This invaluable reference is the first to present the general theory of algebras of operators on a Hilbert space, and the modules over such algebras. The new theory of operator spaces is presented early on and the text assembles the basic concepts, theory and methodologies needed to equip a beginning researcher in this area. A major trend in modern mathematics, inspired largely by physics, is toward `noncommutative' or `quantized' phenomena. In functional analysis, this has appeared notably under the name of `operator spaces', which is a variant of Banach spaces which is particularly appropriate for solving problems concerning spaces or algebras of operators on Hilbert space arising in 'noncommutative mathematics'. The category of operator spaces includes operator algebras, selfadjoint (that is, C*-algebras) or otherwise. Also, most of the important modules over operator algebras are operator spaces. A common treatment of the subjects of C*-algebras, Non-selfadjoint operator algebras, and modules over such algebras (such as Hilbert C*-modules), together under the umbrella of operator space theory, is the main topic of the book. A general theory of operator algebras, and their modules, naturally develops out of the operator space methodology. Indeed, operator space theory is a sensitive enough medium to reflect accurately many important non-commutative phenomena. Using recent advances in the field, the book shows how the underlying operator space structure captures, very precisely, the profound relations between the algebraic and the functional analytic structures involved. The rich interplay between spectral theory, operator theory, C*-algebra and von Neumann algebra techniques, and the influx of important ideas from related disciplines, such as pure algebra, Banach space theory, Banach algebras, and abstract function theory is highlighted. Each chapter ends with a lengthy section of notes containing a wealth of additional information.
Author(s): David P. Blecher, Christian Le Merdy
Series: London Mathematical Society Monographs New Series 30
Publisher: Oxford University Press, USA
Year: 2005
Language: English
Pages: 398
Contents......Page 10
1.1 Notation and conventions......Page 12
1.2 Basic facts, constructions, and examples......Page 15
1.3 Completely positive maps......Page 27
1.4 Operator space duality......Page 33
1.5 Operator space tensor products......Page 38
1.6 Duality and tensor products......Page 49
1.7 Notes and historical remarks......Page 56
2.1 Introducing operator algebras and unitizations......Page 60
2.2 A few basic constructions......Page 68
2.3 The abstract characterization of operator algebras......Page 73
2.4 Universal constructions of operator algebras......Page 79
2.5 The second dual algebra......Page 89
2.6 Multiplier algebras and corners......Page 93
2.7 Dual operator algebras......Page 99
2.8 Notes and historical remarks......Page 107
3.1 Introduction to operator modules......Page 113
3.2 Hilbert modules......Page 120
3.3 Operator modules over operator algebras......Page 126
3.4 Two module tensor products......Page 130
3.5 Module maps......Page 134
3.6 Module map extension theorems......Page 139
3.7 Function modules......Page 142
3.8 Dual operator modules......Page 147
3.9 Notes and historical remarks......Page 153
4.1 The Choquet boundary and boundary representations......Page 158
4.2 The injective envelope......Page 163
4.3 The C*-envelope......Page 167
4.4 The injective envelope, the triple envelope, and TROs......Page 172
4.5 The multiplier algebra of an operator space......Page 178
4.6 Multipliers and the ‘characterization theorems’......Page 186
4.7 Multipliers and duality......Page 191
4.8 Noncommutative M-ideals......Page 194
4.9 Notes and historical remarks......Page 199
5.1 Homomorphisms of operator algebras......Page 206
5.2 Completely bounded characterizations......Page 211
5.3 Examples of operator algebra structures......Page 220
5.4 Q-algebras......Page 226
5.5 Applications to the isomorphic theory......Page 235
5.6 Notes and historical remarks......Page 239
6.1 The maximal and normal tensor products......Page 243
6.2 Joint dilations and the disc algebra......Page 250
6.3 Tensor products with triangular algebras......Page 252
6.4 Pisier’s delta norm......Page 259
6.5 Factorization through matrix spaces......Page 265
6.6 Nuclearity and semidiscreteness for linear operators......Page 270
6.7 Notes and historical remarks......Page 276
7.1 OS-nuclear maps and the weak expectation property......Page 280
7.2 Hilbert module characterizations......Page 285
7.3 Tensor product characterizations......Page 290
7.4 Amenability and virtual diagonals......Page 293
7.5 Notes and historical remarks......Page 303
8 C*-modules and operator spaces......Page 307
8.1 Hilbert C*-modules—the basic theory......Page 308
8.2 C*-modules as operator spaces.......Page 319
8.3 Triples, and the noncommutative Shilov boundary......Page 333
8.4 C*-module maps and operator space multipliers......Page 339
8.5 W*-modules......Page 342
8.6 A sample application to operator spaces......Page 359
8.7 Notes and historical remarks......Page 361
A.1 Operators on Hilbert space......Page 370
A.2 Duality of Banach spaces......Page 371
A.3 Tensor products of Banach spaces......Page 372
A.4 Banach algebras......Page 374
A.5 C*-algebras......Page 375
A.6 Modules and Cohen’s factorization theorem......Page 378
References......Page 380
H......Page 396
R......Page 397
X......Page 398
