Convexity Theory and its Applications in Functional Analysis

1st Edition - November 12, 1980
Author: L. Asimow
Language: English
eBook ISBN:
9 7 8 - 1 - 4 8 3 2 - 9 4 6 9 - 8

Convexity Theory and its Applications in Functional Analysis is a five-chapter text that provides a geometric perspective of the convexity theory and its practical applications.… Read more

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Convexity Theory and its Applications in Functional Analysis is a five-chapter text that provides a geometric perspective of the convexity theory and its practical applications. Chapter 1 reviews the functional analytic preliminaries, including the Krein-Smulyan Theorem, the basic Choquet Theory, and the Bishop-Phelps Theorem. Chapter 2 gives the basic duality results, lattice theory and concrete representation theorems for order unit spaces and Banach lattices of type Mand L. Chapters 3 and 4 deal with the real affine function spaces through examining the Choquet simplex and the application of the study of real A(K) spaces to complex-values function spaces by means of a complex state space. Chapter 5 highlights the application of the theory to the study of non-commutative Banach algebras. This book will prove useful to mathematicians, engineers, and physicists.

Preface

1. Preliminaries

1. Separation and Polar Calculus

2. Krein-Smulyan Theorem and Corollaries

3. Gauge Lemma and Completeness

4. Subspaces of C(X) and Affine Functions

5. Representing Measures

6. Maximal Measures and the Choquet-Bishop-de Leeuw Theorem

7. Bishop-Phelps Theorems

2. Duality in Ordered Banach Spaces

1. Positive Generation and Normality

2. Order Unit and Base Norm Spaces

3. Directedness and Additivity

4. Homogeneous Functionals on Cones and the Decomposition Property

5. Banach Lattices and the Riesz Interpolation Property

6. Abstract L and M Spaces

7. Choquet Simplexes

8. Decomposition of Dual Cones

9. Stability of Split Sets and the Characterization of Complemented Subcones

10. Application to A(K)

11. The Riesz Norm

3. Simplex Spaces

1. The Facial Topology for K and the Centre of A(K)

2. Facial Characterizations of Simplexes

3. Ideals in Simplex Spaces

4. Stone-Weierstrass Theorems for Simplex Spaces

5. Prime Simplexes and Anti-Lattices

6. Topological Characterization of the Extreme Boundary of a Compact Simplex

7. Poulsen’s Simplex—Its Uniqueness and Universality

8. Non-Compact Simplexes and Convex Sets

4. Complex Function Spaces

0. The Complex State Space

1. Complex Representing Measures

2. Interpolation Sets

3. Gauge Dominated Extensions and Complex State Space

4. Peak Sets and M-Sets

5. Dominated Interpolation

6. Decomposability and Exact Interpolation

7. M-Hulls and Function Algebras

8. Facial Topologies and Decompositions

9. Lindenstrauss Spaces

5. Convexity Theory for C*-Algebras

1. The Structure Topology and Primitive Ideas in A(K)

2. The Ideal Structure of a Unital C*-Algebra and the Facial Structure of its State Space

3. Commutativity and Order in C*-Algebras

4. The Centre of a C*-Algebra, Weakly Central Algebras, Prime Algebras

5. Unit Traces for C*-Algebras

6. The State Spaces of Jordan Operator Algebras and of C*-Algebras

References

Index of Definitions

Subject Index