Convexity Theory and its Applications in Functional Analysis (eBook)

Front Cover 1
Convexity Theory and its Applications in Functional Analysis 4
Copyright Page 
5 
Table of Contents 10
Dedication 6
Preface 8
Chapter 1. Preliminaries 12
1. Separation and Polar Calculus 12
2. Krein-Smulyan Theorem and Corollaries 15
3. Gauge Lemma and Completeness 18
4. Subspaces of C(X) and Affine Functions 22
5. Representing Measures 26
6. Maximal Measures and the Choquet-Bishop-de Leeuw Theorem 29
7. Bishop-Phelps Theorems 37
Chapter 2. Duality in Ordered Banach Spaces 41
1. Positive Generation and Normality 42
2. Order Unit and Base Norm Spaces 46
3. Directedness and Additivity 50
4. Homogeneous Functionals on Cones and the Decomposition Property 55
5. Banach Lattices and the Riesz Interpolation Property 59
6. Abstract L and M Spaces 68
7. Choquet Simplexes 80
8. Decomposition of Dual Cones 85
9. Stability of Split Sets and the Characterization of Complemented Subcones 94
10. Application to A(K) 102
11. The Riesz Norm 110
Chapter 3. Simplex Spaces 113
1. The Facial Topology for K, and the Centre for A(K) 113
2. Facial Characterizations of Simplexes 119
3. Ideals in Simplex Spaces 128
4. Stone-Weierstrass Theorems for Simplex Spaces 133
5. Prime Simplexes and Anti-Lattices 135
6. Topological Characterization of the Extreme Boundary of a Compact Simplex 137
7. Poulsen's Simplex—its Uniqueness and Universality 141
8. Non-Compact Simplexes and Convex Sets 149
Chapter 4. Complex Function Spaces 157
0. The Complex State Space 157
1. Complex Representing Measures 159
2. Interpolation Sets 165
3. Gauge Dominated Extensions and Complex State Space 170
4. Peak Sets and M-sets 175
5. Dominated Interpolation 181
6. Decomposability and Exact Interpolation 191
7. M-Hulls and Function Algebras 204
8. Facial Topologies and Decompositions 219
9. Lindenstrauss Spaces 230
Chapter 5. Convexity Theory for C*-Algebras 235
1. The Structure Topology and Primitive Ideas in A(K) 235
2. The Ideal Structure of a Unital C*-Algebra and the Facial Structure of its State Space 237
3. Commutativity and Order in C*-Algebras 243
4. The Centre of a C*-Algebra, Weakly Central Algebras, Prime Algebras 245
5. Unit Traces for C*-Algebras 250
6. The State Spaces of Jordan Operator Algebras and of C*- Algebras 253
References 263
Index of Definitions 272
Subject Index 274