L

E

I,

t;

:b,xp,lx

0 = (
= P,<, =

<,

A€ 1

Hence

< = 0 and

fL)LElis a Fourier basis of H .

Theorem 5.6.4.7 ( 8 ) W e use the notation and the hypotheses of Proposition 5.6.4.6 and assume i n addition that H , is self-dual for each L E I . a ) H is self-dual.

b)

a ) H , is a Banach subspace of H dense i n H,, LEI

c) For every J

c I , the map

belongs to Pr L E ( H ) and for every u 71

E

LE(H)

= lim 7r~u7r.~ , J , ~ I

where the limit as taken i n ( L E ( H ) ) ,

392

5. Hilbert Spaces

d ) Put G : = { " ‘ E L E ( H ) I u ( ~L) EHIL ) C La)H., EI

U * ( ~ H . )

OH.}

c

LEI

and

for every u E G . Then G is a unital C*-subalgebra of L E ( H ) , dense i n (LE(H)), , generatzng t E ( H ) as W*-algebra ( a ) and Theorem 5.6.3.5

a)),

ii E

(

tE 0 H ~ )for every u E G , and the m a p LEI

is a n isometry of C'algebras.

a) By Proposition 5.6.3.3 b + a , it is sufficient to prove that pact. Let 5 be an ultrafilter on H # . Put

where the limit is taken in (H,),, (Proposition 5.6.3.3 a J E p J ( I ) . Then

HZ

+ b) . Let

is com-

a E E and

IIvJI/

I((V.I~VJ)~~)~ I((<~VJ)>~)~ 5 llall

(Proposition 5.6.1.2 d ) ) so that

1 1 ~ ~ =1 1 II(VJIVJ)~~ ~ = SUP ~((vJIvJ),~)~ I1 1 ~ ~ 1 1 ~ a€E*

Since .I is arbitrary, 7 E H # (Proposition 5.6.3.7, Corollary 4.2.1.17 a a b). Take a E E , ( E H , and E > 0 . By Proposition 5.6.4.6 c), there is a J E PJ(I)such that

for every

< E H# . It follows

5.6 Hilbert right C' -Modules

5 I((v - C I t ~ ) , a ) l+ 2~ for every

Since

E

< E H#

and so

is arbitrary,

'jB (=)(o = E ) ( V )

HH#

<

arc also arbitrary, 5 converges t o 71 in H'. Hence is Since a and H compact. b) follows from Proposition 5.6.4.6 c) . c) It is easy t o see that T J E P r L E ( H ) for every J c I . Take (a,<,71) E E x H x H and E > 0 . By Proposition 5.6.1.6 c), there is a J E y f ( I ) such that, for every E H # and K E Vf ( I ) ,J c K ,

<

l ( ( < l ( ~ * ~ ) r \ t i )< . ~2(1 )l Hence for every K E (Pf ( I ) ,J C K ,

E +

l l f l l .)

394

5. Hilbert Spaces

so that

Hence

in ( L E ( H ) ) ~ . is a unital C'-subalgebra of L E ( H ) . By c) , G d ) It is easy to see that is dense in (LE(H))" , SO that G generates C E ( H ) as W* -algebra (Theorem 5.6.3.5 b) and Corollary 4.4.4.12 a)). Let u E G . For [, 7 E a ) H L , LEI

so that ii E LE(

a> H,)

and ii* = i*. In particular, cp is involutive. It is easy

LE l

to see that cp is an injective unital algebra homomorphism. We want to show that cp is surjective. Let u E LE( H,) and let 3 be an ultrafilter on v f ( I ) finer than 3 1 .

a>

LEI

Then

for every ( E H and J E ?,(I). By a) and Proposition 5.6.3.3 a define

* b , we may

where the limits are considered in HH . Take <, 7 E H . Then for every a E E ,

-

= (a*,O(wv) =

-

1jy ( a * ,E ) ( u * v ~=)

5.6 Hilbert right C*-Modules

lim ( ( u * v ~ \ ) ,

J,3

= lim lim(((JIu8qK),a ) = l i m ( ( f J I w v ) ,a ) = 5,s

K,3

J,3

Hence

and so v E L E ( H ) and v* = w . For every

<E

a) H , , LEI

so that

T h u s v E E and

Therefore cp is surjective and so it is an isometry o f G*-algebras. Proposition 5.6.4.8 P r E (in P r E ) . Put

for every

L

H

Let E be a W*-algebra and ( P , ) ~ ~a, family i n Ec r-

E I.

a ) H , is a von Neumann (right) unztal E-module for every

L

E I

396

5. Hilbert Spaces

W

b) H :=

a)H, is a uon Neumann (right) unital E-module. For every x E E

LEI

(resp. x E E C )put

Z :H

4 H

c ++xE

,

(resp. lx)

c) Z E C E ( H ) for every x E E (resp. x E EC). Put q : E (resp. E C )-+ C E ( H ) , x ct Z

d ) cp is a W*-homomorphism (Theorem 5.6.3.5 b), Corollary 4.4.4.12 e)) and

for every ( a ,[, 17) E E x H x H , where i : EC + E denotes the inclusion map and @ the pretranspose of cp .

e) Let

c,11 E H

such that

for all distinct and

L,

X E I . Then

(<,),,I

f ) Assume p,px = 0 for all distinct

L,

and

( T / , ) ,are ~ ~summable

X E I and put

(Corollary 5.6.3.21). Then the map

is an isomorphism of uon Neumann (right) E-modules.

in E,

5.6 Hilbert right C*-Modules

397

a) By Example 5.6.1.5, H, is a Hilbert (right) unital E-module and by Proposition 5.6.2.3, it is self-dual. b) By a) and Proposition 5.6.4.6 b),d) , H is a Hilbert (right) unital Emodule and by Theorem 5.6.4.7 a), it is self-dual. c) was proved in Proposition 5.6.1.8 h ) . d ) By Proposition 5.6.1.8 h), cp is an involutive algebra homomorphism. For every x E E , (resp. x E EC),

By Theorem 3.6.3.5 d ) , cp is a W*-homomorphism and

e) By Proposition 5.6.3.20 b + a, (<,),,, and (qL)lElare summable in E, . For J E

Y J ( ~ 3

By Proposition 5.6.3.19 a) (and Proposition 4.4.1.3),

f ) For every <, 7 E H and distinct

L,

XE I,

398

5. Hilbert Spaces

By e ) ,

and

(<,),,I

( v ' ) , are ~ ~summable in

(5 (5L ) LEI

Let

CE

%) *

LEI

p E . Then pLC E HL for every

for every J E

yr( I ). Hence

L

E, and

=

E I and

(pLC),EI E H . Moreover

(Proposition 5.6.3.17 d ) ) , so that the map

LEI

is surjective. Example 5.6.4.9

Let I be a n infinite set. Put

(Proposition 5.6.4.8 a), b)), := { U E

L E ( H ) I u ( t 2 ( IE , ) ) C e2(1,E ) ,

u * ( t 2 ( 1E, ) ) C t 2 ( 1 ,E ) ) ,

a n d f o r e v e y < E H and i , A E I ,

Fix a poznt

for every

LO

i n I and define for every

<EH

and A, p E I .

L

E I a m a p p, : H

-+ H

by putting

5.6 Hilbert right C*-Modules

a) If L ,X E I \ { L ~ ) then p, E Pr G and

b) The family ( P , ) , , ~ is unbounded i n G

c ) C E ( e 2 ( IE , ) ) is not a W *-algebra. a) For < , V E H and p~ I ,

=

1 -

)

(
+

"El

=

1

-

"El

<",

(&LO

tLL)

6', (6UL0 + 6UL)Siu, =

+ 6"') 6',

(v',,' + Sill) =

so that

Hence p, E C E ( H ) and p, = pt . It is easy to see that p E For E H and p , v E I ,

<

G for every

1

E I.

4 00

5. Hilbert Spaces

so that

P~E P r G .

b) Let u be an upper bound of the family (p,),,~ in

< := (6,,,er),,, Then for

L

E I \ { L ~ )and A, j~ E

1 (P~<)A, = 2(
G. Put

E e2(1,E ) .

I, 1

+ 6~1)= 26L11(6~L0 +~ A I )

so that 1

-

-

(PL<~<)F = ~ ( P L < ) A P ~ X5 , 6 (
.

XE I

By Theorem 5.6.1.11 c,

+ c2 ,

Hence the family ((u
and this is a contradiction. is c) By a), (p,),,,\(,,) is a family in Pr G . By b) and Theorem 4.4.1.8 i), not a We-algebra. By Theorem 5.6.4.7 d), L ~ ( l ' ( 1 ,E)) is not a W*-algebra.

5.6 Hzlbert right G*-Modules

401

Proposition 5.6.4.10 ( 8 ) Let E be a W*-algebra, H a von Neumann right E module, and A a Fourier basis of H . For every E A denote by

<

the con Neumann right E -module (Proposition 5.6.2.3, Proposition 5.6.4.8 a ) ) and put

a) For every 17 E H

,

and the map

zs a n zsomorphism of von Neumann right E-modules (Theorem 5.6.4.7

a)).

b) If @ denotes the pretranspose of cp then

for every ( a , C) E E x G

c) For every p E Pr E put

and define

If E is finite-dimensional then H is isomorphic t o

a) By Propositiori 5.6.3.9 a

+ b , for every 6 E A ,

(vl<) = (vl<(
4 02

5. Hilbert Spaces

is surnmable in EE , SO that

47 is obviously linear and injective (Theorem 5.6.3.13 b).

Take

BE

C

E G . By Corollary 4.2.2.3 (and Proposition 5.6.1.2 b)), for every

Pf(A),

By Proposition 5.6.3.8 a), the family ({(F)FEAis summable in H H . P u t

Then

for every toE A . Hence

and p is surjective. Take 9, E H . By Theorem 5.6.3.13 f),

<

Hence p is an isomorphism of von Neurrlarin right E-modules. b) For every q E H ,

5.6 Hilbert right C*-Modules

so that

) . Theorem 5.6.3.13 a ) , ( < X ~ )
in H . Put

Let 17 E H and put

By 'Theorem 5.6.3.13 d),f), y E

a) e2(Tp,pE) and

p€Pr S

so that the map

is surjective. By Thcorerri 5.6.3.13 f ) ;

for all x, y E

a) e2(TP,pE). Hence the above map is an isomorphism of p€Pr E

Hilbert right E-modules.

Corollary 5.6.4.11 Fourier basis of H . a ) For every 17 E H

(8 ) ,

Let H be a Hilbert right M-module and A a

4 04

5. Hilbert Spaces

b) The map

is a n isomorphzsm of Hzlbert right M-modules.

c) All Fourier bases of H have the same cardinality and H m a y be endowed wzth the structure of a unztal Hilbert M-module. d) T w o Hilbert right M - modules are isomorphic iff their Fourier bases have the same cardinality. e) If H denotes the real Hilbert space obtained by endowing the underlying real vector space of H wzth the scalar product

(Proposition 5.6.2.5 a),e)) then A U ( A i )U ( A j )U ( A k ) is a n orthonormal basts of H .

a ) By Proposition 5.6.2.3 f), H is self-dual so that the assertion follows from Theorem 5.6.3.13 d),f) . b) follows from a ) and Proposition 5.6.4.10 c). c) and d) follow from b) . e) is easy to see. Proposition 5.6.4.12 ( 8 ) Take E E {R,C, M} , n E I N , and H a Hilbert right En,, module. For every z E IN,, put

a ) pl zs a mznzmal element of Pr E,,,\{O}

for every i E IN,.

b ) If 2L denotes the set of Fourzer sets A of H such that

for every ( E A , then every mazzmal element of 2L (U ordered by incluszon) is a Fourzer basis of H .

c) There zs a unzque cardznal number N such that H is isomorphic to e2(N,P I E,,,,) .

5.G Hilbert right C*-Modules

a) It is easy to see that p, E Pr E,,,\{O}. Then

(Corollary 4.2.7.6 a

Let q E Pr E,,,\{O}

4 05

with q 5 p, .

+ f) so that

for all j , k E INn. Hence q = p and p is a minimal element of P r E,,,\{O}. b) Let A be a maximal element of 2L. Assume A is not a Fourier basis of H . Then there is an Q E H such that

and

Since

there is an i E Nn with q p , # 0 . It follows ~ 1 ( 0 1 1 7 ) ~=1 ( V P I I V P I )

so that there is an

Put

and

For j , k E N, ,

(Y

> 0 with

f 0,

4 06

5. Halbert Spaces

so that

and

for every ( E A . This contradicts the maximality of A . c ) The existence follows from b) and Proposition 5.6.4.10 c). The uniqueness is obvious.

Remark. N is a kind of Hilbert dimension for H Proposition 5.6.4.13 ( 8 ) Assume E is the C'-direct product of a finite famzly ofunital C*-algebras and for every L E I , denote by e, the unit of E, . Further let G , H be Hilbert right E-modules. a) For every L E I , H e , is a Hzlbert right E-module and a Hzlbert right EL--modulein a natural way. b)

The map

is an isomorphzsm of Hilbert right E-modules and

is its inverse c) For every

5.6 Hzlbert right C*-Modules

Then ii E LE(G,H ) and i' = 71' for every

d) The map

is an isomorphism of Banach spaces e) The map of d) is an isomorphism of the C*-direct product of the family (Ls(He,))LEI of C*-algebras (Theorem 5.6.1.11 a)) onto L E ( H ). a) and b) are obvious.

c) ?or

((,I))

EGx H,

so that

d) For every v E L E ( G ,H ) and V , : Gel

Then for

(<,TI

E

L

E I , put

+ He,,

< 4 (v()e,

(Ge,) x ( H e , ) ,

= (
so that v, E LE,(Ge,,He,). Putting 7' := ( v J L E I ,

it follows

4 08

5. Halbert Spaces

for every

<

G , i.e.

-

u=v and the map is surjective. The other assertions are easy t o see. e ) follows from a),c),d).

Example 5.6.4.14 ( 0 ) Let G , H be Hilbert right E-modules. Then L E ( G ,H ) endowed with the bilinear map

and with the sesquilinear map ) L E ( G ,H ) x L E ( G ,H ) + L E ( G ) , ( u , ~cf

(uI~)

:= V*OU

is a Hilbert righ,t L E ( G )-module (Theorem 5.6.1.11 a)). Take u , v E L E ( G ,H ) and w, w l , w2 E L E ( G ) . Then

(Proposition 5.6.1.8 a ) , e ) ) , (ulu) = u'ou E LE(G)+ (Corollary 5.6.1.12),

and

(Proposition 5.6.1.8 f)). B y Proposition 5.6.1.2 g ) ; L E ( G :H ) is a Hilbert right E-module in the complex case.

5.6 Halbert right C*-Modules

4 09

Assume now IK = R . In the sequel we use the notation o f Proposition 5.6.1.9. By this proposition, L o ( G ,H ) and may be identified with the E coniplexifications o f L E ( G ,H ) and L E ( G ) ,respectively. Let u , v E L E ( G ,H ) . By the above, 0

-*

0

~ ~ ( 8 )

--

- -

0 5 ( u ,v ) o ( u ,v ) = ( u , -v1)0(u, v ) =

(Proposition 5.6.1.9, Lemma 2.1.5.3). Hence L E ( G ,H ) is a Hilbert right LE(G)-module. Definition 5.6.4.15 ( 0 ) Let F := ( F t ) t E rand G := ( G s ) s E Sbe families of Hilbert right E-modules. We denote by NGSFthe vector space of families (u.~,~)(S,~)ESXT

such that

for every ( s ,t ) E S x T and such that

is finite. For every u E N G , F , define u* E NF,G by

for all ( t ,s ) E T x S . If H := ( H r ) t E Ris a third family of Hilbert right E modules then for all u E N G , F and v E N H , G define vu E NII,Fby

for all ( r ,t ) E R x T . It is easy t o see that the above u* and vu are well-defined. If S := N, , T := N, for some m, n E IN and

for all ( s ,t ) E S x T then

5. Hzlbert Spaces

410

and U * and v u coincide with the corresponding notation in the matrix case (Definitions 2.1.4.23, 2.3.1.30). Hence NG,F has to be seen as a generalized matrix. The more natural notation will be reserved for the completion of N G , F (Definition 5.6.4.18).

Proposition 5.6.4.16 ( 0 ) Let F := (Ft)tET and G := (Gs)sESbe families of Hilbert right E-modules and for every u E N G , F define

LET

SES

a ) For every u E N G , F ,

LET

~ E S

(Proposition 5.6.4.1 c)), and

where for every s E S , us,. is the element of

NG,Fdefined by

for all (s', t ) E S x T

b) The map

zs linear and injective (bijective zf S and T are finite).

C) If H := (Hr)rER is a famzly of Hilbert right E-modules then for all u E NG,Fand v E N H , ~ ,

5.6 Hilbert right C*-Modules

d) If T is finite then

NF,F endowed with the norm

is a C*-algebra and the map

is an isomorphism of C'algebras.

a) Take

E a) Ft and 77 E

=

)x(~~ttt1strtl.l~) L E T s€S

Hence

Moreover,

a) G, . Then

SES

LET

=

1) ( F L I ~ ~ V ~=) LET sES

5. Hilbert Spaces

41.2

(Proposition 5.6.1.2 d), Proposition 5.6.4.1 b)). Hence

Ilu112 5 C

C lIustII

llusr

ll

=

SES t.tlET

C SES

From

it follows

Fix (s, t ) E S x T and take

tt E F? . Define <'E a) Ftc by t1ET

for every t' E T . By Proposition 5.6.4.1 b ) ,

<'E

a) Ftl (t.ET

,"

and so

It follows

( aF ~ ). #

Now fix s E S and take ( 6

tET

By Proposition 5.6.4.1 b),

5.6 Hilbert right C*-Modules

so that

It follows

b) It is obvious that the map is linear. By a ) , it is injective. Assume S and T finite and take

uE

CE

( ~ aE T l ~ ~ ~ S@ E SG S )

<

.

Fix (s, t) E S x T . For every E Ft and q E G , denote by Fp and by q' the element of a)G,, defined by

a

L'ET

<' the element of

s'ES

for every t' E T and s' E S . Define

Then for ( E Ft and

E G,,

Hence us, E L E ( F tG , , ) and it is easy to see that 6 = v . Therefore the map

LET

is surjective. c) Take

<E

a) Ft . Then tET

SES

5. Hilbert Spaces

414

so that

H

d) follows from a),b),c).

Corollary 5.6.4.17 Let H be a n infinite-dimensional Hilbert space and n E I N . T h e n the C'-algebras C(H) and K ( H ) are isomorphic to the C*algebras L(H),,, and K(H),,, , respectively (Proposition 5.6.4.16 d)).

Let A be an orthonormal basis of H . Since A is infinite, there is a bijection A -t A x IN,, so H is isomorphic to a)H (Proposition 5.5.1.22). Hence kEN"

C ( H ) (resp. K ( H ) ) is isomorphic to C

( m )( m P

K( QH)

kEN,

which is isomorphic to C(H),,,

kEN,

)

(resp. K(H),,, ) (Proposition 5.6.4.16 d)). H

Proposition 5.6.4.18 ( 0 ) In this proposition we use the notation of Proposition 5.5.7.10. For every E H , put

<

and zdentify H and e2(T) x e2(T) via the m a p

H

4

e 2 ( ~x) e 2 ( T ) ,

< c-f (FlrEz)

For each v E C(e2(T))2,2put :H

-+

H,

< c-f (UIIFI+ ~12F2,

UZI
+ ~22<2)

a) For every u E C(e2(T))2,2,

and the map L(e2(T))2,2-+ L ( H ) , v

c-f

11

is a n isomorphism of 6' -algebras (Propositzon 5.6.4.16 d)).

5.6 Hzlbert right C*-Modules

b) For every


(ull+~2i+u~j+u4k)(<~l+~i+~j+<~k).

If we define c~ : ( L ( ~ ' ( T , I R ) ) -+) ~ F ( L ( H ) ),

4

u 4u ,

then cpo$-' is an isomorphism of real C*-algebras (Example 5.6.4.2 a), Theorem 5.6.1.11 a)) and

p o $ - 1 ( L M ( e 2 ( ~ , Mn) K ( 1 2 ( T ,I H ) ) ) = F ( K ( H ) ) . c)

The complexification of L s ( e 2 ( T , M ) ) (of L H ( e 2 ( TM , ) ) n K ( e 2 ( T .M))) is isomorphic to L ( H ) (to K ( H ) ) .

a ) By Proposition 5.6.4.16 d),

ii E L ( H ) for every u E L ( e 2 ( T ) ) 2 , 2and the map

is an isomorphism o f C*-algebras. Take u E L ( ! 2 ( T ) ) 2 , 2and

< E H . Then

5. Hilbert Spaces

Since 6 E F ( C ( H ) ) iff Ga: = sz,it follows G E F ( C ( H ) ) iff

b) By a ) and Proposition 5.6.4.5,

for every u E (C(e2(T,R ) ) ) h n d cp and 4 are bijective and linear. Hence cp 0 I/-' is bijective, linear, axid involutive. Take a, b E Lm (12(T,El)) and put

+

c := cp(u)cp(v)=

with c E C(e2(T))2,2.Then

c22 = ~ 1 v -1 u2v2

-

u3v3 - uqv4

+ i(-uIv2

- UZVI -

u3v4

+ u4v3)= w1

-

iw2.

Hence

and cp o is an algebra homomorphism, i.e. an isomorphism of real C*algebras. The last relation is easy to see. c) follows from b) and Proposition 5.5.7.10 a ) .

5.6 Hilbert right C*-Modules

417

Definition 5.6.4.19 ( 8 ) Let F := ( F t ) t E TGr := ( G s ) s E Sand , H := ( H r ) r Ebe ~ families of Hilbert right E-modules. W e endow the vector spaces of the J o n N G , F with the norms

(Proposition 5.6.4.16 b)) and denote by we define the maps

MG,F

M G . ~ + L E ( tET Q F ~ .~c.7 0

the completion of N G , F . Further

~

~C ~ ).

U

by extending continuously the corresponding maps of Proposition 5.6.4.16 a)).

N (Definition 5.6.4.15,

T h e first m a p is linear and norm preserving, the second one is conjugate linear and norm preserving, and the third one is bilinear and continuoub.

Proposition 5.6.4.20 ( 8 ) Let F := ( F I ) I , =G~ ,:= ( G s ) s , = s ,H := ( H r ' ) r E Rbe families of Hilbert right E - m o d ~ l e s .

~ is a unzque famzly a ) For each u E M G , there

( ~ , , t ) ( ~ , t ) ~ . such $x~

for every (s,t ) E S x T and such that

for every

<E

b) For every u

a) Ft . LET

E MG,F

and with the notation of a ) ,

that

5. Hilbert Spaces

418

~ by where for every s E S , us,. is the element of M G , defined

0Ft

+

LET

0Gsc , F

((~F)s'~s.s')~'Es

s'tS

Moreover,

for all ( s ,t ) E S x T and

c ) For all u E M G , F and v E M H , ~ ,

COG = Z . d) M F , endowed ~ with the multzplzcation

M F , Fx M F , F+ M F , F , ( u , v )H uv and wzth the involution M F , F+ M F , F

uH

U*

is a C*-algebra. If T is finite then the map M F , F + C ~ ( ~ $ F LI ) u - 6 is as isomorphism of C*-algebras.

e ) M G , endowed ~ wzth the bilinear map M G , FX M F , F-+ MG.Fr

(u?)

UV

and with the sesguzlznear map

is a Halbert right MF,F-module.

f ) If we put 0

0

F := ( F L ) L E TG, := ( G s ) s ~ s (Propositzon 5.6.1.6) then the complexi~catzonof the C*-algebra M F , F is isomorphic to M;,;.

5.6 Halbert right C*-Modules

g)

For all u E M G , F A, such that

c S , and

B

c T , there is a unique

419

UA,B

E

MGVF

if ( s ,t ) E A x B (uA,B),,~

ij(s,t)$AxB

0

Moreover, the map

is a n operator (of n o r m 1 i f A and B are nonempty). h ) Put

order Then

by inclusion, and denote by

5 the upper setion filter of

1)32.

u = lim u A , ~ AxB,S

for every u E M G , ~ . i) Assume E is a W*-algebra and for every A E p f ( S ) and B E ' P f ( T )

put

Then

TA

E Pr LE

(5

G.) ,

TB

E ~r

( kt)for every A c s LET

and B c T ,

A W

for every u1 E a) Ft i n the topology of pointwise convergence o n LET

W

LE ( % LETF L

s@ ES G S )

,

5. Hilbert Spaces

and u = lim

for euery u E CE

+ . .

( a)Ft , W

W

LET

SES

a)G,)

TAOUOTB

in the topology of pointwise conver-

gence on a ) Ft . LET

j) Let (S,),,I and ( T L ) , E Ibe disjoint families in v ( S ) and v ( T ) , respec-

tively, and put

for every

L

E I . Let

such that (llIL1l)tclE % ( I ) and define vSt :=

u ~ if, ( s~,t )~E S , x T , for some L E I 0 if ( s ,t ) E S x T\ U ( S , x T,) LEI

for every (s,t ) E S x T , Then there is a unique v E MG,Fsuch that

for every

<E

a) Ft . Moreover, LET

a) The uniqueness is obvious. By Proposition 5.6.4.16 a ) , the m a p

is coritinuous for every ( s ,t ) E S x T . We may extend this m a p continuo~~sly to M G , ~getting , in this way a family ( T L , , ~ with ) ~ ~ ~ US.^ E L E ( F ~G,) ,

for every ( s ,t ) E S x T and

5.6 Hilbert right C*-Modules

< E t a) Ft with { t E T I EL # 0) holds for arbitrary < E a) Ft . for every

finite. By continuity, this relation

ET

LET

b),c), and d) follow from a) and Proposition 5.6.4.16. e ) follows from c),d), and Example 5.6.4.14. f ) is easy to see.

g) Take v E MG,E.and assume

UA,J

exists. We want to prove the inequa-

lity

Take

< E (t$Ft)X

and define q E

n Ft by

LET

By Proposition 5.6.4.1 b), 11 E ( t $ ~ t ) w

The linearity of the rnap

is obvious. By the above, its norm is one if A and B are nonempty. Now we prove the first assertion. The uniqueness and the case u E &,F are trivial. Let be a sequence in NG,F converging to v . By the above, ( ~ l ; l ? k is) ~a ~Cauchy ~ sequence in Nc,F. Put

v := lim u F b E M,,F n+m

BY b),

4 22

5. Halbert Spaces

for all (s, t) E S x T . It follows

Vs,t

=

us,,

if (s, t) E A x B

0

if (s, t ) $ A x B

for all (s, t ) E S x T . h) Take E > 0 . There is a v E NG,F such that IIu

-

vII

E

< -. 2

Take A E Y f ( S ) and B E Y f ( T ) such that

Then

""d by g) ,

Hence lim

UA,B

=u.

AxB,3

i) It is easy to see that n~ E Pr C E

( &c,) and

n~ E Pr CE

sFS

for every A

c S and B c T . For the assertion concerning

where

By Proposition 5.6.3.1 e) , h -

<, 7 ) " ~= ( a ,T B < , T A V )

m ~ ( a ,

for every A

c S and B c T , so that

we have to prove

. tFT .

U'

, we may assume

5.6 Hilbert right C*-Modules

4 23

-

T 9) .

lim (a, <e,, 9e*) - (a,

AxB,S

Take v E LE

(5F ~5, c ~ )

and

LET

E


> 0. Since the family

SES

(((ti I ( v * ~ ) Q ~ ) ), ~ E T is summable, there is a Bo E p j ( T ) such that

for every B E p j ( T ) , Bo c B . For every B c T ,

(Corollary 4.2.1.18). By Proposition 5.6.4.6 c), there is an AIJ E p j ( S ) such that I((v<ei I 9e2\*). a) I <

E

5

for every B c T and A E p j ( S ) , A. C A . Take A € p j ( S ) , A 0 c A , a n d B € p j ( T ) , B o c B . T h e n

4 24

5. Hilbert Spaces

so that

in the topology of pointwise convergence on

(

T h e assertion concerning u follows. j) For every L E I , we identify a) FL and LET,

a subspace of a) Ft and LET

W

W

IET

sES

LE @ Ft. O C , ) . a) G, in a natural way with ~ES,

a) G , , respectively. For

L

E

I and

<,

E

sES

(Corollary 5.6.1.12). Moreover, for every X E I\{&) and


E a) Ft , IET*

(GLCL
<E

a) Ft and for every

L

E

I , define

LET

2 := ( < L ) L E TE, LOFt ET Then for every J E Y , ( I ) ,

a) Ft ,

LET,

5.6 Hilbert right C*-Modules

For every A E

pf ( S x T ) and v ~ , $:= t

( s ,t ) E S x T , put

vst if ( s ,t ) E A if ( s ,t ) E S x T\A

0

Since (llll,ll),El E c O ( I ) it , follows from the above that

exists, where

for every

5

denotes the upper section filter of p f ( S x T ) ,

< E a) Ft , and tET

Since the reverse inequality is trivial, we have

ll;ll

= SUP Il'iiLll . LEI

Example 5.6.4.21 Assume that E is the C*-direct sum of a family (E,),,I of C*-algebras and for every L E I let 5, be an approximate unit of E L .Let H be a Hilbert right E-module and for each L E I put H,

:= {< E

H

I < = lim
Define

a ) G endowed with the right multiplzcation

and with the inner-product

is a Hilbert right E-module such that

for every

<EG

426

5. Hzlbert Spaces

b)


E H for every (' E G and the map

LEI

G--+H,

<-Ccl LEI

is an isomorphism of Hilbert right E-modules. c ) H is self-dual iff H, is sev-dual for every

Proposition 5.6.4.22

i

E I.

Let E be a C*-algebra and x

E such that

p:=x*x~PrE, q:=xxg€PrE. Define cp:pE--tqE,

Y++XY,

$:qE--+pE,

z-x'z.

Then 9 and i are isomorphisms of Hilbert right E-modules (Example 5.6.1.5), and E

L E ( P E ,q E ) , i E L E ( Q EP, E ) , P*= @ = c p - I .

First remark that

for every y E pE (Corollary 4.1.2.22 a + c), so that cp (and For y E pE and z E q E ,

i )is well-defined.

so that cp

L E ( P E ,q E ) ,

+ E L E ( Q EP, E ) ,

P* = * .

Moreover,

and so

E yE E, For y1,y2 E ~ and

( V Y II'pY2) =

Thus

ip

(XYI 1 ~ ~ = 2 Y;x'xYl )

= Y;YI = (91ly2)

and @ are isomorphisms o f Hilbert right E-modules.

5.6 Hilbert right C*-Modules

Proposition 5.6.4.23 such that

for all distinct

1,

427

Let E be a W8-algebra and ( p J r E l a family i n Pr E

X E I and

Put

and

for every x 6 E (Proposition 5.6.3.20 b + a). a) 5 E L E ( H ) for every x E E and 9:E

k' = 2.Define

+L E ( H ) ,

x

*+ Z

b) y is a n isomorphism of W*-algebras (Theorem 5.6.9.5 b), Theorem 5.6.4.7 a), Proposition 5.6.2.3) such that

for every ( a ,t,17) E E x H x H (Proposition 5.6.3.17 d)), where (;j denotes the pretranspose of p . Moreover, 27 we put P := ( P ~ ) ' E I

then

for every u E L E ( H )

ff

4 28

5. Hilbert Spaces

c) If there is a family ( x , ) , ~in~ E such that

for every

L

E I , then the W *-algebras E and

are isomorphic. a) For <,q E H ,

=

5 (5 E l

LEI

17:)

5 $

X ~ A= ~ A XEI

%) *
XEI

(Proposition 5.6.3.19 a), Proposition 5.6.3.17 c)). Hence Z E C E ( H ) and %' = x* . b) By a), cp is involutive. For x, y E E and E H ,

so that

i.e. cp is an algebra homomorphism. cp is obviously injective. In order to prove that cp is surjective take u E L E ( H ) and put

5.6 Hilbert right C ' M o d u l e s

Then for

< E H and

L

E I, E

(?Or

=P L X )

E
= PL C

XEI

E

1=
( ~ p ) f i

fie1

AEI

so that

Hence cp is surjective and E

cp-'"

=

C("P), LEI

For

X E

E,

LEI

XEI

E

=

)((9:s )< A , a ) = ) LEI

XEI

(17:x<x7a) =

LEIX E ~

E

=

):

E

( x l( ~ ~ 1 7 : = ) (x,

LEI XEl


,

LEI XEI

so that

LEI XEI

c) By Proposition 5.6.4.22, the Hilbert right E-modules E and pLE are rn isomorphic for every L E I and the assertion follows from b).

4 30

5. Hilbert Spaces

Definition 5.6.5.1 ( 8 for every (<, 9) E H x G ,

)

Let G, H be Hilbert right E-modules. W e define

Proposition 5.6.5.2

(8)

-

C

<(.19) : G -+ H ,

J(C19).

Let F, G, H be Hilbert right E-modules.

a ) For everg ( c , ~E) F x G , 9(.I<) E LE(F,G) ,

(9(.1<))*= <('19)

9

b ) T h e map

zs sesquilinear and continuous. C)

(<, 9) E F x G , x E E

* ( ~ x ) ( . l <=) ~ ( . l < x *. )

d) (<, 9) E F x G , u E LE(G, H) =+ u o ( ~ ( . I < )=) (v)(.I<). e ) (9, C) E G x H , u E LE(F,G) f)

* (C(.Iq)))ou = C(.Iufv) .

( F , ~ I ) E F ~ G , ( ' ~ , C ) E G ~ H * =+ (C(.192~)0(91(~1~)) = (C(91l92))(.l<).

g)

(<>v) E F x (2 =+ (9(l.<))*o(v(.l<))= ( < ( 9 1 9 ) 9 ( . 1 < ( 9 1 9 ) ~ ) . 0

0

h) If F, G denote the complexifications0 of F and G , respectively (Proposition 5.6.1.6), then for all (<,,
+

(71,92)(.1(<1,<2))= (VI(.I
where we identified Lg(;,

8) wzth E E ( ~G),

i)

< E F * <( . I < )

j)

If F, G are Hilbert E-modules then for (<, 9)

(Proposition 5.6.1.9).

E L E ( F ) +.

x(7(.l<)) = (x9)(.l<)

7

F x G and x E E ,

(9(.IO)x = 9(.lx-<)

where we used the notation of Proposition 5.6.1.8 g ) .

7

5.6 Hilbert right C*-Modules

431

k ) If E is finite-dimensional then

v(.l<) E L f ( F , G ) for every ( 1 , ~E) F x G .

1) Let (71) r (52,712)E F

X

G such that 771(.111) = 712(.1<2)

and such that (ql lvl) and (t2(F2) are invertible in E

EI = <2x,

~2

= VIX*

.

Then

,

where

x := ((1 It2)(<21<2)-' . rn) Let

( F t ) t E ~ and

OF,,

ttT

(G,),vEs be families of Hilbert rzght E-moduls and

<E

V E a>Gs. Put SES

order TI by inclusion, and denote by Then

5 the upper section filter of

Dl.

= (
(Proposition 5.6.1.2 a)), so that o(.I<) E LE(F,G ) , (v(.\<))*= <(.b?). Moreover, by Proposition 5.6.1.2 a),e), l l ~ V ~ ~ l 0 ~= 1 0llv~
4 32

5. Hilbert Spaces

so that, by Proposition 5.6.2.1 a) and Proposition 5.6.1.2 e),

b) follows from a ) . toE F ,

c) For every

= v(
(Proposition 5.6.1.2 a)), so that ( ~ x ) ( . I o= v(.I<x*). d ) For every

toE F ,

(vo(v(.lO))<~ = v(v(<~lr))= (vv)(<~l<) = ((vv)(.l:))
toE F , ((C(.lll))ou)
so that

5.6 Halbert right C' -Modules

f ) follows from d) . g) BY a), f ) , and c ) ,

= (vI(.I
(Proposition 5.6.1.9), so that (71, 7l2)(.I(
i) For every 7 E F ,

so that

by a) and Theorem 5.6.1.11 cz j) For <' E F ,

3

cl .

433

4 34

5. Hilbert Spaces

so that

The other relation is easy to see. k) We have

Dim Im q(.1<)

< Dim E < m .

1) We have

vz

= V I ( G I < I ) ( < Z ( < ~ ) - ~= V I X * .

It follows by c),

v1(.1<1)= '72(.1<2)

= 711~*(.l<2) = ' 7 1 ( . 1 < 2 ~ ) ,

m) By Proposition 5.6.4.1 e), there are A. E that

pf(S) and

Bo E

pf( T ) such

5.6 Halbert right C*-Modules

4 J,'s'

for every A E Q,(S), A. c A , and B E p,(T), B o c B . By a ) (and Corollary 4.2.1.18) for such A and B ,

Definition 5.6.5.3 ( 8 ) Let G , H be Hilbert rzght E -modules. We denote by K E ( G ,H ) the closed vector subspace of L E ( G ,H ) generated by

(Proposition 5.6.5.2 a ) ) and put

Corollary 5.6.5.4

a ) If

uE

(8)

L E ( KG ), v U*

E

Let F,G , H , I be Hilbert right E modules.

K E ( G ,H ) , w E L E ( H ,I ) then

E K,y(H,G ) ,

wouou E K,y(F, I ) .

b) K , y ( H ) is a closed ideal of L E ( H ) and so a hereditary C*-subalgebra of

LE(H). c ) If G , H are Hilbert E-modules then K E ( G ,H ) is an E-submodule of L E ( G ,H ) (Proposition 5.6.1.8 g)). d ) If E is finite-dimensional then

5. Hilbert Spaces

436

0

0

0

f) If we identify c E ( G , H ) with LE(G, H ) using the m a p

of Proposition 5.6.1.9,t h e n 0

0

0

ip(KE(G, H ) ) = Kh(G, H ) > 0

0

i.e. KE(G, H ) and Ki(G, H ) m a y be identified. a) b) c) d) e)

follows from Proposition 5.6.5.2 a),d),e) . follows from a), Theorem 5.6.1.11 a ) , and Proposition 4.3.4.5 e) follows from Proposition 5.6.5.2 j) . follows from Proposition 5.6.5.2 k) . For every v E K E ( H ) , llvou - u1I2 = II(v0u - u)o(vou - u)*II =

(Proposition 5.6.1.8 c),e),f)). By b), K E ( H ) has an approximate unit

5 so that

lim Ilvou - ull = 0 v.3

(Proposition 2.3.4.9). By a), u = limvou E KE(G, H ) . v.3

f ) For (E, 71)E G x H , P(~I(.Ic). 0) = ( ( 0 . 0 ) ( I

K, 0)))

K;(&.

h)

so that m ( k E ( ~H)) .

c~

h.

~ ( 6 .

0

Take ((1, (2) E G and ( q , 71~) E H . Then

and by a simple calculation,

+ v2(.1<2),7I2(.IR?)(.l(<17<2)). ~(7Il(.l
0

P ( ~ E ( GH, ) = x;(G. H ) .

5.6 Hilbert right C*-Modules

(8)

Proposition 5.6.5.5

a ) For every finite family ((x,, Y,)),,~ in E x E ,

l I L1E I I1 l I L E I I1 x, (. 1 y,)

=

):x, y:

(Example 5.6.1.5).

b) KE(E) = {x(.(y)Ix,y 6 E ) (Example 5.6.1.5) c)

The map

is an isometry of C*-algebras (Corollary 5.6.5.4 b)). :r)

1

-&(.ly,)

1 1 (5 '1 I I/ I = sup z€Eff

= sup Z(E#

x~""~))

Cx,(z1y,) .(I

= sup

zcbff

=

):x,y:z

I/

=

(Proposition 4.1.1.12). 11) and c) . Let 3 be the vector subspace of C E ( E )generated by

By a), the linear rnap

is wc:ll-defined and preserves the nornis. 'Take x E E . By Corollary 4.2.4.14, there is a y E E such that x = y1x1; . Then

so that cp is surjective. In particular, 3 is complete so that

438

5. Halbert Spaces

K E ( E )= 3 and K E ( E ) = {x(.IY)Iz,

Y 6E).

For a , b , c , d E E , ip((a(.lb)o(c(.Id)))= cp(a(clb)(.ld))= = p(ab'c(.ld)) = ab'cd' = p(a(.lb))p(c(.ld))

(Proposition 5.6.5.2 f ) ) , p ( ( a ( . l b ) ) ' )= p(b(.la)) = ba* = (ab')' = ( p ( a ( . ( b ) ) ) * (Proposition 5.6.5.2 a ) ) , so that p is an isometry o f CW-algebras.

Proposition 5.6.5.6 a ) ( < , x )E H x E b ) For every

(8)

Let H be a Hzlbert right E-module.

J x(.l<) =

(.I<x*).

<EH, (.I<) = 1 : ~ ~ ( . 1 <E )K E ( H ,E )

where c)

5

is an approximate unit of E .

The map

c * ('It)

-+ K E ( H , E ) ,

is a conjugate lznear isometry of real Banach spaces with inverse

K E ( H ,E ) where

5

+H

, u

c-,lirn

r.3

u'x,

is an approximate unit of E .

d ) Put

C:E-+H, for every

~c-t<x

< E H . Then :=

(.I<)*

5.6 Hilbert right C*-Modules

for every

<E H

and the map H

---t

W E , H) ,

<

-

F

is an isometry of Banach spaces with inverse K E ( E , H ) 4 H , u++limux, 1,3

where

5 is an approximate unit of E .

e) If E is unital then KE(E, H ) = C E ( E ,H ) = L E ( ~H, ) ,

a ) For every 17 E H , (x(.lO)v = ~(7110= ( ~ I t x * ) (Proposition 5.6.1.2 a)), so that x(.IC) = (.I<.*) . 1)) By a) and Proposition 5.6.2.1 a),

lIx(.lO - (.loll = ll(.ltx*) - (.loll = = ll(.ICx* -

9 ,11= IKx* -
for every x E E so that

(Proposition 2.3.4.9, Proposition 5.6.1.2 f)). (1) By b) (and Theorem 4.2.8.2), the map is well-defined. It is obviously conjugate linear and by Proposition 5.6.2.1 a), it preserves the norms. By a ) , the map is surjective. Take u E K E ( H ;E ) and let E H such that

<

For X E E and Q E H ,

440

5. Hzlbert Spaces

so that

llu'x - tll

I l l t x - Jll .

It follows lim u * x =

<

2,s

d ) By Proposition 5.6.2.1 a ) ,

for every J E H . By Proposition 5.6.1.8 a),b),d) and Corollary 5.6.5.4 a ) , the map

is a conjugate linear isometry of real Banach spaces, and so the assertion follows from c) . e) The equality

follows from c) and Proposition 5.6.2.1 b). This equality implies

K d E , H ) = C E ( E ,H ) The equality

C E ( E ,H ) = & ( E , H) follows from the Proposition 5.6.2.3 and 5.6.2.4. Remark. The equality

does not hold in general as it was shown in the remark of Proposition 5.6.2.4.

5.6 Hilbert right C*-Modules

441

Proposition 5.6.5.7 Let H be a Hilbert right E module, <,17 E H , and a E IK\{O) . W e denote by F the C*-algebra L E ( H ) (Theorem 5.6.1.11 a)). T h e following are equivalent:

b) a $ aF(<(.Iv)) (Proposition 5.6.5.2 a)).

If these condztions are fulfilled then

( a 1 - (
1

+ ((14)

where ( := ( a 1 - <(.Iv))-'<.

First assurrie that a) holds. By Proposition 5.6.5.2 f ) , 1 ( a 1 - <(.lv));(l

=1

Hence cr

-

+ < ( a 1 - (
1 -<(I - ( a 1 - (
4 U F ( < ( . I ~ )and ) (a1

-

1 <(.Iv))-l = -(I a

Now assume that b) holds. Then

+ < ( a 1 - (
442

5. Hilbert Spaces

s = ( a 1 - <(.Iv))C = ffc

-

<(CIv)

so that

1 (a1 - (Clv));(l

1

1

+ (clv)) = 1 - ;(slv) + (Clv) - ;(
=1

By Proposition 2.3.1.14 and Proposition 5.6.5.2 a ) , 54 a~(v(.lr)).

Put

6 := ( a 1 - v(.ls))-Lv. Then

so that (
-

1 (EIv)) = 1 - ,(51v)

1

+ (Elio) - ;(El
4 aE((
Proposition 5.6.5.8 Let H be a Hilbert right E-module. Put 71

f

V

:= UOU

for all u , u 6 L E ( H ) and denote by F the C*-algebra obtained from the C * algebra C E ( H ) (Theorem 5.6.1.1 1 a)) by reversing the multiplzcation i.e. h y endowing F with the multiplication

5.6 Hzlbert right C*-Modules

443

a ) H endowed with the right multiplication

and with the inner-product

H x H --+ F ,

( < , q ) ++ (C

t 7 ) := <(.Iv)

+

is a Hilbert right F-module, denoted by H , such that its norm coincides with the initial norm of H . b) For every x E E , the map +

F:H+H,

+

<-<x

+

belongs to L F ( H ) and the map

is an involutive algebra homomorphism with kernel

If there are <,g E H sucll that (Elg) is invertible i n E then this map is surjective. +

c)

Let G be the C*-algebra obtained from the C*-algebra L F ( H ) by reversing the multiplication, i.e. by endowzng G with the multiplicatzon

Then H endowed ulzth the right multiplzcation

and wzth the inner-product

t

is a Hilbert right G -module, denoted by H , such that its norm coincides with the initial norm of H ,

<

t

for all <,q , E H , und L E ( H ) is a unital C*-subalgebra of L G ( H ). C Moreover, for every u E & ( H ) and <, g, E H ,

<

444

5. Hilbert Spaces

C((~t171)- (
In particular, if {x E El< E H

==+[x = 0)

= (0)

then

a ) We have

for all [ E H and u , v E F . By Proposition 5.6.5.2 a), the map

is sesquilinear and Her~nititansuch that llC(.l~)Il= ll([lOIl

=

11<112

for every [ E H and by Proposition 5.6.5.2 i),

< E ff * <(.I<) E F+. By Proposition 5.6.5.2 d ) ,

for all [, 7 E H and u E F . Take [, q E H . By the above and Proposition 5.6.5.2 h),

5.6 Hilbert right C*-Modules

(Proposition 5.6.5.2 c)), so that --t

I E L F ( ~ ) , Z* Take x, y E E . For every

-

= X*

.

<EH,

.ye = < b y ) = (@)Y = @l= ( I -r i4< so that

.y=.tfj Hence the map + cp: E + L v ( H ) ,

XMI

is an involutive algebra homomorphism. The relation Kercp= {x E E I J E H = = + - < x = 0 ) is easy to see. + We prove now the last assertion. Take u E Lr..(H) and put x := (
CEH, (uC)(.I7) = ( 4 T 7) = ( I T ~ ~ 1 = 7 C(.lllt71), )

Hence u = I and cp is surjective. t c) By a ) , H is an Hilbert right G-module such that its norm coincides with the initial norm of H . Take u E L E ( H ) . Then u E F , so that by b), ii E L c ( k ) . But

<

t

for rvcry E H . Hence 6 = u and L E ( H ) is a subset of L G ( H ) . By b) , t L E ( H ) is a unital C* subalgebra of LG(H). We have

446

5. Htlbert Spaces

+

Hence for every u E L G ( H ) ,

Proposition 5.6.5.9 ( 8 ) Let F := (Ft)LET and G := (Gs)sESbe families of Hilbert right E modules and

Put

H:= a)Ft,

OGs, sES

LET

and

for every u E

.

Define

EM:={Ax B I A E p f ( S ) , B E P f ( T ) } , order EM by znclusion, and denote by

a ) Let

<E H

5

the upper section filter of EM.

and 17 E K and for every A x B E EM define

Then GA,B= (veS1)(.I<ez) for any A x B E EM and 71(.1<)

=

lirn i i ~ , ~ . Axfl.3

UA,B

PC,F by

5.6 Hilbert right C*-Modules

44 7

b) KE(H, K ) is the closed vector subspace of CE(H,K ) generated by {ulu E

Pc,F}. a ) For AX BE^ and ( E H ,

so that

By Propositiori 5.6.5.2 m), 17(.1<) = lim hArB . Ax&3

b) Let F be the closed vector subspace of LE(H, K ) generated by {%\uE (Proposition 5.6.4.16 a)). By a),

Take ( a , b ) ~ S x Tand ( < , v ) E F , x G b . D e f i n e and u E

by

for all (s, t) E S x T . Then for any ( E a) Ft , L E T

so that

FE

a ) F t , i j € a)Gs, tET SES

5. Hzlbert Spaces

448

It follows

LET

for every v E

sES

such t h a t

for all ( s , t ) E S x T . We deduce further

( a ~ tmcs) , .

~ C K EtET

sES

Proposition 5.6.5.10

(8)

W e use the notatzon of Proposition 5.6.4.13.

a) 12 E K E ( G ,H ) for eve1 y

and the m a p

n

x L , ( G e , , He,)

4 K E ( G ,H

)

LEI

zs a n isomorphism of Banach spaces

K , ( G e , , He,) = &,(GeL, H e L )n K(Ge,, He,) for every

a ) Fix

L

E

L

E

I then

I , let ( ( ( A v, ~ ) ) ~be, ,a finite family in (Ge,) x ( H e , ) and put

u, = 0 for cvery n E I \ { L,)and

5.6 Hzlbert right C*-Modules

Then

11 :=

V X ( . I < XE) KE(G,H ) . XEL

By Proposition 5.6.4.13 d), 11 E KE(G,H ) for every

~ family in G x H and put Now let ( ( [ A v, ~ ) ) bex a~ finite v := ~ V X ( . l < X7 ) XEL

U L :=

2V A ~ ~ ( . I < A ~ L ) XEL

for every

L

E I and

Then for every [ E G ,

LEI

LEI XEL

so that

and the map is surjective. The other assertions are easy to see. b) follows from a ) and Proposition 5.6.4.13 d).

Proposition 5.6.5.11 ( 8 ) Let H be a Hzlbert right E - module, p E P r C E ( H ) , and A a finite Fourier set of I m p (which obviously is a Hilbert right E -module).

5. Hilbert Spaces

450

b) If E is C-order a-complete and p is a minimal element of P r LE(H)\{O), then there is a E I m p with

<

c) If E is C-order a-complete and A is a Fourier basis of I m p then

a ) By Proposition 5.6.5.2 a),f) and Proposition 5.6.3.9 a

+ b,

<(.I<) FEA

selfadjoint and

so that

Moreover,

(Proposition 5.6.5.2 d ) ) , so that by Corollary 4.2.7.6 d

b) follows from a ) and Proposition 5.6.3.10 a),d) c) Take 17 E H . For every E A ,

<

+a,

is

5.6 Hilbert right C*-Modules

= (7710 - (7710 = 0

so that by Theorem 5.6.3.13 b ) , (P-

,

,(.)

=0

It follows

Proposition 5.6.5.12 ( 8 ) Assume IK = R and take E :=C ( E := M). Let G and H be Halbert rzght E-modules, [ E G , 77 E H , and put (1

a) For s , t

:= (,

(2

:= ( i ,

771 := 7 ,

[ I := ( ,

(2 := ( i ,

6 := (3,

F4 := [ k

771 := 77,

;r12 := 772,

q3 := v j ,

q4 := qk

Ri2 (s, t E IN4) ,

a) is easy to see. b) We do the proof only for E = M . For

Hence

C E G:

771

re(CI(1) = ore(CI0

772

re(CIG) = vi re(-i(ClO),

773 re(Cl(3)

so that

q2 := qi

>

= 773 re(-j(CIE))

,

452

5. Halbert Spaces

Proposition 5.6.5.13 Let E be a W*-algebra and H a von Neumann right E - module.

a) K E ( H ) is C-order complete b) For every u E K E ( H ) , the space

(Definition 5.6.3.2) is compact.

c) Re K E ( H ) and K E ( H ) + are closed i n (KE(H)), . d ) A family ( u , ) , ~i n~ KE(H)+ zs summable i n ( K E ( H ) ) ~iff

is upper bounded i n K E ( H ) and zn this case

e) If A is a Fourier basis of H then

for every u E C E ( H ) , where the lzmits are considered i n (LE(H))/,

a ) C E ( H ) is a W * -algebra (Theorem 5.6.3.5 b)) and so it is C-order complete (Theorem 4.4.1.8 c)). Since K E ( H ) is a hereditary C*-subalgebra of C E ( H ) (Corollary 5.6.5.4 b)), it is also C-order complete (Theorem 4.4.1.2, Corollary 5.6.5.4 f)). b) Let v E L E ( H ) with v'v < u'u. Since K E ( H ) is a hereditary C*subalgebra of L E ( H ) (Corollary 5.6.5.4 b)), we have v'v E K E ( H ) . By Corollary 5.6.5.4 e), v E K E ( H ) . Hence

By Proposition 4.4.2.23 and Theorem 5.6.3.5 b),

5.6 Hilbert right C* Modules

is compact so that

is also compact. c) By Proposition 4.4.1.3 (and Theorem 5.6.3.5 b)), ReLE(H) and L s ( H ) + are closed in ( L E ( H ) ) H. Hence ReKE(H) and K E ( H ) + are closed in (KE(H)),, d ) If ( u , ) , ~is~s~immablein (KE(H)),, then by c),

for every J E

Pf( I ) . Hence

is upper bounded. Kow rrsslime

is upper bounded. Let 3 , 6 be liltrafilters on !Qf (I) finer than 8 1 .By b), and 6 converge in (KE(H)) to u arid v , respectively. By c),

3

for every J E P r ( I ) SO that u 2 v . By symmetry u = v . It follows that converges in (KE(H)) , i.e. the family (u,),,~ is summable in (KE(H)) and

,,

e) Take (a, g , C) 6 E x H x H . Then for every B E !J3 (A) ,

,,

4 54

5. Hilbert Spaces

(Proposition 5.6.5.2 d ) , e ) ) .B y Theoreni 5.6.3.13 f ) ,

so that

Proposition 5.6.5.14 Let E be a W'-algebra, H a von Neumann right E-module, and ( G L ) r EaI family of Hilbert right KE(H)-modules (Corollary 5.6.5.4 b)). Put

a ) If <, 17 E G then ( K E ( H ) ) ,. Put

<+

TI

E G and the famzly ((
5.6 Hilbert right C*-Modules

b) G endowed with the right multiplicatzon

and with the inner-product

is an inner-product right KE(H)-module. If G, is self-dual for every L

E I , the it is even a Hilbert right KE(H)-module.

a ) By Proposition 5.6.1.2 b),

<

for every L E I . By Proposition 5.6.5.13 d), + 17 E G . If IK = C then by the Polarization Identity (Proposition 2.3.3.2 d ) ) , the family ((
for every J E P f ( I ) (Proposition 5.1.5.13 d ) , Corollary 4.2.2.3) so that Moreover,


G.

456

5. Hilbert Spaces

(Proposition 5.6.5.13 c)), (([I<)

+ (717)

7

(vl<) -

(
(Proposition 4.4.1.3, Theorem 5.6.3.5 b), Corollary 5.6.5.4 f)), = ( < L ~ ) ~=E (<()LEI I =

<

1

Now we want to show that G is complete. By Proposition 5.6.2.12 (and Corollary 5.6.5.4b)), for every L E I it is possible to define a right multiplication

such that G, becomes a Hilbert right LE(H)-module. Put

By Proposition 5.6.4.6 b), F is a Hilbert right &E(H)-module so that it is sufficient to show that G is closed in F . Let E F and let (<(")),,N be a sequence in G converging to ( . Since K E ( H ) is closed in LE(H) it is s~ifficent to show that converges to (
<

= lI(<(")l<(")- <)I1 + Il(<(") -


116")
(Proposition 5.6.1.2d)) so that

(
=

lim (<(")I<("))E K E ( H ) .

n+m

-

5.6 Htlbert right C*-Modules

457

Proposition 5.6.5.15 Let E be a W*-algebra, H a v o n Neumann right E module, G a self-dual Hilbert right K E ( H )-module ( C o r o l l a y 5.6.5.4 b)), and A a Fourier basis of G . For every

and denote by G the closed vector subspace of G' generated by

Define (Proposition 5.6.5.14, Proposition 5.6.2.3)

:=

{x E

n

( < l f ) K a ( H )I ( X ; X < ) ~ , A 2s smrmable i n ( K E ( H ) ) ~ ,

EEA

a) G# zs compact. G

b ) For every

the family ( < x ( ) < is ~ ~summable i n G

G

,

for every 7 E A , the map

is a n isomorphism of Hilbert rzgttt K E ( H )-modules, and

5. Halbert Spaces

458

W'

G

aI

7I

FEA

-

((~II<))[€A

is its inverse. In particular, for 71,C E G ,

c ) If F is a self-dual Hilbert right Ks(H)-module and

then the map G -tF G

F

,

J-u<

is continuous. In particular,

for every 71 E G . a ) Let

for every

5 be an ultrafilter on G# . Take

< E G . Then

E G# (Proposition 5.6.1.2 c ) ) . By Proposition 5.6.5.13 b),

<- := lim ((1~) r1>3 exists in ( K s ( H ) ) and

I I ~ I I 5 11<11

. Hence for

5.6 Hzlbert right C*-Modules

= 1;y

(((ll")"lP)?

a) .

It follows for u E K E ( H ) ,

((GQlb), a) = lim (((Euln)alP),a ) = vs

- -


Since G is self-dual, there is a

< E G# (El0

for every

=

< E G . For every ( a , a , P , [ )E

so that

with

5 converges

to

< in G

b) For every B E q f ( A ) ,

G

so that (Proposition 5.6.5.13 d))

E X

H x H x G,

. Thus G# is compact. G

4 60

5. Hzlbert Spaces

Let 5 be an ultrafilter on !J?,(A) finer than z A . By a ) , there is a such that

CE G

in C . For q~ A and ( a , a , P ) E E x H x H , G

so that

Since q is arbitrary and K E ( H ) is C-order complete (Proposition 5.6.5.13 a)), ib ~ summable A does riot depend on 5 (Theorem 5.6.3.13 b)). Hence ( < x ~ ) ~ in G and C

for every 77 E A . Now take q E G . Then

for every

<EA

(Proposition 5.6.3.9 a

+ c) and

for every B E p f ( A ) (Proposition 5.6.3.12 b)). By Proposition 5.6.5.13 d),

5.6 Hilbert right C*-Modules

By the above, G

0 = a)<(01<). EEA

Hence the map

is bijective and

is its inverse. Nowtake o , ( ~ G . F o r ( a , c u , / 3 ) ~E x H x H ,

Hence

In particular the map

is an isomorphism of Hilbert right KE(H)-modules. c) Take

Then for ( E G ,

4 62

5. Halbert Spaces

and this proves the continuity. T h e final assertion follows from b ) . Proposition 5.6.5.16 Let ( F ) F E 3 be a family i n {R,C, M) , ( T F ) F Ea~ family of sets, and E the C ' d i r e c t s u m of the family

For every F E 5, we identify K ~ ( ~ ! ~ (FT)F) ,with the corresponding closed ideal of E . For F E 5 and s , t E T F , put

a) E is (C-order complete. b ) If F, G E

5,

q, r E T F , and s , t E

e:e:

TG then

= brcb,,e;,

c ) If IK = C ( I K = R), F E 5 , and t E T F then e i is a minimal element of Pr E\{O) and e{ E e{ (e{(Re E ) e & ) is one-dimensional. for every d ) Denote by T the disjoint union of the family of sets t E T denote by Ft the element F of 5 with t E T F and put

If 6 is the upper section filter of p f ( T ) and if cp denotes the m a p

then 4 6 ) is an approximate unit of E .

5.6 Hilbert right C*-Modules

4 63

a) By Proposition 5.6.5.13 a ) (and Proposition 5.6.2.5 f)), KF(e2(TF,F ) ) is C-order complete for every F E 5 so that E is C-order complete. b) is easy to see (Proposition 5.6.5.2 a),f). c) By b) , e& E P r E\{O) . Take q E Pr E with q 5 e & . Then q = eiqef; = (ef (.le:'))oqo(ef (.lef)) =

= (er(qefleF))(.Ief) = (ef (qef lqef ))(.Ief)

(Corollary 4.2.7.6 a

there is an cr E

+ f, Proposition 5.6.5.2 d),f)). Since

R+ such that q = @eft

Then

(Proposition 4.1.2.21). Hence e: is a minimal element of Pr E\{O) . By a ) and Corollary 4.3.2.12 a + d , e&E e& ( e & ( ~~e) e ; ) is one-dimensional. d) Take F E 3, 1E e2(Tp,F) , and denote by TT, the upper section filter of CJ3/(TF). For every t E TF ,

so that (xe;) tES

for every S E ~ , ( T F ) It . follows

(Example 5.6.4.2 e)). In particular

for all

t,71 E e2(TF,F) . Hence

t = tea

4 64

5. Halbert Spaces

for every u E KF(e2(TF,F ) ) . The assertion follows. Remark. A C'-algebra E is of the above form iff there is a faithful representation ( H , p ) of E such that cp(E) c K ( H ) (Theorem 6.3.6.14). Proposition 5.6.5.17 W e use the notation of Proposition 5.6.5.16 and denote by H a Hilbert right E-module. Take

and denote by M the set of Fourier sets A of H such that

Let A be a maximal element of M and put

for every F E 5 , where positzon 5.6.2.3).

(
a) A is a Fourier basis of H and H is isomorphic to

(Example 5.6.4.21). b ) For every

there is a A E em(5) with

JIuII = llXlloo

and

5.6 Hilbert right C*-Modules

for every 17 E H , where

for any F

E

5. In particular, if Card 5 = 1 then

where 1 is the identity map of H . a) Assume A is not a Fourier basis of H . Then there is an 7 E H with

By Proposition 5.6.5.16 d) and Propositiorl 5.6.1.2 f), there is an F E an s E TF with

17e;: # 0 . Then

eL(17l.l)e:; = (WS, I 17e,F,)# 0 . By Proposition 5.6.5.16 c), there is an a > 0 such that F

I.'

ess(71117)e:;= ae,, Put 1 < := -77e" &

","

'

By Proposition 5.6.5.16 b),c), 1 ,. I nett,) = ;ef~,.(nln)e:,, 1 ((10 = ;(71e,5,t, -

For < E A ,

F

eFeF

e t F , ~S S

S,fF

-

- efF.LF

'

=

5 and

4 66

5. Hilbert Spaces

Hencc A U {C) E 2l and this contradicts the maximality of A . By Proposition 5.6.5.13 b) and Example 3.6.4.21, H is isomorphic to

b) Put

B

:= {( E

H 1 ([I<)

E PIE)

and

xc := (u
for every

< E B (Proposition 5.6.3.9 a * b) . For < , q E B , (~<)(.171) = ~0(<(.177)) = (t(.Iv))ou = J(.lu*v)

(Proposition 5.6.5.2 d),e)) so that (uE)(vIv) = <(vIu*q)= 5 ( ~ 1 =~J )x ~ .

If (
-

x,) = <x<- (x,, = U(

Xc

Let F E 5 and

=0,

.

< E AF. By the above, we may define xr: := xc ,

Then

= x,,

- U<

1

Xp

:= (xFez

I e;)

E

F.

5.6 Halbert right C8-Modules

=

Take p

F with

(eL(x~ef, I eL))('le;)

F

= ( ~ L X F )('leiF).

IIpII = 1 and put v

:=

t ((4;~)(.leE))

.

Then

('Iv)

=

((.fFp) (.Ie&))' (
= ( e & l I ~ I l ~(. ) I ef;,)

= e&.tF

.

It follows U7] = ~ X F = v ((e:XF)

=

(' I )e:

< ( e r F p b ) (. I e[.)

=

.

On the other side, ~ 7 1 =( 2 1 0

((e:.~)

(. I )e:

=

=

4 68

5. Hilbert Spaces

=

< ( ~ G X F P (.) I e % )

so that, successivley,

I (e,F,pXF) (. I

e f , ) = f ( ~ ~ X F (.P l )e r F )

uf=fxF

By Proposition 5.6.5.15 b),c), for q E H ,

In particular,

lXFl 5

llull . Since

5.6 Hzlbert right C*-Modules

The relation

where

is easy to see. Proposition 5.6.5.18 note by F F ,the ~ set of U'

( 8 ) W e use the notation of Proposition

:= ( ~ L : , ) ( ~ , S ) E T ~ SE

n

Kt.(Ft,

5.6.5.9, de-

Gs)'

(t,s)E7'xS

such that the map

{~Iu

E

PG,F) +K ,

(us[,uis)

4 (s,t)ESxT

is continuous (Proposition 5.6.4.16 b)), and for every u' E FFIcdenote by ;I' the extension of this map to LET

stS

(Proposztion 5.6.5.9 b)) and set

a) T h e map

is linear and bijective.

b) For every u' E

3)7,G1

ul* E F G , F and

c) Let u' E .FG,,v and A C T x

and Then v' E FG,F

- -

u'* = u'

S and define

1121'11 1 111711

*

4 70

5. Hilbert Spaces

a ) Only the surjectivity needs a proof. For every ( a ,b) E S x T and u,b E KE(F~ G a>)

P U ~

cab

a) F t LET

a)

G s r

SES

c

(bsa~ab
Take

LET

sES

and define

for every ( t ,s) E T x S and

For u E PC,Fand [ E a) F t , LET

so t h a t

It follows

Hence for every u E PC,F.

-

0 = u' and the m a p of a ) is s~irjective.

5.6 Hilbert right C*-Modules

b) For u E P F , ~ ,

-

- - = (u*,u') = ( 2 ui) , = (G,

-

Z*) .

-*

Hence u'* E 3G,F and u1*= u' . c ) In fact, v' is the restriction o f ;I' to

Proposition 5.6.5.19

( 8 ) Let

S , T be sets. For every z E E ( ~put~ ~ )

and denote 69 3 the set of x' E ( E ' ) T X Ssuch that the map

is con.tinuous (Proposition 5.6.1.8 h), Proposition 5.6.4.16 6)) and for e v e q x' E 3 denote by 2 the extension of this map to K E ( ~ ' ( TE, ) , ['(STE ) ) (Proposition 5.6.5.6 d), Propositzon 5.6.5.9 6)). Then ( E ' ) ( T X SC) 3 and

(2I x'

E (E')~~')

is dense in

The inclusion ( E ' ) ( T X SC) 3 is easy t o see. Take

and E > 0 . By Proposition 5.6.5.18 a ) (and Proposition 5.6.5.6 d ) ) ,there is an x' E 3 with x' = 0 . For A c T x S put

4 72

5. Hilbert Spaces

xi, x',:TxS-+Et,

if ( t , s ) E A

( t , s ) ~ 0

if (t, s) E T x S\A

Since {Zlx E E(sxT)) is dense in

(Proposition 5.6.5.9 b)) it is sufficient to show that there is an A E y r ( T x S ) such that 11xT11 5 E for every B E yr(Tx S\A) . Indeed, it follows from this that the limit

exists in

where 5 denotes the upper section filter of Y I ( T x S ) , and Bo = 0 on {SIXE ~(sxr)), Case 1 T is a one-point set (e.g. T = K I ) .4ssumc the contrary. Ey induction, we may construct a disjoint sequence ( S n ) , , ~ in P r ( S ) such that

for evcry n E IS. For every n E I N , there is an xn E E ( ~such ~ that ~ )

By Proposition 5.6.5.6 d),

so that

Take p E IS and put

5.6 Hilbert right C*-Modules

Then y E E ( S x T ) ,

and by Proposition 5.6.5.6 d),

This contradicts x' E

F

Case 2 S is a one- point set The assertion follows from Case 1 and Proposition 5.6.5.18 b) Case 3 S and T are arbitrary Step 1 There are SoE C Q (S) and To E 9 (T) such that 1~; 11 I for every A E Vf((T\To) x (S\So)) . Assume the contrary. By Case 1 and Case 2, we may assume S and T infinite. By induction, we may construct disjoint sequences ( S n ) n E and ~ (Tn)n€~' in V f ( S ) and Vf (T) , respectively, such that

5

for every n E I 5 (Proposition 5.6.5.18 c)). For every n E I N , there is an ~ ( " 7 ' ) with

(s, t) E S x T\Sn x Tn & x,,,,,= 0 . Take p E DJ and put D

Then y E E ( S X T ) ,

X,

E

4 74

5. Hilbert Spaces

and by Proposition 5.6.4.20 j),

This contradicts x' E 3 Step 2 The assertion By Case 1, for every t E T o ,there is an St E p f ( S ) such that

IlxxA+ll E

< 3(1 Card To)

for every A E pf(S\S,). By Case 2, for every s E S O ,there is a Ts E P f ( T ) such that

llx1z)+ <

for every B E Y,(T\To) . Put

and take D E

and

It follows

p f (T x

S\C) . Then

3(1

E

Card So)

5.6 Halbert right C*-Modules

Proposition 5.6.5.20

Let E be a W*-algebra, T a set, and

For every x E Ec , put

Then

2: E L E ( H ) ' for every x E E C , and the map

is a n isomorphism of W*-algebras (Theorem 5.6.4.7 a), Theorem 5.6.3.5 b), Corollary 4.4.4.12 e)). By Proposition 5.6.4.8 c),d),

S E L E ( H ) for every x

is a W'-homomorphism. Let x E E C and u E L E ( H ) . For every

E

E C arid the map

<E H,

so that

uI = du,

Since

Ls(H)" C KE(H)', it is sufficient to show that for every u E K E ( H ) ' there is an x E Ec with

-

x=u. For every a, b E T , put

4 76

5. Hilbert Spaces

~ , b:=

(ue,leb) E E .

Fix a, b E T and y E E and define v := eoy(.leb) E K . y ( H ) .

Then, for every c E T , uaay6bc = ~ ( e , ~ b b ~=) a

=

We deduce

for b # c . Putting b = c , weget uaay

=

YUbb,

so that

u,, = UM E EC. Moreover, (ue,leb) = babuaa = (eauaa(eb)= (Ul,,e,leb) ,

-

ue, = ua,e, For < E H , H

<=1 LET

(Proposition 5.6.4.6 c)), so that by Proposition 5.6.3.4 c),

Thus

-u,,

=u

.

5.6 Hzlbert right C* -Modules

5.6.6 Matrices over C*-algebras

Theorem 5.6.6.1

(0)

Let F be a closed ideal of E such that

for every x E E (since E is quasiurlital (Theorem 4.2.8.2) this condition is automatically fulfilled if F = E (Proposition 2.3.4.9)). Given n E !S, let Fn := e 2 ( I N n , F ) be the Hilbert (unital) E-module defined i n Example 5.6.4.2 c). Take m, n , p E IN and for a E Ern,, define

a) For all a E Em,, ,

-

ii E L E ( F n , F m ) , Zi* = a * ,

If F = E then

Z E K E ( E r ' E, m )

b ) The map

is linear and injective and its image is a closed vector subspace of C E ( F n ,F m ) . C)

If the map of b) is surjective for m = n = 1 (this is the case if F = E and E zs unital), then it is bijective for all m, n E N . In particular,

d ) Ern,, endowed with the norm

and with the maps

4 78

5. Hilbert Spaces

(Definition 2.2.7.23)) is a (unital) E-module and the map of b) is a homomorphism of E-modules (Proposition 5.6.1.8 g)). e ) If a E Em,, and b E En,, then

-

-

ab = Eob

f ) En,, is an E-C'-algebra. If E

ts

unital then the map

zs an zsomorphism of E-C*-algebras (Theorem 5.6.1.11 d)) g) Let a E (En,,)+. Then a,, E E+ for every i E

N, and

h) Em,, endowed with the bzlinear map

Em.

X

E n , +E m ,

(a,b)

ab

and with the sesguzlznear map

is a Hilbert right En,,-module. i)

The norm of Em,, (defined in d)) does not depend on F .

j)

Let p E INm and 9 E I N , . Take a E E,,, and define b E Em,, by b,,

:=

a,,

0

Then

if ( 4 3 ) E IN, x N q otherwise

5.6 Halbert right C' -Modules

k) If E is unital then for all (<,17) E En x E m ,

-

v(.l<)= a

1

where

a ) Take

< E Fn and v E Fm. Then

Hence iT E L E ( F n F , m ) and a' = Example 5.6.4.2 b),

2 . By

Proposition 5.6.1.2 d),e),f) and

480

5. Hilbert Spaces

Hence

By Proposition 5.6.4.16 a),

Let j E IN,. Then for every x E F# ,

so that

By our hypothesis on F ,

It follows

(Example 5.6.4.2 b)). Now assume F = E and take x E E . Then for every ( E E n ,

5.6 Hilbert right C*-Modules

By the above

and therefore

-

a = lim E E K E ( E n rE m ) , r,3

where 5 denotes an approximate unit of E (Theorem 4.2.8.2) b) follows from a ) . c) Let u E C E ( F n ,F m ) . Take (i,j ) E IN, x IN, and pllt

v,,

:

F +F , Y

(u*(~~,Y)~EIN,)~.

For x , y E F , (u,,x)Y) = Y*( u ( ~ , ~ x ) ~ E N =, )t

= ( u * ( ~ P ~ Y ) P ~ ~=. ,(vz,y)'x );x = (xlv,,~)

Hence u,, E C E ( F ) and so there is an a,, E E with

4 82

5. Hilbert Spaces

for every x E F . Take

< E Fn . Then

so t h a t for i E

IN,,

Hence uc = E<

and

-

u=a. Thus the m a p

is surjective. Iiuw assume F =

E and E unital. Take u E L E ( E ) and put

Then

for every x E E (Proposition 5.6.1.8 d)), so t h a t

-a = u and the m a p

is surjective. The relation

5.6 Hilbert right C'-Modules

follows from the final assertion of a). d) By b) and Proposition 5.6.1.8 b), Em,, is a Banach space. Take x E E and a E Em,, . Then for all E Fn,

<

IlZtIl = ll~(x<)llI l l ~ l llx
The final assertion is easy to see. e) Put c : = a b . For < E EP,

so that

-

-

ab = Zob

f ) For x, y E E and a, b E E,,,, ,

-

-

( ~ a )= ' a * ~ ' , (ax)' = x'a* , xa = xa ,

-ax =-a x ,

by a),b), and Proposition 5.6.1.8 b). Hence En,, is an E-C*-algebra and by c) and d), the mentioned map is an isomorphism of E-C*-algebras. g) Put

x :=

C a,,

4 84

5. Hilbert Spaces

Take b E En,, with

We obtain successively for all i , j E IN,,

a,,=

b;,bJ, 5

x

bkt' b kt E E , ,

b;,bkl

= a,,

5

5 2,

llxl/

~ ~ b = l~ ~~b ~~l b~J t 2~

~

(Corollary 4.2.1.18),

Hence by a ) ,

h) We may assume E rinital. In this case the assertion follows from b),c),e),f), and Example 5.6.4.14. i) Take a E Em,, and denote by llZllE the norm of a in the case E = F . By f ) , Proposition 5.6.1.8 f), and Corollary 4.1.1.21,

j) is easy to see. k) By c), there is an a E Em,, with

5.6 Hilbert right C*-Modules

Put

for every j E IN,. Then for ( i ,j ) E IN, x IN,,

so that a = [q1<;1( z , J ) ~ ~ m x. N n

(8)

Corollary 5.6.6.2

If E is unital then

for all m, n E N . By Proposition 5.6.2.3, E as Hilbert right E--module is self dual and so Em and En arc also self-dual (Theorem 5.6.4.7 a)). By Proposition 5.6.2.4,

(0)

Corollary 5.6.6.3

a ) M,,, b)

Let

71

E IN

is isomorphic to F(C2n,2n)(Proposition 5.5.7.10).

The complexification of El,,,, zs isomorphic to Q12n,2n The assertions follow from Proposition 5.6.4.18 b),c) and Theorern 5.6.6.1

fl. Remark. b) was already proved in Example 2.3.1.46. Proposition 5.6.6.4 F = E and define

(0)

W e use the notation of Theorem 5.6.6.1 with

for every I' E Elrn and

for every a' E EL,,

.

Then for all m, n E N :

5. Hilbert Spaces

486

a)

If a'

E E;,, , then

-*

2 E (Em,,)', a'

-

= a " , and

(Proposition 5.6.4.3). b ) The map

-

E m

4E

n )

a' c+ a'

is linear and bijective.

c ) EL,, endowed with the norm EA,,

4

m+,

4

Eh,, ,

a'

4

11211

and with the maps

E;,, x E

(a',x) ct a'x

(Definition 2.2.7.23) is an E -module. d ) The map

E m

E n )

a'

is an isometry of E-modules. a) Take a E Ern,, . Then by Theorem 5.6.6.1 a),

so that

2 E (Em,,)'

and

-

ct a'

5.6 Hilbert right C*-Modules

(Proposition 5.6.4.3). Let k E INn with

Take 0 E ]0,1[ and x E ( E m ) # with ax.(.)

2 e11aT.11

and define

for all ( i , j ) E K m x IS,. Then ii E E:,

(Theorem 5.6.6.1 a)) and

Since 0 is arbitrary, we get

(Proposition 5.6.4.3). Given a E En,, , m

i*(a)=

=

n

1 ((a*),,. a;.)

=

1=1 ,=1

so that -*

-

a1 = al*.

b) follows from Theorem 5.6.6.1 a) and Proposition 2.1.4.25.

5. Hilbert Spaces

488

c) and d) . By b),

-

E m +(

m

n

a' ++a'

is an isomorphism of vector spaces. Take x E E and a' E E;,, . Then for every

a E Em,,

C

(a1 ~ 2 = ) C ( a z l ,xa;,) =

( a , a%) =

C E ( a , z , a;,) = ( a x , a')- = ( a ,xa')- ,

C ~ ( a , , a;,x) , = C C ( x a , , , a;,) = ( x a , a') = ( a , a'x) ,

so that

-

-

-

-

xa' = xa' , a'x = a'x ,

Theorem 5.6.6.5 ( 0 ) Let E be a W*-algebra. Take n E I N , endow En,, wzth zts canonzcal structure of a C' -algebra (Theorem 5.6.6.1f ) ) and put

for every a

E

En,,

.

a ) ii E (En,,)' for every a E En,, , the m a p

is injective and involutive, and Im u is a n involutive En,,-submodule of (En,,)' . W e zdentzfy En,, wzth Irn u .

b)

(E,,,)L, is compact

c)

The m a p

zs u isometry of Banach spaces. In particular, En,, is a W*-algebra with En,, as predual.

5.6 Hzlbert right C*-Modules

489

d ) If we identify En,, with the predual of En,, (like in c)) and En,, with LE(E") (Theorem 5.6.6.1 c)) then for every a E E and <,17 E En ,

a) By Proposition 5.6.6.4 a), E E (En,,)' and u is injective. For every X

E En,,

, E*(x) = .(.*)=

x

((x*),,, all) =

so that

and u is involl~tive. Let be a sequence in En,, such that (ii,),,,,~ X' E (En,,)'. By Proposition 5.6.6.4 a), lim

m

too

converges to an

= x:,

so that xi, E E for all i, j E N n . Hence x' E Im TL and Im u is closed in (En,,)'. Kow it can be easily seen that I m u is an involutive En,,-submodule of (En,n)' . b) Let

5 be an ultrafilter on (En,,)#. Put

for all i , j E N n . By Proposition 5.6.6.4 a), p,,(5) is an ultrafilter on E # . By to an element y,, . Put the Alaoglu-Bourbaki Theorem, it converges in

EE#

By Proposition 5.6.6.4 a),

-

"(Y) =

):(Y,,. a,I) = lim ,,,=I

for every a E En., . Hence

x(v,,

(2). a,,) =?!I

1,,=1

5 converges to y in

Z(X)

490

5. Hilbert Spaces

.. #

We have to show that y E (En,,)#. Take f E e2(IN, E)# , a E E , and u, v E (En,,)#. Then u<, v f E e2(W,, E)# so that

(Proposition 5.6.1.2 d)). Thus

It follows

=

I( Y<~Y<)~

41.

Since a is arbitrary, ( Y < ~ Y < )E E # ,

5.6 Hilbert right C*--Modules

<

Since is also arbitrary it follows y E (En,,)# c) follows from h) and Proposition 1.3.6.27 d ) For x E En,,

Corollary 5.6.6.6 Let T be a nonempty compact hyperstonian space and n E I N . For every p E (C(T)"),,, , put

Then C(T),,,

is a W*-algebra with

as predual The assertion follows from Theorem 4.4.4.22 c 3 a & d and Theorem 5.6.6.5 c)

.

Theorem 5.6.6.7 ( (Theorem 5.6.6.1 f)). a)

0 )

Take n E IN and consider the C*-algebra En,,

If E is finite-dimensional then n(7~ - 1) Dim Re En,, = DimE 2

b) E simple

+ En,,

+ nDimReE.

simple

a ) An element a E En,, is selfadjoint iff

for all i , j E IN,. It follows n(n - 1) dim Re En,, = --2 dimE+ndiniReE.

4 92

5. Hilbert Spaces

b) Let F be a closed ideal of En,,, E.'

# { 0 ) . Let a

E

F\{O) , p, q E IN,,

and

G is a closed ideal of E (Theorem 5.6.6.1 a)). Let x, y E E and r , s E N n such that a,, # 0 . Put

Then (ba)c E F and

so that

Since E is quasiunital (Theorem 4.2.8.2), we get

Hcnce G # ( 0 ) and therefore G = E . Thus

for every x ,:E E . Sincc p and q are arbitrary, it follows F = En,, , and so En,, is simple. c) is easy to see. Corollary 5.6.6.8

(0)

Take n , p E IS:

a ) T h e map

obtazned by ur~bi7tdir~g the matrices of En,,, zs an zsomorphisnl of E C * algebras.

4 93

5.6 Hilbert right C*-Modules

b) If E is the direct s u m (product) of the family ( F L ) L EofI C f algebras, then the natural map of En,,, into the direct s u m (product) of ( ( E ) n , n ) L E ~ is a n isomorphzsm of E X * -algebras.

c) T h e m a p

is a n zsomorphism of E X *algebras. d) If E is a simple purely real Cb-algebra, then so is En,, . d) By Proposition 4.3.5.3 1)

+ a,

is simple, so by Theorem 5.6.6.7 b), 0

A

En,,, and En,, are simple. By c), En,, is simple, so by Proposition 4.3.5.3 a b , En,, is a purely real C*-algebra.

*

Corollary 5.6.6.9 ( 0 ) Take 71 E I N . R,,, and M,,, purely real (real and complex) CC'algebrasand

(a,,,) are simple

+

n ( n 1) Dim Re R,,, = 2 ' Dim Re M,,, = 2n2 - n

I n particular,

and M,,,

are not isomorphic.

R , C , and M are division algebras (Corollary 2.1.4.17), so they are simple C*-algebras and w.e have

m c = m , M ~ = I R ~ c, C = c . By Theorem 5.6.6.7, IR,,,, , en,,, and M,,, are simple and

Dim Re Kt,,,

=n

n(n 1) +2 -

-

+

n ( n-1) 2

'

4 94

In particular, isomorphic.

5. Hilbert Spaces

k,, and M,,, are purely real and R2n,2n and M,,, are not W

Remark. Put

Then

so that IR2,2 endowed with the maps x

R2.2

+ R2,2,

R2,2x a: + R2,,,

( ( a + i B ) ,x ) 6 (a1 + B a ) x ,

(x,( a + 20))* 4 0 1 + pa)

is a purely real C-C*-algebra.

Corollary 5.6.6.10

( 0 )

Take a function p : IN

E be the C' -direct product ( s u m ) of the family

(C$)

-+

N U (0) and let of complex C*-

)nEN

a r ~ dE algetras, where C:,, := (01, and F a real C*-algebra such that are isomorphic. T h e n there are (uniquely) f , g , h 6 (IN U (0))" such that h vanishes at the odd natural numbers,

and F is isomorphic to the direct product (sum) of the family

of real C' --algebras. By Proposition 4.3.5.6 (and Corollary 5.6.6.9), F is the C*-direct product 0 (sum) of a family (Fn)nENof real C'-algebras such that F is isomorphic to a$$')for every n E IN. By Corollary 5.5.7.15 and Corollary 5.6.6.3 a), for every n E N , there are (uniquely defined)

such that h vanishes if n is odd,

5.6 Hilbert right C*-Modules

is isomorphic to Fn, and

Proposition 5.6.6.11

(0)

b) T h e real C*-algebra C,,, RP,P

Let n , p E N . T h e following are equivalent:

is isomorphic to a unital C'--subalgebra of

.

c) There is a unztal algebra homomorphism d ) M,,,

an,,-t El+,,,

is isomorphic to a unital real C'-subalgebra of C,,, .

e) There is a unital algebra homomorphism M,,, +C,,, . a +b & d . and M are isomorphic to a unital C*-subalgebra of R2,2 and C2,2, respectively, so that an,, and M,,, are isomorphic to a unital 6'subalgebra of

(m2,2)n,n = R ~ n , 2 n and (C2,2)n,n= C2n,2n, respectively (Corollary 5.6.6.8 a)). By a), lK2n,2nis isomorphic to a unital C*-subalgebra of lKp,p. Hence C,,, and M,,, are isomorphic to a linital C*subalgebra of R,,, and C,,, . respectively. b + c and d + e are trivial. c + a (resp. e + a). By Corollary 2.1.4.13 a),b), there is a family (x,,),,,,~, in R,,, such that

for all i , j,k, e E INn and

Let

be a unital algebra homomorphism. Put

4 96

5. Hilbert Spaces

for all i , j E IN,. Thrn

for all i, j , k , e E

IN, and

By Lemma 2.1.4.11, Dim Im y,, = Dim Irn yll for all i , j E IN,. It follows Dim Im y,, =

P n

-

for all z E N, . In particular, we get a unital algebra homomorphism

G

-4

IRE,:,

(resp. M --tC.e,:)

By Proposition 2.1.4.26,

-PE N 2n Remark. The set

is a unital C* subalgebra of

Rz,, , which

is isomorphic to C . The set

si a unital C*-subalgebra of C2,2r which is isomorphic to

Corollary 5.6.6.12

b)

(0)

T h e M -C*-algebra M,,, MP.P

.

M.

Let n , p E DJ . T h e following are equivalent:

is zsornorphic to a unital M C "

subalgebra of

5.6 Hilbert right C*-Modules

c) There is a unital algebra homomorphism

M,,, -+ M,,,

a + b + c is trivial. c + a . By Proposition 5.6.6.11 a 3 e , there is a unital algebra homomor. Proposition 5.6.6.11 c + a , phism M,,, -+ C 2 p , 2 pBy

Remark. Of course, the above corollary can be proved directly like Corollary 2.1.4.13 c). In fact, it holds for an arbitrary finite dimensional C*-algebra (Proposition 6.3.6.19), but it fails in thr infinite-dimensional case (Corollary 5.6.4.17).

Proposition 5.6.6.13

(0)

Let n , p E I N . T h e following are equivalent:

b) M,,, is isomorphic to a unital C*-subalgebra of IR, c) There zs a unztal algebra homomorphism M,,,

-+& , p .

a + b . H is isomorphic to a unital C*-subalgebra of C 2 , 2 .By Proposition 5.6.6.11 a + b , IH is isomorphic to a unital C*-subalgebra of R4,4. Hence M,,, is isomorphic to a unital C* subalgebra of

is isornorphic to a unital C* subalgebra (Corollary 5.6.6.8 a)). By a), lR4n,4,, of Q,, . Thus Mn,,is ison~orphicto a unital C*-subalgcbra. of Q,,. b + c is trivial. c + a . By Corollary 2.1.4.13 a),b), there is a family ( x , , ) , , , , ~ ~in IR,,, such that

for all z , j , k , e E DJ, and

Let

4 98

5. Hilbert Spaces

be a unital algebra homomorphism. Put Ytj := U X t j for all z, j E IN,. Then YtJYk! = 6 J k ~ z ( for all z , ~ k, , e E IN, and

By Lemma 2.1.4.11, Dim Im y,, = Dim Im yJJ for all i, j E IN,. It follows Dim Im y,, = P n for all i E N,

.

In particular we get a unital algebra homomorphism

By Proposition 2.1.4.26,

-P E N 4n

Remark. The set

is a unital C*-subalgebra of R4,,, which is isomorphic to Proposition 5.6.6.14

for every p E N and put

(0)

M

Take n E IN. W e recurszvely define

5.6 Hzlbert right C*-Modules

for all p E I N , a E E:;,

and x E En,, and endow E!$

with the norm

defined inductively for all p 6 N and with the maps En,, x E $ '

Then, given p E N a)

a6

b) E:!,

4 E::;

,

(x, a) +--+ xa ,

:

E~P; * t2jENn sup Ilallll i llall i

5

lla,lll.

t,]=I

zs an ~nvolutzveEn,, -module (Theorem 5.6.6.1 f)) and the map

is an isometry of involutive En,,-modules By Proposition 2.3.6.7 and Theorem 5.6.6.1, a) holds, E!$ is a n involutive E-module, and the map

is a n isometry of involutive E-modules. Take x E En,, and a E E::. Then for every b E E!iP;",

5. Hilbert Spaces

500

The relations

are proved similarly. The other assertions are easy to check.

Proposition 5.6.6.15 Let T be a locally compact space and 71 E fi. W e identzjy as usual M b ( T ) ,the involutive Banach space of bounded Radon measures o n T with Co(T)' and then Mb(T),,, wzth (Co(T),,,)' as i n Proposition 5.6.6.4 a), 6). Take p E Mb(T),,, , put

and let a E Lm(v),,, such that p,, = a,,.v

for all z , E~ I N , T h e n the followzng are equzualent: a) P L O .

b) a > O . a

+ b.

For z,j E IN,,

so that

a E R e Lm(v),,, Let 5 be a filter on (C0(T),.,), converging to a - in ( L w ( u ) n , n ) ~ l ( V . ),,, By Theorem 5.6.6.1 g), it follows successively

5.6 Hilbert right C*-Modules

a b

+a.

> 0.

There is a b E LW(v),,, such that a = b'b

Then

for all i, j E K,

Thus

/L

. Take

x ?:E Co(T),,,, T h e n

>0

Proposition 5.6.6.16

Let n E

IN. For every x

E

( b y Example 4.2.3.2, 2 E (K,,,)') . Take x E Kn,, .

K,,, put

5. Hilbert Spaces

502

a) For every p E PrIK,,, , p is the right carrier of x iff p is the right carrier of 2 .

-

b) For every y E IK,,, , (y,1x1) is the polar representation of x iff (y,1x1) is the polar representation of Z . c)

-

121 = 1x1. a) By Proposition 5.6.6.14 b),

for every y E IK,,,. The assertion follows immediately from this remark. b) First assume that (y, 1x1) is the polar representation of x . By Proposition 4.2.3.2 a + b , 1x1 E (IK,,,);. By Proposition 5.6.6.14 b)),

Take z E

IK,,, with

-

lxlz = 0 By Proposition 5.6.6.14 b),

so that

-

-

Hence y'y is the carrier of 1x1. Thus (y,1x1) is the polar represelltation of E . Now assume that (y,1x1) is the polar representation of E . By Proposition 5.6.6.14b) and Theorem 4.4.3.5 b),

-

2 = ylxl = YIXI

3

x = ylxl,

5.6 Hzlbert right C*Modules

2 = yy*?

-,

= yy'x

x = yy'x .

Take z E En,, with

By Proposition 5.6.6.14 b) and Theorem 4.4.3.5 b):

Hence ( y , 1x1) is the polar representation of x . c) follows from b) . Remark. relatiori

By the above proposition and by the remark of Theorem 4.2.5.6, the

does not hold and there is no ru E a, b E

R+ such that the relation

E ==+ la + 61 5 ru(la1 + Ibl)

is true for all W* algebras E .

Proposition 5.6.6.17 ( 8 ) Take T L E IK and zdentzfy E canonzcally wzth the C*subalgebra {[xb,,],,,,~, lx E E ) of En,, .

T h e n 9 is a n involutioe continuous linear map such that:

504

5. Hzlbert Spaces

b) If H zs a Hilbert right En,,-module then H endowed wzth the right multiplication

and with the inner-product

is a Hilbert rzght E module, denoted by H , and CE,,,(H) is a unital

C*-subalgebra of

cE(H).

c)

If G and H are Hilbert right En,, -modules then

d)

If all Hilbert rzght E--modules are self-dual (reflexzve) then all Hilbert right En,, -modules are self-dual (reflexive). a ) 1 ) is obvious. 2 ) and 4 ) follow from Theorem 5.6.6.1 g ) . Take ( x ,y ) E

(a+ 0

. There is an ( a , b) E En,, such t h a t ( x , y) = ( a ,b)'(a, b) = (a'a

Then

for every j E IN, so that

+ b'b, a'b - b'a)

5.6 Hilbert right C*-Modules

This proves 3 ) . b ) , ~ )and , d ) follow from a ) and Proposition 5.6.2.5 a),b),c),d) .

W

Proposition 5.6.6.18 ( 8 ) W e use the notation of Proposition 5.6.6.17 a n d assume E E {R,C, M} . Moreover, we identzfy Enan with L E ( E n ) (Theorem 5.6.6.1 c)). Take E E n with ll
<

b) A n element a

En,, belongs to p E n , i f l there is a n

71

E E n such that

-

C) {<(.le,)li E E n ) is a F o ~ z e rbasis of the Hilbert right E-module pEn,,, .

d ) For ever?)

< E En

there is a n z E Un En,, with

for every t E K n - l . a) By Proposition 5.6.5.2 a),f), p E Pr E ,,,,, \{O} . Let q E Pr E,,,,, with O < q p . By Proposition 5.6.5.2 d),c) and Corollary 4.2.7.6 a =+ c & d ,

<

(1((.10 = qo(<(.IO) = 4 = (<(.IO)oq = <(.~q<).

By Proposition 5.6.5.2 I),

506

5. Hilbert Spaces

(s
R+

we obtain successively

Hence p is a minimal element of Pr En,,\{O) b) For 17 E E n ,

Now assume a E pEn,, . Then

(Proposition 5.6.5.2 e)). c)

BY b), {t(.leI)la E K n )

C

pEn,n. For

Z,.I

E Nn,

-

(Proposition 5.6.5.2 a),f), Theorem 5.6.6.1 k)). Hence {<(.le,)li E I N , ) Fourier set of pEn,, . Take a E pEn,, . By b), there is an 17 E En such that

a

= <(.Iv)

For every z E IS,, 'P((al<(.le,)) = 'P((e,(.1<))0(<(.117)))=

(Proposition 5.6.5.2 a),f), Theore111 5.6.6.1 k)). Thus if

is a

5.6 Hilbert right C*-Modules

for every z E I N , then a = 0 . It follows that

{[(.le,)lz

507

E N n ) is a Fourier basis

of pEn,n. d) Put

Then y E Un En,, and yJ E INn. Hence there is a iz E UnR,,,

for every t E

with

Then x := z y has the desired properties.

Proposition 5.6.6.19 ( 8 ) Assume E unital and take n E N . Let G , H be Hilbert right En,,-modules and ( E G , 7 E H , E En such that

<

Put

and

Ft := F x , ,

:= VX,

for every z E K, .

a) z E K n

b)

2

C)

<1,<2

,

+ (
E~ INn

+ P(F,IF]) = ( ~ ( ~ 1 1 % = ) 61,

E En

*

~(<1(.1<2))

n

d)

7/(.10

=

C V t ~ ( . 1 < ~.)

,=I

=

(<;I<;).

.

5. Halbert Spaces

508

e)

If IK

= IR and E = C

( E = M) then

(Proposition 5.6.5.2 f ) ) . b) BY a),

(Proposition 5.6.1.2 a), Proposition 5.6.5.2 f), Theorem 5.6.6.1 k)),

c) By Throrcm 5.6.6.1 k ) ,

d ) For

<' E G , by c) ,

5.6 Hzlbert right C* -Modules

509

(Propositior~5.6.1.2 a), Proposition 3.6.5.2 a),e)), whcre we idcritified the element ((
e) follows from d ) and Proposition 5.6.5.12 b) (ar~tf Propositiorl 5.6.2.5 a),e)). Proposition 5.6.6.20

(8)

Let n E I S .

a) If P is a subset of PrM,,,,\{O}

such that

for all p, q E P then Card P

b) For every x E Re M,,,


, Card u ( z )


5. Halbert Spaces

51 0

c) If E is a Gelfand C*-subalgebra of M,,,, then

a) By Theorem 5.6.6.1 b), for every p E P there is a PCP

<,

E IHn such that

#0

Let (aP),,p E IH(?) such that

Then for every q E P ,

Hence (P<,),,~ is linearly independent and therefore Card P


b) Since the complexification of Eln,, is isomorphic to C2,,,2,, (Corollary 5.6.6.3 b)): a(%) is finite. By Theorem 4.1.4.6 a + b & c , there is a family (Pn)n,m(z) in PrMn,n\{O}, such that

Card a ( x ) 5 n . c) follows from b) and Corollary 4.1.2.6.

Corollary 5.6.6.21

Let

(AnlntLN ( ~ n ) n k(un)neN ~ 3

and put

9

(fi u { o } ) ' ~ )

5.6 Hilbert right C*-Modules

a) For every x E Re E ,

b ) If F is Gelfund C*-subalgebra of E then Dim F 5

1*(An + p, + v,) nER

a) follows from Proposition 5.6.6.20 b ) . b) follows from a) and Corollary 4.1.2.6. Proposition 5.6.6.22

(8)

Let E be a unital C'-algebra, n E N , and

for all i, j E N, . For T a set, i , j E IN, define c,, E t2( T , E ) by

<E

e 2 ( T ,En,,) (Theorem 5.6.6.1 f ) ) , and

Take S , T sets and for every

a) If u E ( L E ( e 2 ( SE , ) , e 2 ( T ,E))),,, and a

En,, then

-

2L E LE,,, ( e 2 ( S En,,), , E2(T,En,,)) 6' = U *

(Example 5.6.4.2 c), Proposition 5.6.1.8 g)) b) The map

is bijective

.

1

51 2

5. Hilbert Spaces

a) Take

< E t 2 ( S ,En,,) and 17 E t 2 ( T ,En,,). Then for i, j

E Kn ,

5.6 Halbert right C*-Modules

so that

and

~X=lln, E=aC. b) Take ZJ

E

CE,,, (e2(s,En,,) , P2(7', En,,,)) .

For i, j,k , P E Din put

For

< E k"(S, E,,,n) and i, j

Take

< E P2(S,E,,,,,)

It follows

E K n,

arid p, q E N, . Thcn for k,P E N n ,

5. Halbert Spaces

514

and

for all i, j E IN,. Take 7 E e2(T,E) and i , j , a , b E

Enand

put

Then

b ~ b u z ~ , a= q b]q~,p,ab.

Put u,] := V , l , ] l for all z, j E IN,. Then &p,ab

Since

j

= bpbvzl,al

is arbitrary, Z'Lp.ab = b p b ~ t a

We get for

< E t 2 ( S ,En,,) and

2,

j E

K,

so that

and the map is surjective. Since it is obviously injective it is bijective.

5.6 Halbert right C*-Modules

5.6.7 Type I W*-algebras

Throughout this subsection E denotes a W'-algebra.

Proposition 5.6.7.1 Let p E Pr E , F := p E p , and ( P , ) ~ ~a ,family in Pr E such that

for every L E I . For every L E I denote by H, (by K , ) the von Neumann right E-module (right F-module) associated to p,E (to p,F) (Propositions 4.4.2.2 and 5.6.2.3) and put W

H := Q H ' ,

W

K := Q K ,

LEI

LEI

a) H (resp. K ) is a von Neumann right E-module (right F-module). b) K = { < E H I < p = < l .

c ) If u c L E ( H ) then u ( K ) C K . Put

for every u E L E ( H ); then u~ E L F ( K ) with ( u K ) *= ( u * ) ~ . d ) The map

is an isomorphism of W'-algebras. a ) follows froni Theorem 5.6.4.7 a) (and Propositior~5.6.2.3). b) For every E K and L E I , <, E K1 = p,F, so that <,= <,p. It follows < p = . Cor~versely,take E H with ( p = < . For every L E I ,

<

<

Hence ( E K . c ) By b), for every

<

< E K , < =

5. Htlbert Spaces

51 6'

(Propositiorl 5.6.1.8 d)). Hence, by b), u ( K ) C K . For <, 7 E K ,

d) By a) and Theorem 5.6.3.5 b), L E ( H ) and Lc.(K) are W* -algebras with preduals H and K t rcspectivcly. By c), the map

y : L E ( H ) +L F ( K ) , u

ILK

is well-defir~ed and is involutive. It is obviously an algebra homomorphism

For every

<E H

and J

cI

T J :

For every

L

put

H--t H ,


E I put

* ~ A P ;,

fl : I + E , obviously f, E K . Take u E Kcr p and

< E 11. Then

uf,=uKf,=O and so uJ(c) = tl(fr<') = (~fL)
L

E

I (Proposition 5.6.1.8 d)). It follows IL<.,

=

0

5.6 Hilbert right C*-Modules

for every J E

PI([).By Theorem

5.6.4.7 c),

u = lim TJUT,, = 0 . 5,31

Thus cp is injective. Step 2 cp(LE(H)) is a W*-subalgebra of C p ( K ) Take u E L E ( H ) , u E F , and ( , T I E K . By Corollary 4.4.4.12 c), F is a W*-subalgebra of E and so there is a b E E with blF = a (Corollary 4.4.4.9). From

-

(cpu, ( a . 7)) ~ = ( ( ( v u ) E ~ ~a) I ) ,= (((cpu)
it follows by Theorem 5.6.3.5 d), that v is a We-homomorphisnl. By Corollary ) a Mp-subalgebra of L p ( K ) . 4.4.4.8 b), ~ ( C E ( H ) is Stcp 3 p is surjective Put

for every J

c I . Take u

E L F ( K ) and

L

E I . The map

. a) and Proposition 5.6.2.4, it belorlgs to C E ( H ) . For belongs to Z E ( ~ )By CEK,

so that

Moreover, P{A)uP{')= ( T { A ) ~ )EK'P(LE(H)) for every X E I (Theorem 5.6.4.7 c)). It follows

5. Hilbert Spaces

518

for every J E p j ( I ) . By Theorem 3.6.4.7 c),

u =lirnp~up~ J,3,

in (LF(K))K. By Step 2,

u E v(CE(H))

1

i.e. cp is surjective.

Theorem 5.6.7.2

for all

1, X

Let p E Pr E and let ( x , ) , ~be~ a family i n E such that

E I and

For every L E I denote by H , the won Neumann rzght E-module associated to p E (Proposition 5.6.2.3) and put

a) C E H , (
b) For every x E E the map

belongs to LE(H) with Z* = 2-' . Define

c) For every u E L E ( H ) put

G := (uCl() and define

Then cp and

)I

are isomorphisms of W*-algebras and $I = cp-'

5.6 Hilbert right C*-Modules

51 9

+

and 11, denote the pretranspose of cp and q h , respectively (Theorem d) If 5.6.4.7 a), Theorem 5.6.3.5 b)), then

-

$ ( a , <,v ) = (IIOa((1v) ,

for all a E E and [,v E H .

f ) If

L,

X E I and

<EH

then

g) If p E p is commutative then the map

is an isomorphism of W*-algebras (Corollary 4.4.2.5). a ) By Corollary 4.1.2.22 b

for every

L

E

+c,

I . Since

it follows ( E H arid

(
= 1 . Moreover, for E

((<(.I<))J)L = <'(I10 = 5:

1

XX<X

XEI

b) By a), Z is well-defined. For

=

5E

H and

L

E

E

Xtl

A€ I

E

I,

C X:XXIX = C ~ ~ X P=< IX r,

I,7 E H ,

5. Hilbert Spaces

520

Thus 5 E L E ( H ) and Z' = 2 . c) For x , E~E and E H , by a),

<

= C ~ ( C I C ) Y ( < I C )= Cxy(
so that

and p is an algebra homomorphism. By b), it is involutive. For X E E , b y a ) :

= (CIC)x(CIC)= x .

For u E L E ( H ) and

< E H , by a ) ,

= C(.(C(
= <(u
so that

Hence p and 4 are isomorphisms of W8-algebras and 4 = cp-' . d ) For x E E h -

( p x , (a.<, 7 ) ) = ((Z
Thus

For

IL E

LE(H)

5.6 Hzlbert right C*-Modules

-0.

Thus

* a = (a, <,

f ) We have

g) It is easy to see that the nlap is an involutive algebra homomorphism. Take x E Ec with xp = 0 . Then

for every

L

E

I , SO that

Hence the map is injective Put

For every L E I denote by ICL the von Neunlanrl right F-module associat,ed to F (Propositiori 5.6.2.3) and defino

5. Hilbert Spaces

522

Let y E F and put

<ey<=[y.

u:K+K, Then

(uFlv) = (FyIv) = (FIP)Y = ~(Flv) = (thy*) for all

<, 7

E K , so that u E L F ( K ) . For every

v E L F ( K ) and

EK,

By c) and Proposition 5.6.7.1 d ) , there is an x E Ec with

z< = u< for every ( E K . By e), Z<

for every

< E K . Take

L

-

= x< = UF = Y<

E I and

< :-

(~,AP)XE~

Then

Hence the map is surjective. By the above, it is an isomorphism of LV*-algebras. H

Proposition 5.6.7.3 Let p, q E Pr E such that pEp is commutative and 0 < q <_ p . Further let H be a von Neumann right E-module, A a Fourier basis of H , and B a Fourzer set of H such that

for all

<EA

and 7 E B .

5.6 Hzlbert right C*-Modules

C) Card B 5 Card A a) By Proposition 5.6.3.9 a

+ b and Proposition 5.6.1.2 a),

(tlv) = (<(tlOIv(vlv))= (vlv)(tlv)(tlt) =

Since pEp is commutative, it follows

(
c) Case 1 A is finite Let K be the vector subspace of H generated by A U B . By a),

(
E

K . Take x' A'

E

u(pEp) with (q, x') # 0 and put

:= {< E

A I ( ( t l t ) , ~ '#) 0 ) .

The map

K

X

K +K ,

(<,,<2)* ((<1I<2),x1)

being a positive sesquilinear form, we get by Schwarz inequality,

l((
<E K

and

< E A\A1.

Define

?: A' + IK, t 6 ((CIt),x1) for every

< E K . Then C E ['(A')

for every

<EK

and

524

5. Hilbert Spaces

for all (I, C2 E K (Theorem 5.6.3.13 f)). In particular,

-

for all '71,% E B . Thus (inqGBis linearly independent. Since (<)c€A,generates ['(A') (in fact, it is an orthonormal basis of ['(A') ), Card B 5 Card A' Case 2

5 Card A .

A is infinite

Take a E E+ with (g, a) # 0 and put

f

: A ---t P ( B ) ,

t

I-+

{D E B I ((FV)(171C),a) f 0)

By a ) and Theorem 5.6.3.13 c), for every

<

A,

so that f (<) is countable. Take 7 E B . By Theorem 5.6.3.13 f),

Hence

vc Uf(<)> B c Uf(<), €€A

CtA

Card B 5 Card (A x No) = Card A .

5.6 Hilbert right C*-Modules

525

Corollary 5.6.7.4 Let p E P r E be such that pEp is commutative. Further let H be a von Neumann right E-module and A, B Fourier bases of H such that

(771 1771) = (7721772)

for all

&,(2

1P

E A and q1,q2 E B . Then

Card A = Card B

and

for all (<,q) E A x B .

Card A

= Card

B

follows from Propnsition 5.6.7.3 c ) . By Proposition 5.6.7.3 a),b) and Proposition 5.6.3.9 a + b , for all 77 E B and 5 E A ,

so that (77177)

5 (CIC)

By symmetry,

((10= (1717) for all (<, 17) E A x B .

526

5. Halbert Spaces

Proposition 5.6.7.5 Let p E Pr E such that pEp is commutative. Further let H be a von Neumann right E-module, A a Fourier basis of H such that

for every

b)

< E A , and

( u l A ) ~ l , A ~a, ~family ,l in L E ( H ) such that

There is a Fourier famzly ( q ) l EznI H such that

for all

L,

XE I.

c ) Card 1 5 Card A

.

a) By 1) and 2 ) ,

b) Take p E I and Proposition 5.6.3.4 c ) ,

Hcnce there is a

<EA

C

E Imn,,\{O).

By a), Theorem 5.6.3.13 f ) , and

with upp<# 0 . From

(< - ? ~ p p < l ? ~ p p <= )

- up,<)lt) = 0

5.6 Halbert right C* -Modules

it follows

(ufip
then (17117) E Pr E\{O) .

Moreover,

(17117) = x*(u,,
E PEP

Put 171

for every

L

E I . Then for all

L,

:= U',17

XEI,

= ~ l x ( ~ l u , ,=~ 6,~(17117) ) 5 P.

c) follows from b) and Proposition 5.6.7.3 c).

Proposition 5.6.7.6

Let p, q , r E P r E such that p E p is commutative and O
O
Further, let ( x , ) , , ~ ,( Y , , ) , ~ ~ be families i n E such that

5. Hzlbert Spaces

528

X:XA

for all

L,

= 6,,4g,

Y>Y" =

X E I and p, Y E M and

Then

Card I = Card M , q = r . Put

By Theorem 5.6.7.2 c), E is isomorphic to CE(H) and C E ( K ) . Put U L := ~ xLx;

for all

L,

X E I . Then

for all

1,

A, p, v E I . By Proposition 5.6.7.5 c),

Card I 5 Card M . By symmetry, Card I

= Card

M

By Theorem 5.6.7.2 g), qEq and r E r are isomorphic, so that

Proposition 5.6.7.7 Let I be a set and p E I . Put

and define

for evenj

L

E I

5.6 Hilbert right C*-Modules

a) uLE LE(H) f o r every

I.

E I and

for every 7 E H .

a) For

<,TI

E H,

Hence u, E LE(H) and

for every

< E H , so that

c) follows from a) and b) . d ) By a ) , for L E I , ( , ~ H E , and a~ E ,

529

5. Hilbert Spaces

530

Hence

Definition 5.6.7.8 T h e W'-algebra E zst called homomogeneoua if there ezist a p E P r E and a family ( x , ) , , ~i n E such that p E p is commutative,

for all

L,

X E I , and

Every commutative W' algebra is homomogeneous. We have

for every

L

E

I , SO that p # 0

Theorem 5.6.7.9 W * algebra F put

For every cardznal number N and for every commutative

-

-c

a ) ( N , F ) is a homogeneous W*-algebra and F is isomorphic to ( N , F )

b) For every homogeneous WW-algebra E there is a pair ( N , F ) , s&that N is a cardinal number, F is a commutatzve W *algebra, and ( N , F ) is isomorphic to E . C)

If N , N' are cardinal numbers and F , F' are commutative W*-algebras such that

- -

( N , F ) = (N', F')

then N = N' and F and F' are isomorphic

5.6 Hilbert right C*-Modules

531

d) Let F be a commutative W8-algebra, N a cardinal number, n E N and

-

-

T h e n (N, F),,, is isomorphic to (N', F ) . In particular En,, is homogeneous for every homogeneous W*-algebra E . a) follows from Proposition 5.6.7.7 and Theorem 5.6.7.2 g). b) There exist a p E Pr E and a family (x,),,, in E such that F := p E p is commutative.

for all

1,

X E I , and

-

By Theorem 5.6.7.2.c) and Proposition 5.6.7.1 d), E is isomorphic to Card I, F )

-

c) By Proposition 5.6.7.7, there is a family ( u ~ ) , in , ~ (N, F ) such that for all L , X E I ,

-

U:UA

= ~ L X: P

where p E Pr (N, F)\{O) . Put

for every

1,

X E N . Then for

I,,

A, p, v E N ,

By Proposition 5.6.7.5 c) (and Proposition 5.6.4.6 f ) ) ,

5. Hilbert Spaces

532

By symmetry,

H = N'. By a ) , F and F' are isomorphic. d ) It is easy to see that W

a>

F and

a)

W

a) F

are isomorphic, so that the assertion follows from Proposition 5.6.4.16 d).

W

P r o p o s i t i o n 5.6.7.10 Assume E is the C *-direct product of the family (Ex)x€r, of We-algebras and let I be a set. Put

and

for el~ery X E L and denote by L the C*-direct product of the family

(LE,(Hx))xEL. a ) The elements of H may be zdentified wzth the functions [ defined on I x L such that

for every X E L and such that

c ) For every u E L put

Then ii E L E ( H ) for every u E L and the map

is an isomorphism of W*-algebras

5.6 Hilbert right C*-Modules

5,?9

Definition 5.6.7.11 E is called a type I W*-algebra if it is isomorphic to the C*-direct product of a family of homomogeneous W *-algebras. This is not the usual definition of a type I W*-algebra (in the complex case), but is equivalent to it. Theorem 5.6.7.12 Let 5 be a class of commutative W*-algebras such that any commutative W*-algebra is isomorphic t o exactly one element of 5 . Let further 3 be the class of families (FN)NEIi n 5 for which I is a set of cardinal numbers. For every F = (FN)NEI E 3 denote by F the C*- direct product of the family

( L F ~ ( ~L EFNN ) )

a)

F

is a type I IV'--algebra for every F

N ~ I

E

3

b) For every type I W - *algebra E there is exactly one F E 3 such that E is isomorphic t o F . c) En,,is a type I W*-algebra for every type I W*-algebra E and for every

EN. a ) follows from Theorem 5.6.7.9 a). b) By definition, E is isomorphic to the C*-direct product of a family ( G x ) A ~ofL ho~nogeneotis IV*-algebra. By Theorem 5.6.7.9 t)), for every A E L , there is a cardinal number NA and an Fi E 5 such that Gx is isomorphic to

Put

I := {NA / X

E

L) .

For every N E I put

IN:= { A E L I Nx = N) and denote by FN the element of 5 which is isomorphic to the C*-direct product of the family (Fi)hEI,,. By Proposition 5.6.7.10 c):

5. Hilbert Spaces

is isomorphic to the C'-direct product of the family

Hence, if we put

then F E

G

F and F is isomorphic to E .

Let F := ( F N ) N ~and I G := (GN)NELbe elements of F such that are isomorphic. Then there is a bijection rp : I + L such that

are isomorphic for every

N

E

I . By Theorem 5.6.7.9 c),

for every N E I , i.e. F = G . c) follows immediately from Theorem 5.6.7.9 d)

F

and

Name Index

Name Index Alaoglu, L. 1.2.8.1 Arens,R.F. 1.5.2.10,2.2.7.13 Arzela, C. 1.1.2.16 1.1.2.14, 1.1.2.16 Ascoli, G. Atkinson, F.V. 3.1.3.7, 3.1.3.11, 3.1.3.12, 3.1.3.21, 5.3.3.16 2.3.1.3 Autonne, L. Banach, S. 1.1.1.2, 1.2.8.2, 1.3.1.2, 1.3.2, 1.3.3.1, 1.3.4.1, 1.3.4.10, 1.4.1.2, 1.4.2.3, 1.4.2.19 2.2.5.4 Beurling, A. Bourbaki, S . 1.2.8.1 1.3.5.14 Branges, L. de Calkin, J.W. 5.4.3.5 3.1.3.1 Carlernan, T. Cauchy, A. 1.3.10.6 Choquet, G. 5.4.3.5 1.7.2.1 Dcdekind, R. 1.2.8.2, 3.1.3.9 Dicudonnb, J. Dixrnier, J. 4.4.4.4 4.3.2.13 Drewnoxski, L. 1.1.6.14 Dworetzky, .A. 4.1.3.7 Dye, H.A. Eberlein: W.F. 1.3.7.15 4.2.4.15 Effros, E.G. Enflo,P. 3.1.1.7 Ford, J.W.hl. 2.4.2.4 2.4.6.2 Fourier, J.-B.-J. Frkchet, IM. 1.1.1.2, 1.1.2.13, 5.2.5.2 Fredholm, E. 3.1.6.23 Frobenius, G.F. 2.1.4.21 Fuglede, B. 4.1.4.1 4.2.1.1 Fukarniya, M. 1.4.1.9, 2.2.5.4, 2.2.5.3, 2.4.1.2, 2.4.5.7, 4.1.1.1, 4.1.2.5 Gelfarid, 1.M. 4.1.3.1, 4.2.6.6, 5.4.1.2, 5.4.2.5 3.1.3.12 Gohberg, I. 1.3.6.8 Goldstine. H.H.

Goodearl, K.R. 2.2.1.19. 4.1.1.1, 5.4.2.13 1.2.1.12 Gowers, W.T. Gram, J.P. 5.5.1.18 1.6.1.1,3.1.6.25,4.2.8.13 Grothendieck,A. Hahn, H. 1.1.1.2, 1.2.1.3, 1.3.3.1, 1.3.6.1, 1.3.6.3, 1.3.8.1, 1.4.1.4 Hamilton, W.R. 2.1.4.17 Hellinger, E. 5.2.5.4 1.1.1.2, 1.3.3.13 Helly, E. 2.1.3.1 Hilbert, D. Hirschfeld, R.A. 2.2.5.6 2.1.3.10 Jacobson, N. James, R.C. 1.3.8.1 Jordan, C. 5.1.1.6 Kadison, R.K. 4.3.3.20 Kaplanski,I. 4.1.2.1,4.2.6.5,4.4.2.24,5.6.1.1 Kelley, J.L. 4.2.1.1 1.2.3.11 Kojirna, ?? Kolrnogoroff, A. 1.1.1.2 Kottrnan, C.A. E 1.3.5 Krein, M.G. 1.3.1.10, 1.3.7.3 Laguerre, E.N. 2.2.3.5 Laurent, P.A. 1.3.10.8 5.3.1.3 Lax, P.D. Lt Page, C. 2.2.3 8, 2.2.4.3, 2.2.5.6 Lindenstrauss, J. 1.2.5.13 Liouville, J. 1.3.10.6 3.1.5.10 Lornonsov, V.I. 5.1.1.1 Lowing, H. .Mackey, G.W. 1.3.7.2 .Vazur, S. 2.2.5.5 Mihlin, S.G. 3.1.3.12 Milgrarn, A.S. 5.3.1.3 Milrnan, D.P. 1.3.1.10 1.1.1.2, 1.1.3.4 Minkowski, H. Murray, F.J. 1.2.5.8 2.2.1.1 Nagurno, M. Nairnark, M.A. 4.1.1.1, 4.1.2.5, 4.1.3.1, 4.2.6.6, 5.4.1.2, 5.4.2.5, 5.5.1.24 Ncurnann, C. 2.2.3.5

Name Index

1.1.1.2, 3.1.3.1, 5.1.1.1, 5.1.1.6, 5.2.4 Neurnann, J . von 4.3.2.14, 5.2.1.2 Nikodym, 0. 3.1.3.1 Noether, F. Palmer, T.W. 4.1.1.1 5.6.1.1, 5.6.2.2, 5.6.2.6, 5.6.2.11, 5.6.3.3, 5.6.3.5 Paschke, W.L. Pedersen, G.K. 4.2.4.16 2.2.1.15 Peter, F. Pettis, P.J. 1.3.8.4, 1.3.8.5 1.2.5.14, E 1.3.3, 2.1.4.9 Phillips, R.S. Pierce,B. 2.1.1.1,2.1.3.6 Plancherel, M. 5.5.4.1 Putnam, I.F. 4.1.4.1 Rellich, F. 5.1.1.1 Rickart, C.E. 4.1.1.20, 4.1.2.12 Rieffel, 1M.A. 5.6.1.1 1.1.1.2, 1.1.4.4, 1.2.1.1, 1.2.2.5, 2.2.5.1, 3.1.1.1, Riesz, F. 3.1.3.8, 3.1.3.17, 3.1.5.1, 5.2.1.2, 5.2.5.2, 5.3.3.20 Rogers, C.A. 1.1.6.14 4.1.4.1 Rosenblum, M. Russo, B. 4.1.3.7 Sakai, S. 4.4, 4.4.1.1, 4.4.3.5 Schauder, J.P. 3.1.1.22 Schmidt, E. 1.1.1.2, 5.5.1.18 Schur, I. 1.2.3.11, 1.2.3.12, 1.3.6.11 Schwartz, L. 2.4.6.5, 2.4.6.8 4.2.6.2, 4.2.6.5, 4.2.8.2, 5.4.1.2 Segal, I.E. Shirali, S. 2.4.2.4 Sierpinski, W. 1.1.2.17 Silow, G. 2.2.4.27 Srnulian,V. 1.3.7.3,1.3.7.15 Steinhaus, H.D. 1.4.1.2 1.3.4.10, 1.3.5.16, 2.3.3.12, 4.1.2.5 Stone, M.H. 4.2.6.3 Starrner, E. Toeplitz, 0. 1.2.3.4, 2.3.1.3, 5.2.5.4 Vaught, R.L. 4.2.1.1 Vitushkin, A.G. 2.4.3.7 Volterra, V. 2.2.4.22 1.3.5.16 Weierstrass, K.

Weyl, H . 2.2.1.15 2.2.5.8 Wielandt, H. Wiener, N . 2.4.5.7 Yood, B. 3.1.3.11, 3.1.3.12, 4.1.1.13 ~elazko,W. 2.2.5.6

Subject lndex

Subject Index NT means Notation and Terminology

(A, B,C)-mliltiplication

1.5.1.1

absolute value of a number

1.1.1.1 absolute value of a measure NT absolutely convex 1.2.7.1 absolutely convex closed hull 1.2.7.6 absolutely convex hull 1.2.7.4 absolutely summable family 1.1.6.9 acts irreducibly 5.3.2.19 acts non-degenerately 5.3.2.19 additive group NT adherence, point of NT atihererit point NT adjoint 2.3.1.1,5.6.1.8 adjoint differential operator 3.2.2.3 adjoint kernel 3.1.6.5 adjoint operator 5.3.1.4 adjoint sesquilinear form 2.3.3.1 adjoint sesquilir~earmap 2.3.3.1 adjoiritable 5.6.1.7 algebra 2.1.1.1 algebra, Calkin 3.1.1.13 algebra, complex 2.1.1.1 algebra, dsgerierate 2.1.1.1 algebra, division 2.1.2.1 algebra, Gelfand 2.4.1.1 algebra, Gelfand unital 2.4.2.1 algebra, involutive 2.3.1.3 algebra, involutive Gelfand 2.4.2.1 algebra, irivolutive uriital Gelfand 2.4.2.1 algebra, normed 2.2.1.1 algebra, real 2.1.1.1 algebra, semi-simple 2.1.3.18 algebra, strongly symmetric 2.3.1.26

algebra, symmetric 2.3.1.26 algebra, unital 2.1.1.3 algebra, unital Gelfand 2.4.1.1 algebra homomorphism 2.1.1.6 algebra homomorphism associated to A 5.4.2.3 algebra homomorphism associated to x' 5.4.1.2 algebra homomorphism associated to (x:),,, 5.4.2.2 algebra homomorphism, unital 2.1.1.6 algebra isomorphism 2.1.1.6 algebra isomorphism, unital 2.1.1.6 algebraic dimension 1.1.2.18 algebraic dual 1.1.1.1 algebraic eigenspace 5.3.3.20 algebraic eigenvector 5.3.3.20 algebraic isomorphism, associated 1.2.4.6 algebraic multiplicity 5.3.3.20 algebras, isomorphism of involutive 2.3.1.3 analytic function 1.3.10.1 approximate unit 2.2.1.15 approximate unit of a C*-algebra, canonical 4.2.8.2 Arens multiplication, left 1.5.2.10, 2.2.7.13 Arens mutliplication, right 1.5.2.10, 2.2.7.13 associated algebraic isomorphism 1.2.4.6 associated quadratic form 2.3.3.1 associated quadratic map 2.3.3.1 associated unital C*-algebra 4.1.1.13 atom 4.3.2.20 atomic 4.3.2.20 atomless 4.3.2.20 Baire function 1.7.2.12 Baire set 1.7.2.12 ball, unit 1.1.1.2 Banach Banach Banach Banach Banach Banach

algebra 2.2.1.1 algebra, involutive 2.3.2.1 algebra, quasiunital 2.2.1.15 algebra, rrnital 2.2.1.1 categories, functor of 1.5.2.1 categories, functor of unital 1.5.2.1

Subject Index

Banach Banach Banach Banach Banach Banach Banach Banach Banach

category 1.5.1.1 category, unital 1.5.1.1 space 1.1.1.2 space, complex 1.1.1.2 space, involutive 2.3.2.1 space, ordered 1.7.1.4 space, real 1.1.1.2 subalgebra generated by 2.2.1.9 system 1.5.1.1 1.5.1.9 Banach system, bidual of a Banach system, dual of a 1.5.1.9 Banach systems, isometric 1.5.2.1 band 1.7.2.1 basis, Fourier 5.6.3.1 1 basis, orthonorrrial 5.5.1.1 basis of l 2 ( T ) canonical , orthonormal 5.5.3.1 Bergman kernel 5.2.5.9 Bessel's ideritity 5.5.1.7 Bessel's inequality 5.5.1.8 bicommutant 2.1.1.16 bidual of a Banach systerr~ 1.5.1.9 1.3.6.1 bidual of a normed space bijective NT bilinear map 1.2.9.1 binomial theorem 2.2.3.12 bitranspose 1.3.6.15 bound, lower 1.7.2.1 bound, upper 1.7.2.1 bounded map 1.1.1.2 bounded operator 1.2.1.3 bounded operator, lower 1.2.1.18 bounded sequence 1.1.1.2 bounded set 1.1.1.2 boundedness, principle of uniform 1.4.1.2 bouridedness theorerri, Nikodym's 4.3.2.14 C* algebra 4.1.1.1 C*-algebra, canonical approximate unit of a 4.2.1.2 C*-algebra, carionical order of a

4.2.8.2

C*-algebra, complex 4.1.1.1 C*-algebra, complex unital 4.1.1.1 C*-algebra, Gelfand 4.1.1.1 C*-algebra, purely real 4.1.1.8 C*-algebra, real 4.1.1.1 C*-algebra, real unital 4.1.1.1 C*-algebra, simple 4.3.5.1 C*-algebra, unital 4.1.1.1 C*-algebra associated, unital 4.1.1.13 Calkin algebra 3.1.1.13 Calkin category 3.1.1.12 canonical approximate unit of a C*-algebra 4.2.8.2 canonical involution of EF 2.3.1.1 canonical metric of a normed space 1.1.1.2 canonical norm of a pre-Hilbert space 5.1.1.2 canonical norm of e ( E ,F ) 1.2.1.9 canonical order of a CW-algebra 4.2.1.2 canonical orthonormal basis of e Z ( T ) 5.5.3.1 canonical projection of the tridual of E 1.3.6.19 canonical scalar product of IKn 5.1.2.4 cardinal number NT cardinality, topological NT 4.3.3.1 carrier carrier, left 4.3.3.1 carrier, right 4.3.3.1 carrier of a function NT carrier of a Radon measure NT 1.5.1.1 category, Banach C*-direct product 4.1.1.6 C*-direct sum 4.1.1.6 character 2.4.1.1 characteristic function of a set 1.1.2.1 C*-hull 4.1.1.22 class NT 1.4.2.19 closed graph theorem closed involutivc subalgebra generated by 2.3.2.14, 2.3.2.15 closet1 invollltive unital subalgebra generated by 2.3.2.14, 2.3.2.15 closed subalgebra generated by 2.2.1.9

Subject Index

closed unital subalgebra generated by closed vector subspace generated by C*--module,Hilbert right 5.6.1.4 codimension 1.2.4.1 codomain NT

2.2.1.9 1.1.5.5

cokernel of a linear map 1.2.4.5 commutant 2.1.1.16 commutative 2.1.1.1 commutative monoid E 2.1.1 compact, relatively 1.1.2.9 compact operator 3.1.1.1 compatible, simultaneously 1.5.1.1 compatible (left and right) multiplications complement of a subspace 1.2.5.3 complernented subspace 1.2.5.3 complete, C-order 4.3.2.3

1.5.1.1

complete, order 1.7.2.1 complete norm 1.1.1.2 complete normed space 1.1.1.2 complete ordered set 1.7.2.1 completion of a normed algebra 2.2.1.13 completion of a normed space 1.3.9.1 completion of a pre-Hilbert space 5.1.1.7 complex algebra 2.1.1.1 complex Banach spacc 1.1.1.2 complex C'-algebra 4.1.1.1 complex C*-algebra, unital 4.1.1.1 complex Hilbert space 5.1.1.2 complex normed algebra 2.2.1.1 complex normed space 1.1.1.2 complex pre-Hilbert space 5.1.1.1 complex unital C*--algebra 4.1.1.1 complex universal representation 5.4.2.16 complex Wa-algebra 4.4.1.1 complexification of algebras 2.1.5.7 complexification of Banach algebras 2.2.1.19 complexification of Hilbert spaces 5.3.1.8 complexification of involutive algebras 2.3.1.40

complexification of involutive vector spaces 2.3.1.38 complexification of right C* modules 5.6.1.6 complexification of vector spaces 2.1.5.1 composition of functors 1.5.2.1 composition of maps NT compression of a representation 5.4.2.8 cone 1.3.7.4 cone, sharp 1.3.7.4 conjugacy class 2.2.2.7 conjugate exponent of 1.2.2.1 conjugate exponents 1.2.2.1 conjugate exponents, weakly 1.2.2.1 conjugate involution 2.3.1.3 conjugate linear map 1.3.7.10 conjugate number 1.1.1.1 continuous, order 1.7.2.3 convergence, radius of 1.1.6.22 convcx 1.2.7.1 convex, absolutely 1.2.7.1 convcx closed hull 1.2.7.6 convex closed hull, absolutely 1.2.7.6 convex hull 1.2.7.4 convex hull, absolutely 1.2.7.4 convolution 2.2.2.7,2.2.2.10 C-order complete 4.3.2.3 C-order u-complete 4.3.2.3 C* -subalgebra 4.1.1.1 C * subalgebra, unital 4.1 .I .1 C'-subalgebra generated by 4.1.1.1 C*-subalgebra generated by, hereditary 4.3.4.1 C*-subalgebra generated by, unital 4.1.1.1 cyclic element 5.4.1.1 cyclic representation 5.4.1.1 cyclic vcctor 5.3.2.19, 5.4.1.1 cyclic vector associated to x' 5.4.1.2 decomposition, spectral 4.3.2.19, 5.3.4.7 degenerate algebra 2.1.1.1 derivative 1.1.6.24

Subject Index

diagonalization of u 5.5.6.1 differentiable 1.1.6.24 3.2.2.3 differential operator, adjoint differential operator, selfadjoint 3.2.2.3 dimension, algebraic 1.1.2.18 dimension, Hilbert 5.5.2.2 Dirac measure 1.2.7.14 direct integral of E with respect to p 5.5.2.19 direct sum 1.2.5.3 directed, downward 1.1.6.1 directed, upward 1.1.6.1 disjoint family of sets 1.2.3.9 distance of a point from a set 1.1.4.1 division algebra 2.1.2.1 NT domain downward directed 1.1.6.1 dual, algebraic 1.1.1.1 dual of a Banach system 1.5.1.9 dual of a normed space 1.2.1.3 dual space 1.3.1.11 E-algebra 2.2.7.1 E -algebra, involutive 2.3.6.1 E-algebra, involutive unital 2.3.6.1 E-algebra, unital 2.2.7.1 E-algebras, homomorphisni of 2.2.7.1 E-algebras, homomorphism of involutive 2.3.6.1 E-algebras, homomorphism of involutive unital 2.3.6.1 E- algebras, homomorphism of u~iital 2.2.7.1 E- C*-algebra 5.6.1.10 E--C*-algebra, unital 5.6.1.10 E-C*-algebras, isomorphism of 5.6.1.10 eigenspace 3.1.4.1 eigenspace, algebraic 5.3.3.20 eigenvalue 3.1.4.1 eigenvector 3.1.4.1 eigenvector, algebraic 5.3.3.20 E--module 2.2.7.1 5.6.1.4 E-module, Hilbert

E-module, Hilbert right 5.6.1.4 E-module, inner product right 5.6.1.1 E-module, involutive 2.3.6.1 E-module, involutive unital 2.3.6.1 E-module, semi-inner-product right 5.6.1.1 E-module, weak semi-inner-product right 5.6.1.1 E--module, unital 2.2.7.1 E-module, unital Hilbert 5.6.1.4 E-module, von Neumann (right) 5.6.3.2 E-modules, homomorphism of 2.2.7.1 E-modules, homomorphism of involutive 2.3.6.1 equicontinuous 1.1.2.14 equivalence class NT equivalence class of a point NT equivalence of GNS-triples 5.4.1.2 equivalence of representations 5.4.1.1 equivalence relation NT equivalent G N S - triples 5.4.1.2 equivalent norms 1.1.1.2 equivalent representations 5.4.1.1 essential spectrum 3.1.3.24 E-submodule 2.2.7.1 Euclidean norm 1.1.5.2 evaluation 1.2.1.8 evaluation functor 1.5.2.1 evaluation operator of a normed space 1.3.6.3 E-valued spectral measure 4.3.2.16 exact set 1.7.2.12 expansion, Fourier 5.5.1.15 2.2.3.5 exponential function exponents, conjugate 1.2.2.1 exponents, weakly conjugate 1.2.2.1 extreme point 1.2.7.9 faceofaconvexset 1.2.7.9 factorization of a linear map 1.2.4.6 faithful, order 4.2.2.18 faithful representation 5.4.1.1 family NT

Subject Index

1.1.6.9 family, absolutely srimmable family, sum of a 1.1.6.2 family, summable 1.1.6.2 family of sets, disjoint 1.2.3.9 filter, lower section 1.1.6.1 filter, upper section 1.1.6.1 NT filter of cofinite subsets finite-dimensional 1.1.2.18 F-invariant 3.1.4.4 Fourier basis 5.6.3.11 Fourier expansion 5.5.1.15 Fourier integral 2.4.6.2 Fourier-Plancherel operator 5.5.4.1 Fourier set 5.6.3.11 Fourier transform 2.4.6.2 Frdchet-Ricsz Theorem 5.2.5.2 Fredholm alternative 3.1.6.23 Frcdholm operator 3.1.3.1 Fredholm operator, index of a 3.1.3.1 free ultrafilter NT function NT function, Bairc 1.7.2.12 function, step NT functional calculus 4.1.3 functor 1.5.2.1 functor, identity 1.5.2.1 functor, inclusion 1.5.2.16 functor, isometric 1.5.2.1 functor, quotient 1.5.2.17 functor, transpose of a 1.5.2.3 functor of (unital) Banach categories 1.5.2.1 1.5.2.1 functor of (unital) A-categories functor of (left, right) A-modules 1.5.2.1 1.3.2.1 functors, composition of Gelfand, Theorem of 2.2.5.4 Gelfand algebra 2.4.1.1 Gelfand algebra, involutive 2.4.2.1 Gelfand algebra, involutive unital 2.4.2.1

2.4.1.1 Gelfand algebra, spectrum of a Gelfand algebra, unital 2.4.1.1 Gelfand C' algebra 4.1.1.1 Gelfand-Mazur, Theorern of 2.2.5.5 Gelfand transform 2.4.1.2 GNS-construction 5.4.1.2 5.4.1.2 G N S triple of E associated to x' GNS-triples, equivalence of 5.4.1.2 GNS-triples, equivalent 5.4.1.2 Gram-Schmidt orthonormalization 3.5.1.18 graph NT, 1.4.2.18 Green function 3.2.1.2 group, additive NT Hahn-Banach Theorem 1.3.3.1 hereditary 4.3.4.1 hereditary C*~-subalgebragenerated by 4.3.4.1 Hermitian sesquilinear map 2.3.3.3 Hilbert dimension 5.5.2.2 Hilbert E-module 5.6.1.4 Hilbert E-module, unital 5.6.1.4 Hilbert right C*-module 5.6.1.4 Hilbert right E-module 5.6.1.4 Hilbert right E-modules, isomorphic 5.6.1.7 Hilbert space 5.1.1.2 Hilbert space, complex 5.1.1.2 Hilbert space, complexification 5.3.1.8 Hilbert Hilbert Hilbert Hilbert Hilbert Hilbert

space, involutive 5.5.7.1 space, real 5.1.1.2 5.4.2.3 space associated to A 5.4.1.2 space associated to x' 5.4.2.2 space associated to (x:),,, 5.1.2.3 space of square summable sequences 5.1.3.1 Hilbert sum of a family of Hilbert spaces 5.4.2.1 Hilbert sum of a family of representations

Hiilder inequality 1.2.2.5 homogeneous W*-algebra 5.6.7.8 homomorphism of C*-algebra 4.1.1 .2O honiornorphism of E-algebras 2.2.7.1

Subject Index

homomorphism honiomorphism hornomorphism homomorphism honiomorphism

of of of of of

E- modules

2.2.7.1 involutive E-algebras 2.3.6.1 involutive E-modules 2.3.6.1 2.3.6.1 involutive unital E-algebras uriital E-algebras 2.2.7.1

hyperstonian space ideal 2.1.1.1 ideal, left 2.1.1.1 ideal, maximal proper

1.7.2.12

2.1.1.1

ideal, maxirnal proper left ideal, maximal proper right

2.1.1.1 2.1.1.1

ideal,proper 2.1.1.1 ideal, proper left 2.1.1.1 ideal, proper right 2.1.1.1 ideal, regular maximal proper 2.1.3.17 ideal, regular maximal proper left 2.1.3.17 ideal, regular maximal proper right 2.1.3.17 ideal, right 2.1.1.1 ideal generated by 2.1.1.2 idempotent 2.1.3.6 identical representation 5.5.1.23 identity functor 1.5.2.1 identity map NT identity operator 1.2.1.3 iff NT image of a linear map 1.2.4.5 imaginary part 1.I .1. I , 2.3.1.22 inclusiori functor 1.5.2.16 NT inclusion map index of a Fredholm operator 3.1.3.1 indexofU 3.1.3.21 induced norm 1.1.1.2 infimum 1.7.2.1

infinite-dimensional

1.1.2.18 infinite niatrix 1.2.3.1 injective N'r inner multiplication 1.5.1.1 inner-product 5.6.1.1

inner-product right E--module 5.6.1.1 integral of E with respect t o p , direct 5.5.2.19 interior point NT invariant vector subspace 3.1.4.4 inverse of a bijective m a p NT inverse of a morphism 1.5.1.6 inverse of a n element in a unital algebra 2.1.2.4 inverse operators, principle of 1.4.2.4 1.5.1.5,2.1.2.1 invertible invertible, left 1.5.1.5 invertible, right 1.5.1.5 involution 2.3.1.1 involution, conjugate 2.3.1.3 2.3.1.1 involution of E F , canonical involutive algebra 2.3.1.3 involutive algebra, complexification of an 2.3.1.40 2.3.1.26 Proposition involutive algebra, strongly symmetric involutive algebra, symmetric 2.3.1.26 involutive algebras, isomorphism of 2.3.1.3 involutive Banach algebra 2.3.2.1 2.3.2.1 involutive Banach space involutive Banach unital algebra associated t o 2.3.2.9 involutive E-algebra 2.3.6.1 involutive E-module 2.3.6.1 2.4.2.1 involutive Gelfand algebra involutive Hilbert space 5.5.7.1 involutive map 2.3.1.1 involutive normed algebra 2.3.2.1 involutive normed space 2.3.2.1 involutive normed unital algebra associated t o 2.3.2.9 involutive set 2.3.1.1 involutivcspace 2.3.1.1 involutive subalgebra generated by 2.3.1.18 involutive unital algebra associated t o 2.3.1.9 2.3.6.1 involutive unital E-algebra involutivc unital E -module 2.3.6.1 involutive unital Gelfand algebra 2.4.2.1 involutive unital subalgcbra generated by 2.3.1.18

Subject Index

involutive vector space 2.3.1.3 involutive vector spaces, isomorphism of 2.3.1.3 involutive vector subspace generated by 2.3.1.18 irreducible representation 5.4.1.1 irreducibly, acts 5.3.2.19 isometric Banach systems 1.5.2.1 isometric functor 1.5.2.1 isometric normed algebras 2.2.1.1 isometric normed spaces 1.2.1.12 isometric normed unital algebras 2.2.1.1 isometry of Hilbert spaces, partial 5.3.2.25 isometry of normed algebras 2.2.1.1 4.4.4.5 isometry of W*-algebras isometry of normed spaces 1.2.1.12 isometry of normed unital algebras 2.2.1.1 isomorphic algebras 2.1.1.6 isomorphic Hilbert right E-modules 5.6.1.7 isomorphic normeti algebras 2.2.1.1 isomorphic riormed spaces 1.2.1.12 isomorphic normed unital algebras 2.2.1.1 isomorphic unital algebras 2.1.1.6 isomorphism, algebra 2.1.1.6 isomorphism associated to a linear map, algebraic isomorphism of E-C*-algebras 5.6.1.10 isomorphism of involutive algebras 2.3.1.3 isomorphism of involutive vector spaces 2.3.1.3 isomorphism of normed algebras 2.2.1.1 isomorphism of normed spaces 1.2.1.I2 isomorphism of normed unital algebras 2.2.1.1 kernel, Bergman 5.2.5.9 kernel of a linear map 1.2.4.5 Kronecker's symbol 1.2.2.6 lattice 1.7.2.1 lattice, vector 1.7.2.1 Laurent series 1.3.10.8, 1.3.10.9 Lax-~Milgrarn Theorem 5.3.1.3 left Arens multiplication 1.5.2.10, 2.2.7.13 left carrier 4.3.3.1

1.2.4.6

left ideal 2.1.1.1 left ideal, maximal proper left ideal, proper 2.1.1.1 left left left left

2.1.1.1

ideal, regular maximal proper 2.1.3.17 ideal generated by 2.1.1.2 invertible 1.5.1.5 multiplication 1.5.1 . l , 5.6.1.4

left shift 1.2.2.9, E 1.2.11 left (unital) A - module 1.5.1.10 linear form 1.1.1.1 linear form, positive 1.7.1.9 linear map, conjugate 1.3.7.10 lower bound 1.7.2.1 lower bounded operator 1.2.1.18 lower section filter 1.1.6.1 L2-distributions, in the sense of map NT map, bilinear 1.2.9.1 map, map, map, map,

3.2.2.3

bounded 1.1.1.2 conjugate linear 1.3.7.10 identity NT inclusion NT

map, inverse of a bijective map, involutive 2.3.1.1

NT

map, nuclear 1.6.1.1 map, qrlotier~t 1.2.4.1 maps, composition of KT matrix, infinite 1.2.3.1 maximal proper ideal 2.1.1.1 maximal proper ideal, regular 2.1.3.17 maximal proper left ideal 2.1.1.1 maximal proper left ideal, regular 2.1.3.17 maximal proper right ideal 2.1.1.1 maximal proper right ideal, regular 2.1.3.17 mean ergodic theorem 5.2.4.3 measure, Dirac 1.2.7.14 measlrrc, E-valued spectral 4.3.2.19 measure, Radon NT

Subject Index

measure of x , spectral 4.3.2.19 measure space, u-finite 3.1.6.14 metric of a normed space, canonical 1.1.1.2 module 2.2.7.1 module, involutive 2.3.6.1 module, involutive unital 2.3.6.1 module, unital 2.2.7.1 modriles,homomorphismof 2.2.7.1 modules, homomorphism of involutive 2.3.6.1 modulo NT modulus 4.2.5.1,4.4.3.5 monoid E 2.1.1 E 2.1.1 monoid, commutative morphism 1.5.1.1 morphism, inverse of a 1.5.1.6 multipliable sequence 2.2.4.33 multiplication 2.1.1.1 multiplication, (A,f?, C) 1.5.1.1 multiplication, compatible (left and right) multiplication, inner 1.5.1.1 multiplication, left 1.5.1.1, 5.6.1.4

1.5.1.1

multiplication, left (right) Arens 1.5.2.10, 2.2.7.13 multiplication, right 1.5.1.1, 5.6.1.1 m~lltiplicationoperator 2.2.2.22 multiplicity 3.1.4.1 multiplicity, algebraic 5.3.3.20 negative 1.7.1.1 negative part 4.2.2.9, 4.2.8.13 Nikodym's boundcdncss theorrm 4.3.2.14 nilpotent 2.1.1.1 non-degenerate representation 5.4.1.1 non-degenerately, acts 5.3.2.19 norm 1.1.1.2 norm, complete norm, Euclidean norm, induced norm, quotient norm, supremum

1.1.1.2 1.1.5.2 1.1.1.2 1.2.4.2 1.1.2.2, 1.1.5.2

norm of an operator 1.2.1.3 norm of a pre-Hilbert space, canonical

5.1.1.2 norm of L ( E , F) , canonical 1.2.1.9 norm topology 1.1.1.2 normal 2.3.1.3 normed algebra 2.2.1.1 normed algebra, completion of a 2.2.1.13 normed algebra, complex 2.2.1.1 normed normed normed normed normed normed normed norrned

algebra, involutive 2.3.2.1 algebra, quasiunital 2.2.1.15 algebra, real 2.2.1.1 algebras, isometric 2.2.1.1 algebras, isometry of 2.2.1.1 algebras, isomorphic 2.2.1.1 algebras, isomorphism of 2.2.1.1 space 1.1.1.2

norrned normed normed normed normed normed normed normed normed normed normed normed

space, bidual of a 1.3.6.1 spare, complete 1.1.1.2 space, completion of a 1.3.9.1 space, complex 1.1.1.2 space, involutive 2.3.2.1 space, ordered 1.7.1.4 space, real 1.1.1.2 spaces, isometric 1.2.1.12 spaces, isometry of 1.2.1.12 spaces, isomorphic 1.2.1.12 spaces, isomorphism of 1.2.1.12 unital algebra 2.2.1.1

norrned unital algebras, isometric

2.2.1.1

normed unital algebras, isometry of normed unital algebras, isomorphic normed unital algebras, isomorphism of norms, equivalent 1. l . 1.2 nuclear map 1.6.1.1 number, cardinal NT number, ordinal NT object of a Bariach system 1.5.1.1 onto NT

2.2.1.1 2.2.1.1 2.2.1.1

Subject Index

open mapping principle 1.4.2.3 operator 1.2.1.3 5.3.1.4 operator, adjoint operator, adjoint differential 3.2.2.3 operator, bounded 1.2.1.3 operator, compact 3.1.1.1 5.3.4.1 operator, Fourier-Plancherel operator, Fredholm 3.1.3.1 operator, identity 1.2.1.3 operator, index of a Fredholm 3.1.3.1 operator, lower bounded 1.2.1.18 operator, multiplication 2.2.2.22 operator, order of an 3.1.3.18 operator, selfadjoint differential 3.2.2.3 operators, principle of inverse 1.4.2.4 order complete 1.7.2.1 order continuous 1.7.2.3 order faithful 4.2.2.18 order of a pole 1.3.10.9 order relation of a C*-algebra, canonical 4.2.1.2 order summat)le 1.7.2.10 order a-complete 1.7.2.1 order a-continuous 1.7.2.3 order a-faithful 4.2.2.18 ordered Banach space 1.7.1.4 ordered normed space 1.7.1.4 ordered set, complete 1.7.2.1 NT ordered set, totally ordered set, a-complete 1.7.2.1 ordered vector space 1.7.1.1 IVT ordinal number orthogorial 5.2.2.1 4.1.2.18, 5.2.3.2 orthogonal projection orthogonal set of A 5.2.2.1 orthogonal sets 5.2.2.1 orthogonal vectors 5.2.2.1 orthonormal basis 5.5.1.1 orthonorn~albasis of e2(T), canonical 5.5.3.1

orthonorrnal farnily 5.5.1.1 orthonormal set 5.5.1.1 orthonormalization, Gram-Schmidt parallelogram law 2.3.3.2 Parseval'sEquation 5.5.1.15 partial isometry of Hilbert spaces partition of a set NT pnorm 1.1.2.5,1.1.5.2 point, adherent NT point, extreme 1.2.7.9 point, interior NT

5.5.1.18

5.3.2.25

point of adherence NT point spectrum 3.1.4.1 polar 1.3.5.1 polar representation 4.2.6.9, 4.4.3.1, 4.4.3.5 polarization identity 2.3.3.2 pole (of order) 1.3.10.9 positive 1.7.1.1, 2.3.3.3, 2.3.4.1 positive linear form 1.7.1.9, 2.3.4.1 positive part 4.2.2.9,4.2.8.13 power series 1.1.6.22 precompact 1.1.2.9 predual of a Banach space 1.3.1.11 predual of a W'-algebra 4.4.1.1, 4.4.4.4 pre-Hilbert space 5.1.1.1 pre-Hilbert space, canonical norm of a 5.1.1.2 pre - Hilbert space, completion of a 5.1.1.7 pre Hilbert space, complex 5.1.1.1 pre-Hilbert space, real 5.1.1.1 prepolar 1.3.5.1 pretranspose of an operator 1.3.4.9, 4.4.4.8 principal part 1.3.10.8, 1.3.10.9 principle of inverse operators 1.4.2.4 principle of open mapping 1.4.2.3 principle of uniform boundedness 1.4.1.2 product 2.1.1.1 product, C'-direct 4.1.1.6 product of a farriily of sets NT

Subject Index

product of a sequence 2.2.4.33 product, scalar 5.1.1.1 product associated to f , scalar 5.1.2.9 projection 1.2.5.7 projection, orthogonal

4.1.2.18, 5.2.3.2

projection of the tridual of E , canonical proper ideal 2.1 .l.1

1.3.6.19

proper ideal, maximal 2.1.1.1 proper ideal, regular maximal 2.1.3.17 proper left ideal 2.1.1.1 proper left ideal, maximal

2.1.1.1

proper left ideal, reguar maximal 2.1.3.17 proper right ideal 2.1.1.1 proper right ideal, maximal 2.1.1.1 proper right ideal, regular maximal 2.1.3.17 pure state 2.3.5.1 pure statc space 2.3.5.1 purely rcal C*-algebra 4.1.1.8 Pythagoras' Theorem 5.2.2.3 quadratic form, associated 2.3.3.1 quadratic map, associated 2.3.3.1 quasinilpotent 2.2.4.20 q~rasiunital 2.2.1.15 quaternion 2.1.4.17 quotient functor 1.5.2.17 quotient map NT, 1.2.4.1 quotient norm 1.2.4.2 quotient space 1.2.4.2 quotient A--category 1.5.2.17 quotient A-module 1.5.2.17 Raabe's ratio test 2.2.3.11 radical 2.1.3.18 radius of convergence 1.1.6.22 Radon mcasure NT Radon-Nikodym Theorem 4.4.3.15 range of values KT real algebra 2.1.1.1 real Banach space 1.1.1.2

real C*-algebra 4.1.1.1 real C*-algebra, purely 4.1.1.8 real C'-algebra, unital 4.1.1.1 real Hilbert space 5.1.1.2 real normed space 1.1.1.2 realpart 1.1.1.1,2.3.1.3 real pre-Hilbert space 5.1.1.1 real W*-algebra 4.4.1.1 reduces u reflexive

5.2.3.11 1.3.8.1

regular maximal proper ideal

2.1.3.17

regular maximal proper left ideal 2.1.3.17 regular maximal proper right ideal 2.1.3.17 relatively compact 1.1.2.9 representation 5.4.1.1 representation, associated to A 5.4.2.3 representation, associated to x' 5.4.1.2 representation, associated to ( x , ) , , ~ 5.4.2.2 representation, complex universal 5.4.2.6 representation, compression of a 5.4.2.8 representation, cyclic 5.4.1.1 representation, faithful 5.4.1.1 representation, identical 5.5.1.23 representation, irreducible 5.4.1.1 representation, non-degenerate 5.4.1.1 representation, unital 5.4.1.1 representation, universal 5.4.2.3 representation, 05.4.1.1 representations, equivalence of 5.4.1.1 representations, equivalent 5.4.1.1 representations, Hilbert sum of 5.4.2.1 residue 1.3.10.8, 1.3.10.9 resolvent 2.1.3.1 resolvent equation 2.1.3.9 Riesz, theorem of 2.2.5.1 right Arens mllltiplication 1.5.2.10, 2.2.7.13 right carrier 4.3.3.1 right C*-module, tlilbert 5.6.1.4

Subject Index

right E--module, Hilbert 5.6.1.4 right E--module, inner-product 5.6.1.1 rightE-module,semi-inner-product 5.6.1.1 5.6.3.2 right E-modrlle, von Neumann right E-module, weak semi-inner-product 5.6.1.1 right ideal 2.1.1.1 right ideal, maximal proper 2.1.1.1 right ideal, proper 2.1.1.1 2.1.3.17 right ideal, regular maximal proper right ideal generated by 2.1.1.2 right invertible 1.5.1.5 right multiplication 1.5.1.1, 5.6.1.1 right shift 1.2.2.9, E 1.2.11 1.5.1.10 right (unital) A-module, right W*-module, vori Neumann 5.6.3.2 scalar 1.1.1.1 scalar product 5.1.1.1 scalar product associated to f 5.1.2.9 scalar prodrlct of IKn , canonical 5.1.2.4 2.4.6.5 Schwartz space of rapidly decreasing Cm-functions Schwarz inequality 2.3.3.9, 5.1.1.2 section filter, lower 1.1.6.1 section filter, upper 1.1.6.1 selfadjoint 2.3.1.1 selfadjoint differential operator 3.2.2.3 self-dual 5.6.2.2 self-normal 2.3.1.3 sen~i-inner-product right E module 5.6.1.1 semi-inner-product right E-module, weak 5.6.1.1 seminorm 1.1.1.2 semi-simple algebra 2.1.3.18 separating vector 5.3.2.19, 5.4.4.1 NT sequence series, Laurent 1.3.10.8, 1.3.10.9 series, power 1.1.6.22 sesquiliriear form 2.3.3.1 sesquilinear form, adjoint 2.3.3.1 sesquilinear map 2.3.3.1

sesquilinear map, adjoint sesquilincar map, Hermitian set, Baire 1.7.2.12

2.3.3.1 2.3.3.3

set, bounded 1.1.1.2 set, complete ordered 1.7.2.1 set, exact 1.7.2.12 set, partition of a NT set, totally ordered NT set, p-null NT set, a-complete ordered 1.7.2.1 sharp cone 1.3.7.4 shift, left 1.2.2.9 shift, right 1.2.2.9 simple C*-algebra 4.3.5.1 simultaneously compatible 1.5.1.1 space, Banach 1.1.1.2 space, bidual of a norrned 1.3.6.1 space, complete rlormed 1.1.1.2 space, completion of a normed 1.3.9.1 space, complex Banach 1.1.1.2 space. complex Hilbert 5.1.1.2 space, complex normed 1.1.1.2 space, complex pre-Hilbert 5.1.1.1 space, dual 1.3.1.11 space, Hilbert 5.1.1.2 space, space, space: space, space,

hypcrstonian 1.7.2.12 involutive 2.3.1.1 involutive Banach 2.3.2.1 involutive normed 2.3.2.1 involutive vector 2.3.1.3

space, normed 1.1.1.2 space, ordered Banach 1.7.1.4 space, space, space, spacr, space, space,

ordered norrned 1.7.1.4 ordered vector 1.7.1.1 pre-Hilbert 5.1.1.1 pure state 2.3.5.1 quotient 1.2.4.2 real Banach 1.1.1.2

Subject Index

space,realHilbert 5.1.1.2 space, real normed 1.1.1.2 space, real pre-Hilbert 5.1.1.1 space, state 2.3.5.1 space, Stone 1.7.2.12 space, subspace of a normed space, vector 1.1.1.1 space, 0.-Stone

1.1.1.2

1.7.2.12

space of square summable sequences, Hilbert 5.1.2.3 spaces, isometric normed 1.2.1.12 spaces, isometry of normed 1.2.1.12 spaces, isomorphic normed 1.2.1.12 spaces, isomorphism of involutive vector 2.3.1.3 spaces, isomorphism of normed 1.2.1.12 spectral decomposition 4.3.2.19, 5.3.4.7 spectral rrleasure, E-valued 4.3.2.16 spectral measure of x 4.3.2.19 spectral radius 2.1.3.1 spectrum, essential 3.1.3.24 spectrum, point 3.1.4.1 spectrum of an element 2.1.3.1 spectrum of a Gelfand algebra 2.4.1.1 square summable sequences, Hilbert space of 5.1.2.3 state 2.3.5.1 state, pure 2.3.5.1 state space 2.3.5.1 state space, pure 2.3.5.1 step functiori NT Stone space 1.7.2.12 strongly synimetric invol~rtivealgebra 2.3.1.26 subalgebra 2.1.1.1 subalgebra, unital 2.1.1.3 subalgebra generated by 2.1.1.4 subalgebra generated by, invol~ltive 2.3.1.18 subspace, complemented 1.2.5.3 subspace generated by, closed vect,or 1.1.5.5 subspacc of a normcd space 1.1.1.2 sum, C* direct 4.1.1.6

sum, direct 1.2.5.3 sum of a family 1.1.6.2 5.6.3.6 sum of a family in FA sum of representations, Hilbert 5.4.2.1 summable, absolutely 1.1.6.9 summable, order 1.7.2.10 summable family 1.1.6.2 5.6.3.6 summable in FA support of a function NT NT support of a Radon measure supremum 1.7.2.1 1.1.2.2,1.1.5.2 supremumnorm surjective NT symbol, Kronecker's 1.2.2.6 symmetric involutive algebra 2.3.1.26 symmetric involutivc algebra, strongly 2.3.1.26 5.2.4.3 theorem, mean ergodic Theorem of Alaoglu-Bourbaki 1.2.8.1 Theorem of Banach 1.3.1.2 Theorem of Banach-Steinhaus 1.4.1.2 Theorem of closed graph 1.4.2.19 Theorem of Frkchet-Riesz 5.2.5.2 Theorem of Gelfand 2.2.5.4 2.2.5.5 Theorem of Gelfand-Mazur Theorem of Hahn-Banach 1.3.3.1 Theorem of Laurent 1.3.10.8 5.3.1.3 Theorem of Lax-14ilgram Theorem of Liouville 1.3.10.6 1.3.1.10 Theorem of Krein-Milman 1.3.7.3 Theorem of rein-Smulian Theorem of Minkowski 1.1.3.4 1.2.5.8 Theorem of Murray Theorem of Pythagoras 5.2.2.3 4.4.3.15 Theorem of Radon~-Nikodyni Theorem of Riesz 2.2.5.1 Theorem of Weierstrass- Stone 1.3.5.16 topological cardinality NT topological zero-divisor 2.2.4.24

Sribjcct Index

topology, norm 1.1.1.2 topology, weak 1.3.6.9 NT totally ordered set transpose kernel 3.1.6.5 transpose of a functor 1.5.2.3 1.3.4.1 transpose of an operator transpose unital category of L 1.3.2.2 transposition functor of L 1.5.2.2 triangle inequality 1.1.1.2 tridual of a Banach system 1.5.1.9 tridual of a normcd space 1.3.6.1 type I W*-algebra 5.6.7.11 u-invariant 3.1.4.4 NT ultrafilter, frce uniform boundedness, principle of 1.4.1.2 unit 1.5.1.1, 1.5.1.4, 2.1.1.1 unit, approximate 2.2.1.15 1.1.1.2 unit ball unit of an inner multiplication 1.5.1.1 unital algebra 2.1.1.3 unital algebra, norn~ed 2.2.1.1 unital algebra associated to 2.1.1.8 unital algebra associated to, involutive 2.3.1.9 unital algebra homomorphisni 2.1.1.6 unital algebra isomorphism 2.1.1.6 unital algebras, isometric normed 2.2.1.1 unital algebras, isometry of normed 2.2.1.1 unital algebras, isomorphic 2.1.1.6 rlnital algebras, isomorphic normed 2.2.1.1 unital algebras, isomorphism of normed 2.2.1.1 urlital Banach algebra 2.2.1.1 unital Banach algebra associated to 2.2.1.4 unital Banach algebras, isomorphism of 2.2.1.1 unital Banach category 1.5.1.1 rlnital Banach subalgebra generated by 2.2.1.9 unital C*-algebra 4.1.1.1 unital C' algebra, complcx 4.1.1.1 4.1.1.1 unital C*--algebra,real

unital C*-algebra, associated to a C* algebra 4.1.1.13 unital C*-subalgebra 4.1.1.1 unital C *-subalgebra generated by 4.1.1.1 ur~italE-algebra 2.2.7.1 unital E-algebra, involutive 2.3.6.1 unital E C*-algebra 5.6.1.10 unital E -module 2.2.7.1 unital E-module, involutive 2.3.6.1 unital Gelfand algebra 2.4.1.1 unital Gelfand algebra, involutive 2.4.2.1 unital Hilbert E-module 5.6.1.4 unital involutive algebra associated to 2.3.1.9 unital involutive Banach algebra associated to 2.3.2.9 unital involutive normed algebra associated to 2.3.2.9 unital left A-module 1.5.1.10 unital normed algebra associated to 2.2.1.4 unital normed algebras, isomorphisrri of 2.2.1.1 unital representation 5.4.1.1 unital right A-module 1.5.1.10 unital subalgebra 2.1.1.3 unital subalgebra generated by 2.1.1.4 unital subalgebra generated by, involutive 2.3.1.18 unital W * subalgebra 4.4.4.5 unital IV'--subalgebra generated by 4.4.4.5 unital 11 category 1.5.1.14 unital 11-rr~odule 1.5.1.12 unital (A, A ) -module 1.5.1.12 unitary 2.3.1.3 universal representation 5.4.2.3 uriiversal representation, complex 5.4.2.6 upper bound 1.7.2.1 upper section filter 1.1.6.1 upward directed 1.1.6.1 vector, cyclic 5.3.2.19, 5.4.1.1 vector, separating 5.3.2.19, 5.4.1.1 vector associated to z', cyclic 5.4.1.2 vector lattice 1.7.2.1 vector space 1.1.1.1

Subject Index

vector space, involutive 2.3.1.3 vector spaces, isomorphisms of involutive 2.3.1.3 Volterra integral equation 2.2.4.22 von Neumann (right) E-module 5.6.3.2 von Neurnann (right) W*-module W * algebra 4.4.1.1

5.6.3.2

W* algebra, complex 4.4.1.1 W*-algebra, homogeneous 5.6.7.8 W*-algebra, predual of a 4.4.1.1, 4.4.4.4 W*-algebra, real 4.4.1.1 5.6.7.11 We--algebra,type I W*-algebras, isometry of 4.4.4.5 Mr*-homomorphism 4.4.4.5 W* -module, von Neumann (right) 5.6.3.2 W*-subalgebra 4.4.4.5 W*-subalgebra, unital 4.4.4.5 W*-subalgebra generated by 4.4.4.5 IV*-subalgebra generated by, unital 4.4.4.5 weak semi-inner-product right E module 5.6.1.1 weak topology 1.3.6.9 weakly conjugate exponer~ts 1.2.2.1 zero-divisor 2.1.1.1 zero- divisor, topological 2.2.4.24 A-categories, functor of (unital) 1.5.2.1 A category 1.5.1.14 A-category, quotient 1.5.2.17 A-category, unital 1.5.1.14 A-module 1.5.1.12 A--module, left (right) 1.5.1.10 A -module, quotient 1.5.2.17 11-modlile, unital 1.5.1.12 A module, unital left (right) 1.5.1.10 A modules, functor of left (right) A-subcategory 1.5.2.16 A-si~bmodule 1.5.2.16 (A, A) n~odule 1.5.1.12 1.5.1.12 (A, A)- module, unital p-null set ST

1.5.2.1

a-complete, a-order 4.3.2.3 a-complete order 1.7.2.1 a-complete ordered set 1.7.2.1 a-continuous, order 1.7.2.3 a -faithful, order 4.2.2.18 u-finite measure space 3.1.6.14 a-Stone space 1.7.2.12 0-representation 5.4.1.1

Symbol Index

Symbol Index NT means Notation and Terminology la1 a* A* A' A

4.4.3.5 2.3.1.30 2.3.1.1 5.2.2.1 NT NT "A,A" 1.3.5.1 A", A"", A"" 2.1.1.16 A', A", A"' 1.5.1.9 AAl 1.3.6.9 aa' , a'a 2.2.7.8 ab 2.1.4.23 ax 2.2.7.23 a'z" 2.2.7.11

-

( a ,(, 7) 5.6.3.2 AIt3 1.5.2.17 A +B 1.2.4.1 A\B NT AAB NT AxB NT A +z 1.2.4.1 B 1.1.2.4 C NT c 1.1.2.3,2.1.4.3 co 1.1.2.3,2.1.4.3 c(T) 1.1.2.3,2.1.4.3 cO(T) C(T) C ( T ,E) Co(T)

1.1.2.3,2.1.4.3 1.1.2.4, 2.1.4.4 1.1.2.8 1.2.2.10, 2.1.4.4

NT

Card Coker

1.2.4.5

dA

1.1.4.1

Det NT Dim 1.1.2.18 D ( k , p, v) 3.1.6.15

&(k, E' E" E"' E e~ et e: er ez

U)

3.1.6.1 1.2.1.3 1.3.6.1 1.3.6.1

2.1.5.1, 2.1.5.7, 2.3.1.38, 2.3.1.40, 5.3.1.8 1.1.2.1 1.1.2.1 1.1.2.1 1.1.2.1 2.2.3.5

Ea(u) 3.1.3.18 Et,(u) 3.1.3.18 Em, 2.1.4.23, 2.3.1.30 En, 2.1.4.24, 2.3.1.31, 5.6.6.1 ~ ' r 1.1.2.1 E(') 1.1.2.1 E" 1.7.2.3 E" 1.7.2.3 E 4.4.4.4 E+ 1.7.1.1,4.2.1.1 E# 1.1.1.2 E,# 1.7.1.4 E ; , E: 2.2.7.15 E(x) 2.3.2.15 E ( x , 1) 2.3.2.15

Symbol Index

f' 1.1.6.24 f 2.3.3.1 3,4 1.2.6.1 3(E) 3.1.3.1 3 ( E ,F ) 3.1.3.1

51

fls f (a, .) f (., b) f(A) f(x) f-'

1.1.6.1

NT NT NT NT NT, 4.1.3.1, 4.1.3.2, 4.3.2.5 NT

-1

f (B) f (Y)

NT

-I

h'T f:X+Y NT f : X + Y , x.-tT(x) F [ s ,t ] NT F[t] NT F @G 1.2.5.3 { f =9 ) NT { f # 9) NT if>@) KT g0f NT, 1.5.2.1 4,4 1.7.2.3

EI

6 H .. H irn Im Indu Ind U

NT

2.1.4.17, 2.3.1.46, 4.1.1.31 5.6.1.6 5.6.2.2 5.6.3.2 5.6.3.2 1.1.1.1, 2.3.1.22 1.2.4.5 3.1.3.1 3.1.3.21

n

k

u

k

1.2.3.1

Ker

1.2.3.1 1.2.4.5

C

1.2.1.3, 1.5.1.1, 2.1.4.6

Ln Lr L'

1.5.1.1 1.2.1.3 1.6.1.1,1.6.1.3 1.6.1.13

LE(G,H ) ( H) LE(H)

5.6.1.7 5.6.2.2

5.6.1.7 1.1.2.5 e 2 ( I ,F ) 5.6.4.2 &P

tP(T) t2(T)

1.1.2.5 5.5.7.1

e0

1.1.2.3 P(T) 1.1.2.3

trn

1.1.2.2, 2.1.4.3

&"(T) 1.1.2.2, 2.1.4.3 eys, T ) 1.2.3.2 t r 2 q ( ST, ) 1.2.3.2 log

2.2.3.9, 4.2.4.4

Mb M MI.,^

1.1.2.26

IN

IN,

5.6.4.20 5.6.4.19

NT 1.1.3.3

Symbol Index

N,! No NF,F

NF,G

Pr

p Ip,

2.3.4.1 2.3.1.3 5.6.4.16 5.6.4.15, 5.6.4.19 4.1.2.18 1.1.2.1 1.1.2.1

Q

NT IR NT IR NT re 1.1.1.1,2.3.1.3 Re 2.3.1.1 Re E# 2.3.2.1 ( x )( x ) 2.1.3.1 Sn 2.3.1.3 S(IRn) 2.4.6.5

NT S ~ P fP S~PPP NT T 2.4.4.1 u' 1.3.4.1 ul' 1.3.6.15 u* 2.3.1.1, 3.2.2.3, 5.3.1.4, 5.6.1.8. 5.6.4.12, 5.6.4.14 0 u 2.1.5.11, 2.3.1.41 C'n 2.3.1.3 Ua(t) U,T(t)

1.1.1.2 1.1.1.2 vu 5.6.4.12, 5.6.4.14 XINT x" 2.1.1.1 xa 4.2.4.1, 4.2.4.4 x - ~ 2.1.2.5 xo 2.1.1.3 x-' 1.5.1.6, 2.1.2.4 x* 2.3.1.1 x 5.5.7.1 E

2,2 1x1

2.4.1.1 2.3.5.1 4.2.5.1

x+,x x", X I -

4.2.2.9 4.2.8.13

(xL)LEI

NT

xu 2.2.7.23 x'a" 1.5.2.8 xl'a' 2.2.7.11 xx' , x'x 1.5.2.5 x[ 5.6.1.4 (x,x1),(x',x) 1.2.1.3 (.Jx)y 5.3.2.12 x *y 2.2.2.7 xr'iy",x"~y" 1.5.2.10,2.2.7.13 x8 3.1.1.25

(

x)

5.6.3.1

{x l Pb)} NT { x ~ X l P ( x ) ) YT (.,xl)y 1.3.3.3 Z KT z +A 1.2.4.1 a 1.1.1.1 la1 1.1.1.1 aA 1.2.4.1 () 2.2.3.10 la, A

a[,10,

PI,

a[, [a,01

[a,

NT 1.2.2.6 6, 1.2.7.14 6 ( s ,t ) 1.2.2.6

6,,

1P1 P -*V <x [(.1q) aA

nE, LEE

ST 2.2.2.10 5.5.7.1 5.6.1.1 5.6.5.1 5.2.1.2 2.1.4.1,2.3.1.4

NT

Symbol Index

eL

n=O

5.6.3.6

LEI

c'x,

1.7.2.10

LEI

r ( E ) , TO(E) 2.3.5.1 1 1.2.1.3, 2.1.1.3 1~

1.2.1.3,1.5.1.5

1 a-z

[1+xIa

2.2.3.5 2.2.3.13

0°

+

2.1.1.3 1.2.4.1

x

NT

\

NT

(.;)

1.2.1.3

(.I.) {.I.}

5.1.1.1, 5.6.1.1

-

YT

{.=.},{.#.),{.>.) (modp) KT

,

1 V,A

1 1 . 11 (1 . (Ip ((.((I

NT

1.5.2.10, 2.2.7.13 5.2.2.1 1.7.2.1 1.1.1.2, 1.2.1.3, 5.1.1.1, 5.6.1.4 1.1.2.5 1.6.1.1

(1 . 110 1.1 1

1.1.2.3 1.1.2.2 V , 3,3! NT 0 NT, 1.5.2.1 a> 5.1.3.1, 5.1.3.3, 5.6.4.1 W

a>

@

.I S x dp

5.6.4.6 1.2.5.3

.[, [., .[, I., .] 4.3.2.17

NT