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Topologies and continuous functions on extreme points and pure states

Published online by Cambridge University Press:  24 October 2008

C. J. K. Batty
C. J. K. Batty
Affiliation:
Department of Mathematics, University of Edinburgh, Edinburgh EH9 3JZ
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There are various ways of using the affine geometry of a compact convex set K to topologize its extreme boundary ∂eK (see [2, 7, 10] and the references quoted therein). The best-known method is to take as closed sets the traces on ∂eK of the closed split faces of K. This topology is known as the facial topology [1, 2, 4], and is significant because the facially continuous functions correspond to the order-bounded linear operators on A(K) and to the multipliers in A(K) [1, 3, 20]. Typically, split faces are scarce, so that the facial topology is coarse. The Choquet topology [7, 15, 18], whose closed sets are the traces of compact extremal subsets of K, is finer, but is usually less easily recognized. The Bishop–de Leeuw theorem was extended in [5] by showing that a maximal measure on K induces a measure on a σ-algebra on ∂eK containing the Choquet closed sets as well as the traces of the Baire subsets of K. The maximal topology is finer even than the Choquet topology, and the Bishop-de Leeuw theorem was further extended in [18] to a σ-algebra including the maximally closed sets. Although the definition of the maximal topology in [18] made use of integral representation theory, a geometrical description of it will be given in Proposition 2·1 below. The Choquet and maximally continuous functions on ∂eK were characterized in [18] in terms of the geometry and integral representation theory of K, respectively. It will be shown in Theorem 2·4 that the Choquet and maximally continuous functions coincide if is a union of faces of K, even though the topologies may differ.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 98 , Issue 3 , November 1985 , pp. 501 - 511
Copyright
Copyright © Cambridge Philosophical Society 1985

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References

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