Convexity theory and its applications in functional analysis

Cover......Page 1
S Title......Page 2
L.M.S. MONOGRAPHS......Page 3
Title: CONVEXITY THEORY AND ITS APPLICATIONS IN FUNCTIONAL ANALYSIS......Page 4
Copyright......Page 5
......Page 5
ISSN 0076-0560......Page 5
ISBN 0120653400......Page 5
517'.7  QA640......Page 5
Dedication......Page 6
Preface......Page 8
Contents......Page 10
CHAPTER 1: Preliminaries......Page 12
1. SEPARATION AND POLAR CALCULUS......Page 12
2. KREIN-SMULYAN THEOREM AND COROLLARIES......Page 15
3. GAUGE LEMMA AND COMPLETENESS......Page 18
4. SUBSPACES OF C(X) AND AFFINE FUNCTIONS......Page 22
5. REPRESENTING MEASURES......Page 26
6. MAXIMAL MEASURES AND THE CHOQUET-BISHOP-DE LEEUW THEOREM......Page 29
7. BISHOP-PHELPS THEOREMS......Page 37
Notes......Page 40
CHAPTER 2: Duality in Ordered Banach Spaces......Page 41
1. POSITIVE GENERATION AND NORMALITY......Page 42
2. ORDER UNIT AND BASE NORM SPACES......Page 46
3. DIRECTEDNESS AND ADDITIVITY......Page 50
4. HOMOGENEOUS FUNCTIONALS ON CONES AND THE DECOMPOSITION PROPERTY......Page 55
5. BANACH LATTICES AND THE RIESZ INThRPOLATION PROPERTY......Page 59
6. ABSTRACT L AND M SPACES......Page 68
7. CHOQUET SIMPLEXES......Page 80
8. DECOMPOSITION OF DUAL CONES......Page 85
9. STABILITY OF SPLIT SETS AND THE CHARACTERIZATION OF COMPLEMENTED SUBCONES......Page 94
10. APPLICATION TO A(K)......Page 102
11. THE RIESZ NORM......Page 110
CHAPTER 3: Simplex Spaces......Page 113
1. THE FACIAL TOPOLOGY FOR K, AND THE CENTRE FOR A(K).......Page 113
2. FACIAL CHARACTERIZATIONS OF SIMPLEXES......Page 119
3. IDEALS IN SIMPLEX SPACES......Page 128
4. STONE-WEIERSTRASS THEOREMS FOR SIMPLEX SPACES......Page 133
5. PRIME SIMPLEXES AND ANTILATTICES......Page 135
6. TOPOLOGICAL CHARACTERIZATION OF THE EXTREME BOUNDARY OF A COMPACT SIMPLEX......Page 137
7. POULSEN'S SIMPLEX-ITS UNIQUENESS AND UNIVERSALITY......Page 141
8. NON-COMPACT SIMPLEXES AND CONVEX SETS......Page 149
Notes......Page 156
CHAPTER 4: Complex Function Spaces......Page 157
0. THE COMPLEX STATE SPACE......Page 157
1. COMPLEX REPRESENTING MEASURES......Page 159
2. INTERPOLATION SETS......Page 165
3. GAUGE DOMINATED EXTENSIONS AND COMPLEX STATE SPACE......Page 170
4. PEAK SETS AND M-SETS......Page 175
5. DOMINATED INTERPOLATION......Page 181
6. DECOMPOSABILITY AND EXACT INTERPOLATION......Page 191
7. M-.HULLS AND FUNCTION ALGEBRAS......Page 204
8. FACIAL TOPOLOGIES AND DECOMPOSITIONS......Page 219
9. LINDENSTRAUSS SPACES......Page 230
Notes......Page 234
CHAPTER 5: Convexity Theory for C *..Algebras......Page 235
1. THE STRUCTURE TOPOLOGY AND PRIMITIVE IDEALS IN A(K)......Page 235
2. THE IDEAL STRUCTURE OF A UNITAL C*-ALGEBRA AND THE FACIAL STRUCTURE OF ITS STATE SPACE......Page 237
3. COMMUTATIVITY AND ORDER IN C*-ALGEBRAS......Page 243
4. THE CENTRE OF A C*-ALGEBRA WEAKLY CENTRAL ALGEBRAS, PRIME ALGEBRAS......Page 245
5. UNIT TRACES FOR C*-ALGEBRA......Page 250
6. THE STATE SPACES OF JORDAN OPERATOR ALGEBRAS AND OF C*-ALGEBRA......Page 253
Notes......Page 261
References......Page 263
Index of Definitions......Page 272
Subject Index......Page 274
BOOK REVIEWS......Page 279