The metric theory of tensor products. Grothendieck's resume revisited

Front Cover......Page 1
Title Page......Page 2
Copyright Information......Page 3
Table of Contents......Page 4
Preface......Page 8
The algebraic preliminaries......Page 12
1.1.1. Definitions ......Page 16
1.1.2. Injectivity of \vee and projectivity of \wedge......Page 22
1.1.3. The universal mapping property of \otimes^{\wedge} and the dual of X \otimes^{\wedge} Y......Page 24
1.1.4. Examples: C(K) \otimes^{\vee} X and L¹(μ) \otimes^{\wedge} X......Page 25
1.1.5. Integral bilinear forms and the dual of X \otimes^{\vee} Y ......Page 33
1.2. Definition of \otimes-norms......Page 36
1.2.1. Fundamental operations on \otimes-norms......Page 37
1.2.2. Order relations among \otimes-norms......Page 39
1.3. Extension of \otimes-norms to spaces of infinite dimensions......Page 40
1.3.1. Metric accessibility and accessibility......Page 43
1.4. Bilinear forms and linear operators of type α......Page 51
1.4.1. General properties of α-forms......Page 53
1.4.2. General properties of α-integral operators......Page 58
1.4.3. Composition of α-integral and α^{\vee}-integral operators......Page 59
1.4.4. Accessibility and metric accessibility (continued)......Page 61
1.5. α-nuclear forms and operators......Page 65
1.6.1. The fundamental lemma......Page 70
1.6.2. Consequences......Page 74
2.1. Complements on \wedge and \vee......Page 78
2.1.1. Representability, equimeasurability and nuclearity......Page 83
2.2. Fundamental linear topological properties of C- and L-spaces......Page 87
Notes......Page 92
2.3. Injective and projective \otimes-norms......Page 95
2.4. Formation of new \otimes-norms......Page 101
2.5. Complements on /Λ, Λ\, /Λ\, \V, V/, \V/......Page 112
2.6. A table of natural \otimes-norms......Page 117
3.1. Definitions and generalities about H and H*......Page 122
3.2. Hermitian H-forms......Page 130
3.3. Hermitian H*-forms......Page 133
3.4. Basic relations between H, H*, etc.......Page 142
3.5. The "little" Grothendieck inequality......Page 143
3.6. The classes of α-integral operators between Hilbert spaces......Page 153
4.1. Functions of type α......Page 160
4.2. The Fundamental Theorem (Grothendieck's inequality) and its variants......Page 163
4.3.1. Compositions of operators between spaces of type C, L and H......Page 170
4.3.2. Linear topological characterizations of Hilbert space......Page 171
4.3.3. A theorem of Littlewood......Page 172
4.4. A table of the fourteen natural \otimes-norms......Page 174
4.4.1. A summary with regards to the characterizations and factorization schemes of the various classes of integral operators......Page 175
4.4.2. There are at most 14 natural \otimes-norms......Page 176
4.4.3. There are exactly 14 natural \otimes-norms......Page 177
4.5. Notes and remarks on the natural tensor norms and Banach Algebras......Page 179
Further notes and remarks......Page 186
Glossary of terms......Page 188
A.l. Problem 1: The approximation problem......Page 194
A.2. Problem 2: The possible reduction of the table of "natural" tensor norms......Page 197
A.3. Problem 3: Grothendieck's inequality and the "best" constant......Page 202
A.4. Problem 4: Algebraic-topological properties of C*-algebras......Page 212
A.5. Problem 5: Characterizing classes of spaces by the behavior of tensor products and the action of operators on the spaces......Page 219
A.6. Problem 6: Comparison of the projective and injective tensor products......Page 220
B.l. Blaschke's Selection Principle......Page 222
B.2. Compact sets in Euclidean spaces......Page 223
B.3. Ellipsoids in finite dimensional Banach spaces......Page 227
C.l. The facts, ma'm, just the facts......Page 228
C.2. Some basics about duality in Banach lattices......Page 229
C.3. Lattice homomorphisms......Page 233
C.4. AM-spaces and AL-spaces......Page 236
C.5. Kakutani's vector lattice version of the Stone-Weierstrass theorem......Page 238
C.6. Kakutani's characterization of AM-spaces with unit......Page 239
C.7. AL-spaces: The Freudenthal-Kakutani theorem......Page 240
C.8. Kakutani's characterization of AL-spaces......Page 245
C.9. Grothendieck's inequality for Banach lattices......Page 246
Notes and remarks......Page 251
D.l. The Nakano Stone Theorem......Page 252
D.2. Injective Banach spaces......Page 253
Notes and remarks......Page 264
Epilogue......Page 266
Monographs......Page 270
Papers......Page 271
Bibliography......Page 272
Author Index......Page 282
Index of Notation......Page 284
Index......Page 288
Back Cover......Page 290