Compact Convex Sets And Boundary Integrals (ergebnisse Der Mathematik Und Ihrer Grenzgebiete. 2. Folge)

The importance of convexity arguments in functional analysis has long been realized, but a comprehensive theory of infinite-dimensional convex sets has hardly existed for more than a decade. In fact, the integral representation theorems of Choquet and Bishop -de Leeuw together with the uniqueness theorem of Choquet inaugurated a new epoch in infinite-dimensional convexity. Initially considered curious and tech­ nically difficult, these theorems attracted many mathematicians, and the proofs were gradually simplified and fitted into a general theory. The results can no longer be considered very'deep'or difficult, but they certainly remain all the more important. Today Choquet Theory provides a unified approach to integral representations in fields as diverse as potential theory, probability, function algebras, operator theory, group representations and ergodic theory. At the same time the new concepts and results have made it possible, and relevant, to ask new questions within the abstract theory itself. Such questions pertain to the interplay between compact convex sets K and their associated spaces A(K) of continuous affine functions; to the duality between faces of K and appropriate ideals of A(K); to dominated­ extension problems for continuous affine functions on faces; and to direct convex sum decomposition into faces, as well as to integral for­ mulas generalizing such decompositions. These problems are of geometric interest in their own right, but they are primarily suggested by applica­ tions, in particular to operator theory and function algebras. The importance of convexity arguments in functional analysis has long been realized, but a comprehensive theory of infinite-dimensional convex sets has hardly existed for more than a decade. In fact, the integral representation theorems of Choquet and Bishop-DeLeeuw together with the uniqueness theorem of Choquet inaugurated a new epoch in infinite-dimensional convexity. Initially considered curious and technically difficult, these theorems attracted many mathematicians, and the proofs were gradually simplified and fitted into a general theory. The results can no longer be considered very "deep" or difficult, but they certainly remain important. Today Choquet Theory provides a unified approach to integral representations in fields as diverse as potential theory, probability, function algebras, operator theory, group representations and ergodic theory. At the same time the new concepts and results have made it possible, and relevant, to ask new questions within the abstract theory itself. Such questions pertain to the interplay between compact convex sets [italic]K and their associated spaces [italic]A([italic]K) of continuous affine functions; to the duality between faces of [italic]K and appropriate ideals of [italic]A([italic]K); to dominated extension problems for continuous affine functions on faces; and to direct convex sum decomposition into faces, as well as to integral formulas generalizing such decompositions. These problems are of geometric interest in their own right, but they are primarily suggested by applications, in particular to operator theory and function algebras Title page ......Page 1 Series ......Page 3 Date-line ......Page 4 Preface ......Page 5 Contents ......Page 7 §1. Distinguished Classes of Functions on a Compact Convex Set ......Page 13 §2. Weak Integrals, Moments and Barycenters ......Page 21 §3. Comparison of Measures on a Compact Convex Set ......Page 33 §4. Choquet's Theorem ......Page 43 §5. Abstract Boundaries Defined by Cones of Functions ......Page 56 §6. Unilateral Representation Theorems with Application to Simplicial Boundary Measures ......Page 67 §1. Order-unit and Base-norm Spaces ......Page 79 §2. Elementary Embedding Theorems ......Page 91 §3. Choquet Simplexes ......Page 96 §4. Bauer Simplexes and the Dirichlet Problem of the Extreme Boundary ......Page 115 §5. Order Ideals, Faces, and Parts ......Page 121 §6. Split-faces and Facial Topology ......Page 140 §7. The Concept of Center for $A(K)$ ......Page 165 §8. Existence and Uniqueness of Maximal Central Measures Representing Points of an Arbitrary Compact Convex Set ......Page 183 Appendix ......Page 201 References ......Page 205 Subject Index ......Page 221