TOPOLOGICAL ALGEBRAS SELECTED TOPICS

Anastasios MALLIOS Mathematical Institute University ofAthens Greece

1986

NORTH-HOLLAND-AMSTERDAM • NEW YORK »OXFORD »TOKYO

xiii

Contents

Preface ix

PART I. GENERAL THEORY

CHAPTER I. General Concepts

1. Preliminaries. Definitions 1

2. Examples of topological algebras 9

2. (1). The algebra £(E) 10

2.(2). The algebra 0S(R) 10

2.(3). The algebra C [t] 11

2.(4). The Arens algebra L®([0, 1]) 12

3. Topologies defined by submultiplicative semi-norms 13

4. Continuity of the multiplication. Complete topological

algebras 21

5. Topological algebras admitting locally m-convex topolo­

gies 31

5. (1). Michael's Theorem 31

5. (2). 4-convex algebras 37

6. Certain particular classes of topological algebras 39

6.(1). Locally bounded algebras 39

6. (2). g-algebras 43

6. (3). Advertibly complete algebras 44

CHAPTER II. Spectrum (Local Theory)

1. Spectrum of an element. Spectral radius 47

2. The resolvent set 50

3. Topological algebras with continuous inversion 51

4. Waelbroeck algebras 54

5. Topological division algebras. Gel'fand-Mazur Theorem .... 61

6. Maximal ideals 63

7. Characters. Closed maximal ideals 67

8. Appendix: Schur's Lemma 77

CONTENTS

CHAPTER III. Projective Limit Algebras

1. Initial topologies. Topological subalgebras. Cartesian

products 79

2 . P r o j e c t i v e Systems of t o p o l o g i c a l a l g e b r a s 82

3 . R e p r e s e n t a t i o n s of l o c a l l y m-convex a l g e b r a s a s p r o j e c ­

t i v e l i m i t s . A r e n s - M i c h a e l d e c o m p o s i t i o n 85

4 . A p p l i c a t i o n s of t h e A r e n s - M i c h a e l d e c o m p o s i t i o n 91

5 . A d v e r t i b l y c o m p l e t e l o c a l l y m-convex a l g e b r a s 94

6 . S p e c t r a l p r o p e r t i e s of a d v e r t i b l y c o m p l e t e l o c a l l y

m-convex a l g e b r a s 99

CHAPTER IV. Inductive Limit Algebras

1. Inductive Systems of algebras. Algebraic pireliminaries ... 109

2. Final topologies. Inductive Systems of topological al­

gebras 113

3. Inductive limits of locally m-convex algebras 120

4. Examples of topological inductive limit algebras 127

4.(1). The algebra K(X) 127

4.(2). The algebra S)(X) as a topological subalgebra of

Cf°(X) 129

4. (3). The algebra 0(K) 134

CHAPTER V. Spectrum (Global Theory)

1. Spectrum of a topological algebra 139

2. Spectrum of the completion of a topological algebra 144

3. Spectrum of an inductive limit topological algebra 152

4. Envelopes of holomorphy 160

5. The dual of the Arens-Michael decomposition 164

5.(1). Compactly generated topological Spaces 165

6. The dual of the Arens-Michael decomposition (contn'd.) ... 167

7. Spectrum of a projective limit topological algebra.

Dense projective limit algebras 173

8. Appendix: Generalized spectrum 176

8.(1). General theory 176

8.(2). Generalized spectrum of a topological projective

limit algebra 179

8. (3). Generalized spectrum of a topological inductive

limit algebra 180

CONTENTS XV

CHAPTER VI. The Gel'fand Map

1 . Continuity of the Gel'fand map 181

2 . Boundaries 188

3. Functional calculus. Holomorphic functions of a Single

element in a topological algebra 198

4. Functional calculus (contn'd.). Holomorphic functions

of finite many elements in a topological algebra 207

5. Appendix: Generalized Gel'fand map 212

CHAPTER V I I . Spectra of Certain Particular Topological Algebras

1 . Spec t rum of t h e a l g e b r a C (X) 215

2 . Spec t rum of t h e a l g e b r a C°°(%) 224

3 . Spec t rum of t h e a l g e b r a 0(X). S t e i n a l g e b r a s 228

4 . Spec t rum of t h e a l g e b r a L (G) 231

eoo

(X)(contn d . ) . The Nach­b i n Theorem ( n e c e s s i t y ) 240

6 . The Nachbin Theorem ( s u f f i c i e n c y ) 246

7 . Append ix : V a r i a n t s of N a c h b i n ' s Theorem 251

7. (1). D i f f e r e n t i a b i l i t y of c l a s s C-m 251

7. (2) . C o m p l e x i f i c a t i o n 251

CHAPTER V I I I . Some Special Classes of Topological Algebras

1. Spectrally barrelled algebras (contn'd.) 253

2. Nachbin-Shirota algebras 262

3. Functional representations 265

4. Topological algebras with a given dual 270

5. Uniform topological algebras 274

6. 4-convex algebras (contn'd.) 280

7. Finitely generated topological algebras 283

8. Functional calculus (contn'd.). The Silov-Arens-Calderön-

Waelbroeck theory 294

9. Miscellanea 301

9.(1). £F2-algebras 301

9. (2). Nuclear algebras 302

9.(3) . (Ik) -algebras 304

9. (4). m-infrabarrelled algebras 306

9. (5). Gel'fand-Mazur algebras 308

9. (6). Infra-Ptäk algebras 308

10. Infinite dimensional holomorphy. Spectra of particular

topological algebras 311

xvi CONTENTS

10.(1). The algebra of continuous polynomials *r(E) 311

10. (2). Topological algebras of holomorphic func-

tions on infinite dimensional Spaces 315

10. (3) . Holomorphic functions on infinite dimen­

sional spaces (contn'd.). The algebra H(U)[x ] 321

11 . Convolution algebras of C - functions 325

CHAPTER IX. Structure Theory

1 . Maximal ideal space (structure space) . /z/c-topology 329

2. Regulär, normal and Silov algebras 332

3. Hulls of ideals 335

4. Hulls of ideals: Regulär, normal and Silov algebras

(contn'd.) 338

5 . F u r t h e r r e s u l t s . The L o c a l Theorem 347

6 . S e t s of s p e c t r a l s y n t h e s i s . W i e n e r - T a u b e r a l g e b r a s 348

7. Uniform a l g e b r a s ( c o n t n ' d . ) . Riemann a l g e b r a s 352

PART I I . TOPOLOGICAL TENSOR PRODUCTS

CHAPTER X. Topological Tensor Products of Topological Algebras

1 . A l g e b r a i c p r e l i m i n a r i e s 359

2 . T o p o l o g i c a l t e n s o r p r o d u c t s of l o c a l l y convex s p a c e s 364

2.(1) . The p r o j e c t i v e t e n s o r i a l t o p o l o g y w 364

2. (2). The i n d u c t i v e t e n s o r i a l t o p o l o g y i 369

2. (3). The b i p r o j e c t i v e t e n s o r i a l t o p o l o g y e 370

2.(4) . The e - p r o d u c t of L . Schwar tz 373

3 . Topo log ica l t e n s o r p r o d u c t s of t o p o l o g i c a l a l g e b r a s .

Compatible t o p o l o g i e s 375

4 . Tensor p r o d u c t s of l o c a l l y bounded a l g e b r a s 379

5 . I n f i n i t e t o p o l o g i c a l t e n s o r p r o d u c t a l g e b r a s 383

CHAPTER X I . Topological Tensor Product Algebras. Examples

1 . The a l g e b r a Ca(X,JE) 387

2 . The a l g e b r a CJ^fX, E) 392

3 . The a l g e b r a CJ°(X, E)(contn'd.). N a c h b i n ' s Theorem

( v e c t o r i z a t i o n ) 395

4 . The a l g e b r a 0(X, E) 400

5 . The a l g e b r a L (G, E) ( g e n e r a l i z e d g r o u p a l g e b r a ) 402

CHAPTER XI I . Spectra of Topological Tensor Product Algebras

1 . Spectrum of a tensor product o f t o p o l o g i c a l a lgebras

COHTENTS XVii

(numerical case) 407

2. Spectrum of an infinite topological tensor product

algebra 414

3. Generalized spectrum of a tensor product of topological

algebras. Canonical decomposition . . .• 419

4. Inductive limits and generalized spectra 424

5. Generalized spectra and "point evaluations" 429

CHAPTER XIII. Properties of Permanence of Topological Tensor

Product Algebras

1. B o u n d a r i e s of t o p o l o g i c a l t e n s o r p r o d u c t a l g e b r a s 433

2 . C o n t i n u i t y of t h e G e l ' f a n d map 438

3 . S p e c t r a l l y b a r r e l l e d a l g e b r a s 439

4 . S e m i - s i m p l i c i t y 441

5 . I d e n t i t y e l e m e n t s 447

6 . R e g u l a r i t y . S i l o v a l g e b r a s 450

7 . Wiene r -Taube r c o n d i t i o n 453

8 . Append ix : G e n e r a l i z e d s p e c t r a ( c o n t n ' d . ) . C a n o n i c a l

d e c o m p o s i t i o n 456

CHAPTER XIV. Generalized Spectra in the Presence of Approximate

Ident i t ies. Representation Theory

1. Topological algebras with approximate identities. Repre­

sentation theory 465

2. Elementary measures of representations 474

CHAPTER XV. Topological Algebras with Involution. Representation

Theory (contn'd.)

1 . P r e l i m i n a r i e s 481

2 . C e r t a i n p a r t i c u l a r (commuta t ive) t o p o l o g i c a l * - a l g e b r a s

and t h e i r r e p r e s e n t a t i o n s 483

3 . SNAG Theorem ( t h e c l a s s i c a l c a s e ) 488

4 . R e p r e s e n t a t i o n s of g e n e r a l i z e d g r o u p a l g e b r a s . SNAG Theo­

rem ( e x t e n d e d form) 490

5 . A b s t r a c t forms of "SNAG Theorem" t y p e 493

6 . Append ix : E n v e l o p i n g l o c a l l y m-convex C * - a l g e b r a s 497

BIBLIOGRAPHY 503

List of Symbols 527

INDEX 531