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Compact convex sets and boundary integrals

Author: Erik Magnus Alfsen
Publisher: Berlin : Springer, 1971.
Series: Ergebnisse der Mathematik und ihrer Grenzgebiete, 57; Ergebnisse der Mathematik und Ihrer Grenzgebiete., Neue Folge. ;, 57.
Edition/Format:   Print book : EnglishView all editions and formats
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Document Type: Book
All Authors / Contributors: Erik Magnus Alfsen
ISBN: 0387050906 9780387050904 3540050906 9783540050902
OCLC Number: 489633862
Description: 1 vol. (XII-210 p.) : ill. ; 24 cm.
Contents: I Representations of Points by Boundary Measures.- 1. Distinguished Classes of Functions on a Compact Convex Set.- Classes of continuous and semicontinuous, affine and convex functions.- Uniform and pointwise approximation theorems.-Envelopes.-*Grothendieck's completeness theorem.-Theorems of Banach-Dieudonne and Krein-Smulyan*.- 2. Weak Integrals, Moments and Barycenters.- Preliminaries and notations from integration theory.-An existence theorem for weak integrals.-Vague density of point-measures with prescribed barycenter.-*Choquet's barycenter formula for affine Baire functions of first class, and a counterexample for affine functions of higher class*.- 3. Comparison of Measures on a Compact Convex Set.- Ordering of measures.-The concept of dilation for simple measures.-The fundamental lemma on the existence of majorants.-Characterization of envelopes by integrals.-*Dilation of general measures.-Cartier's Theorem*.- 4. Choquet's Theorem.- A characterization of extreme points by means of envelopes.-The concept of a boundary set.-Herve's theorem on the existence of a strictly convex function on a metrizable compact convex set.-The concept of a boundary measure, and Mokobodzki's characterization of boundary measures.-The integral representation theorem of Choquet and Bishop - de Leeuw.-A maximum principle for superior limits of 1.s.c. convex functions.-Bishop - de Leeuw's integral theorem relatively to a ?-field on the extreme boundary.-*A counterexample based on the "porcupine topology"*.- 5. Abstract Boundaries Defined by Cones of Functions.- The concept of a Choquet boundary.-Bauer's maximum principle.-The Choquet-Edwards theorem that Choquet boundaries are Baire spaces.-The concept of a Silov boundary.-Integral representation by means of measures on the Choquet boundary.- 6. Unilateral Representation Theorems with Application to Simplicial Boundary Measures.- Ordered convex compacts.-Existence of maximal extreme points.-Characterization of the set of maximal extreme points as a Choquet boundary.-Definition and basic properties of simplicial measures.-Existence of simplicial boundary measures, and the Caratheodory Theorem in ?n.-Decomposition of representing boundary measures into simplicial components.- II Structure of Compact Convex Sets.- 1. Order-unit and Base-norm Spaces.- Basic properties of (Archimedean) order-unit spaces.-A representation theorem of Kadison.-The vector-lattice theorem of Stone-Kakutani-KreinYosida.-Duality of order-unit and base-norm spaces.- 2. Elementary Embedding Theorems.- Representation of a closed subspace A of C?(X) as an A(K)-space by the canonical embedding of X in A*.-The concept of an "abstract compact convex" and its regular embedding in a locally convex Hausdorff space.-The connection between compact convex sets and locally compact cones.- 3. Choquet Simplexes.- Riesz' decomposition property and lattice cones.-Choquet's uniqueness theorem.-Choquet-Meyer's characterizations of simplexes by envelopes.-Edward's separation theorem.-Continuous affine extensions of functions defined on compact subsets of the extreme boundary of a simplex.-Affine Borel extensions of functions defined on the extreme boundary of a simplex.-*Examples of "non-metrizable" pathologies in simplexes.*.- 4. Bauer Simplexes and the Dirichlet Problem of the Extreme Boundary.- Bauer's characterizations of simplexes with closed extreme boundary.-The Dirichlet problem of the extreme boundary.-A criterion for the existence of continuous affine extensions of maps defined on extreme boundaries.- 5. Order Ideals, Faces, and Parts.- Elementary properties of order ideals and faces.-Extension property and characteristic number.-Archimedean and strongly Archimedean ideals and faces.-Exposed and relatively exposed faces.-Specialization to simplexes.-The concept of a "part", and an inequality of Harnack type.-Characterization of the parts of a simplex in terms of representing measures.-*An example of an Archimedean face which is not strongly Archimedean.*.- 6. Split-faces and Facial Topology.- Definition and elementary properties of split faces.-Characterization of split faces by relativization of orthogonal measures.-An extension theorem for continuous affine functions defined on a split face.- The facial topology.-Specialization to simplexes.-*Near-lattice ideals, and primitive ideal space.-The connection between facial topology and hull kernel topology.-Compact convex sets with sufficiently many inner automorphisms.-A remark on the applications to C*-algebras.*.- 7. The Concept of Center for A(K).- Extension of facially continuous functions.-The facial topology is Hausdorff for Bauer simplexes only.-The concept of center, and the connections with facially continuous functions and order-bounded operators.-Convex compact sets with trivial center.-*An example of a prime simplex.-Stormer's characterization of Bauer simplexes.*.- 8. Existence and Uniqueness of Maximal Central Measures Representing Points of an Arbitrary Compact Convex Set.- The relation xoy, and the concept of a primary point.-A point x is primary iff the local center at x is trivial.-The concept of a central measure.-s maximum principle.-The Choquet-Edwards theorem that Choquet boundaries are Baire spaces.-The concept of a Silov boundary.-Integral representation by means of measures on the Choquet boundary.- 6. Unilateral Representation Theorems with Application to Simplicial Boundary Measures.- Ordered convex compacts.-Existence of maximal extreme points.-Characterization of the set of maximal extreme points as a Choquet boundary.-Definition and basic properties of simplicial measures.-Existence of simplicial boundary measures, and the Caratheodory Theorem in ?n.-Decomposition of representing boundary measures into simplicial components.- II Structure of Compact Convex Sets.- 1. Order-unit and Base-norm Spaces.- Basic properties of (Archimedean) order-unit spaces.-A representation theorem of Kadison.-The vector-lattice theorem of Stone-Kakutani-KreinYosida.-Duality of order-unit and base-norm spaces.- 2. Elementary Embedding Theorems.- Representation of a closed subspace A of C?(X) as an A(K)-space by the canonical embedding of X in A*.-The concept of an "abstract compact convex" and its regular embedding in a locally convex Hausdorff space.-The connection between compact convex sets and locally compact cones.- 3. Choquet Simplexes.- Riesz' decomposition property and lattice cones.-Choquet's uniqueness theorem.-Choquet-Meyer's characterizations of simplexes by envelopes.-Edward's separation theorem.-Continuous affine extensions of functions defined on compact subsets of the extreme boundary of a simplex.-Affine Borel extensions of functions defined on the extreme boundary of a simplex.-*Examples of "non-metrizable" pathologies in simplexes.*.- 4. Bauer Simplexes and the Dirichlet Problem of the Extreme Boundary.- Bauer's characterizations of simplexes with closed extreme boundary.-The Dirichlet problem of the extreme boundary.-A criterion for the existence of continuous affine extensions of maps defined on extreme boundaries.- 5. Order Ideals, Faces, and Parts.- Elementary properties of order ideals and faces.-Extension property and characteristic number.-Archimedean and strongly Archimedean ideals and faces.-Exposed and relatively exposed faces.-Specialization to simplexes.-The concept of a "part", and an inequality of Harnack type.-Characterization of the parts of a simplex in terms of representing measures.-*An example of an Archimedean face which is not strongly Archimedean.*.- 6. Split-faces and Facial Topology.- Definition and elementary properties of split faces.-Characterization of split faces by relativization of orthogonal measures.-An extension theorem for continuous affine functions defined on a split face.- The facial topology.-Specialization to simplexes.-*Near-lattice ideals, and primitive ideal space.-The connection between facial topology and hull kernel topology.-Compact convex sets with sufficiently many inner automorphisms.-A remark on the applications to C*-algebras.*.- 7. The Concept of Center for A(K).- Extension of facially continuous functions.-The facial topology is Hausdorff for Bauer simplexes only.-The concept of center, and the connections with facially continuous functions and order-bounded operators.-Convex compact sets with trivial center.-*An example of a prime simplex.-Stormer's characterization of Bauer simplexes.*.- 8. Existence and Uniqueness of Maximal Central Measures Representing Points of an Arbitrary Compact Convex Set.- The relation xoy, and the concept of a primary point.-A point x is primary iff the local center at x is trivial.-The concept of a central measure.- Existence and uniqueness of maximal central measures in a special case.-The "lifting" technique.-Wils' theorem on the existence and uniqueness of maximal central measures which are pseudo-carried by the set of primary points.- References.
Series Title: Ergebnisse der Mathematik und ihrer Grenzgebiete, 57; Ergebnisse der Mathematik und Ihrer Grenzgebiete., Neue Folge. ;, 57.
Responsibility: Erik M. Alfsen.

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