TOPOLOGICAL ALGEBRAS SELECTED TOPICS
Transcript of TOPOLOGICAL ALGEBRAS SELECTED TOPICS
TOPOLOGICAL ALGEBRAS SELECTED TOPICS
Anastasios MALLIOS Mathematical Institute University ofAthens Greece
1986
NORTH-HOLLAND-AMSTERDAM • NEW YORK »OXFORD »TOKYO
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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
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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