Non-associative normed algebras 2

Cover......Page 1
Half-title......Page 3
Series information......Page 4
Title page......Page 5
Copyright information......Page 6
Dedication......Page 7
Table of contents......Page 9
Preface......Page 13
  1.1.1 Basic spectral theory......Page 25
  1.1.2 Rickart\'s dense-range-homomorphism theorem......Page 40
  1.1.3 Gelfand\'s theory......Page 44
  1.1.4 Topological divisors of zero......Page 51
  1.1.5 The complexification of a normed real algebra......Page 54
  1.1.6 The unital extension and the completion of a normed algebra......Page 57
  1.1.7 Historical notes and comments......Page 60
  1.2.1 The results......Page 62
  1.2.2 Historical notes and comments......Page 76
  1.3.1 The polynomial and rational functional calculuses......Page 77
  1.3.2 The main results......Page 82
  1.3.3 Historical notes and comments......Page 92
  1.4.1 Operators from a normed space to another......Page 94
  1.4.2 Operators from a normed space to itself......Page 99
  1.4.3 Discussing the inclusion [overline(mathfrak F (X,Y))] subseteq mathfrak K (X,Y) in the non-complete setting......Page 107
  1.4.4 Historical notes and comments......Page 110
  2.1.1 Algebra numerical ranges......Page 118
  2.1.2 Operator numerical ranges......Page 128
  2.1.3 Historical notes and comments......Page 138
  2.2.1 Non-associative results......Page 144
  2.2.2 The Kadison–Paterson–Sinclair theorem......Page 148
  2.2.3 Historical notes and comments......Page 154
 2.3 The associative Vidav–Palmer theorem, starting from a non-associative germ......Page 155
  2.3.1 Natural involutions of V-algebras are algebra involutions......Page 156
  2.3.2 The associative Vidav–Palmer theorem......Page 162
  2.3.3 Complements on C*-algebras......Page 167
  2.3.4 Introducing alternative C*-algebras......Page 175
  2.3.5 Historical notes and comments......Page 180
 2.4 V-algebras are non-commutative Jordan algebras......Page 184
  2.4.1 The main result......Page 185
  2.4.2 Applications to C*-algebras......Page 190
  2.4.3 Historical notes and comments......Page 195
  2.5.1 Introducing quaternions and octonions......Page 200
  2.5.2 The Frobenius–Zorn theorem......Page 201
  2.5.3 The generalized Gelfand–Mazur–Kaplansky theorem......Page 216
  2.5.4 Historical notes and comments......Page 222
  2.6.1 Determining smooth-normed algebras and absolute-valued unital algebras......Page 227
  2.6.2 Unit-free characterizations of smooth-normed algebras, and of absolute-valued unital algebras......Page 236
  2.6.3 Historical notes and comments......Page 240
  2.7.1 Focusing on complex algebras......Page 247
  2.7.2 Involving real scalars......Page 251
  2.7.3 Discussing the results......Page 262
  2.7.4 Historical notes and comments......Page 268
 2.8 Complements on absolute-valued algebras and algebraicity......Page 273
  2.8.1 Continuity of algebra homomorphisms into absolute-valued algebras......Page 274
  2.8.2 Absolute values on H*-algebras......Page 275
  2.8.3 Free non-associative algebras are absolute-valued algebras......Page 281
  2.8.4 Complete normed algebraic algebras are of bounded degree......Page 286
  2.8.5 Absolute-valued algebraic algebras are finite-dimensional......Page 294
  2.8.6 Historical notes and comments......Page 298
 2.9 Complements on numerical ranges......Page 307
  2.9.1 Involving the upper semicontinuity of the duality mapping......Page 308
  2.9.2 The upper semicontinuity of the pre-duality mapping......Page 315
  2.9.3 Involving the strong subdifferentiability of the norm......Page 323
  2.9.4 Historical notes and comments......Page 334
  3.1.1 Isometries of unital JB-algebras......Page 343
  3.1.2 Isometries of non-unital JB-algebras......Page 348
  3.1.3 A metric characterization of derivations of JB-algebras......Page 351
  3.1.4 JB-algebras whose Banach spaces are convex-transitive......Page 356
  3.1.5 Historical notes and comments......Page 360
  3.2.1 The main result......Page 364
 3.3 The non-associative Vidav–Palmer theorem......Page 368
  3.3.1 The main result......Page 369
  3.3.2 A dual version......Page 375
  3.3.3 Historical notes and comments......Page 380
  3.4.1 JB-algebras versus JB*-algebras......Page 383
  3.4.2 Isometries of unital non-commutative JB*-algebras......Page 390
  3.4.3 An interlude: derivations and automorphisms of normed algebras......Page 394
  3.4.4 The structure theorem of isomorphisms of non-commutative JB*-algebras......Page 405
  3.4.5 Historical notes and comments......Page 412
 3.5 The Gelfand–Naimark axiom ||a*a|| =|| a*|| ||a||, and the non-unital non-associative Gelfand–Naimark theorem......Page 416
  3.5.1 Quadratic non-commutative JB*-algebras......Page 417
  3.5.2 The axiom ||a*a|| =|| a*|| ||a|| on unital algebras......Page 421
  3.5.3 An interlude: the bidual and the spacial numerical index of a non-commutative JB*-algebra......Page 428
  3.5.4 The axiom ||a*a|| =|| a*|| ||a|| on non-unital algebras......Page 435
  3.5.5 The non-unital non-associative Gelfand–Naimark theorem......Page 438
  3.5.6 Vowden\'s theorem......Page 442
  3.5.7 Historical notes and comments......Page 445
 3.6 Jordan axioms for C*-algebras......Page 449
  3.6.1 Jacobson\'s representation theory: preliminaries......Page 450
  3.6.2 The main result......Page 456
  3.6.3 Jacobson\'s representation theory continued......Page 459
  3.6.4 Historical notes and comments......Page 468
 4.1 Involving the Jordan inverse......Page 474
  4.1.1 Basic spectral theory for normed Jordan algebras......Page 475
  4.1.2 Topological J-divisors of zero......Page 484
  4.1.3 Non-commutative JB*-algebras are JB*-triples......Page 487
  4.1.4 Extending the Jordan spectral theory to Jordan-admissible algebras......Page 497
  4.1.5 The holomorphic functional calculus for complete normed unital non-commutative Jordan complex algebras......Page 504
  4.1.6 A characterization of smooth-normed algebras......Page 511
  4.1.7 Historical notes and comments......Page 514
 4.2 Unitaries in JB*-triples and in non-commutative JB*-algebras......Page 521
  4.2.1 A commutative Gelfand–Naimark theorem for JB*-triples......Page 522
  4.2.2 The main results......Page 529
  4.2.3 Russo–Dye type theorems for non-commutative JB*-algebras......Page 542
  4.2.4 A touch of real JB*-triples and of real non-commutative JB*-algebras......Page 545
  4.2.5 Historical notes and comments......Page 551
  4.3.1 The case of C*-algebras......Page 560
  4.3.2 The case of JB*-algebras......Page 576
  4.3.3 An application to non-commutative JB*-algebras......Page 584
  4.3.4 Historical notes and comments......Page 586
 4.4 Algebra norms on non-commutative JB*-algebras......Page 589
  4.4.1 The Johnson–Aupetit–Ransford uniqueness-of-norm theorem......Page 590
  4.4.2 A non-complete variant......Page 595
  4.4.3 The main results......Page 597
  4.4.4 The uniqueness-of-norm theorem for general non-associative algebras......Page 601
  4.4.5 Historical notes and comments......Page 616
 4.5 JB*-representations and alternative C*-representations of hermitian algebras......Page 628
  4.5.1 Preliminary results......Page 629
  4.5.2 The main results......Page 635
  4.5.3 A conjecture on non-commutative JB*-equivalent algebras......Page 654
  4.5.4 Historical notes and comments......Page 656
  4.6.1 Stability under the holomorphic functional calculus......Page 660
  4.6.2 Stability under the geometric functional calculus......Page 668
  4.6.3 Historical notes and comments......Page 689
 Papers......Page 695
 Books......Page 720
Symbol index......Page 728
Subject index......Page 731