Compact Convex Sets and Boundary Integrals

  • Erik M. Alfsen

Part of the Ergebnisse der Mathematik und ihrer Grenzgebiete book series (MATHE2, volume 57)

Table of contents

  1. Front Matter
    Pages I-XI
  2. Erik M. Alfsen
    Pages 67-187
  3. Back Matter
    Pages 189-212

About this book

Introduction

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.

Keywords

Boundary Convexity Finite Integral Integrals Konvexe Menge algebra function functional analysis operator theory proof theorem

Authors and affiliations

  • Erik M. Alfsen
    • 1
  1. 1.Department of MathematicsUniversity of OsloBlindern Oslo 3Norway

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-642-65009-3
  • Copyright Information Springer-Verlag Berlin Heidelberg 1971
  • Publisher Name Springer, Berlin, Heidelberg
  • eBook Packages Springer Book Archive
  • Print ISBN 978-3-642-65011-6
  • Online ISBN 978-3-642-65009-3
  • Series Print ISSN 0071-1136
  • About this book