A geometric Jordan decomposition theorem

Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas Pub Date : 2024-03-11 DOI:10.1007/s13398-024-01569-0

引用次数: 0

Abstract

For a compact convex set K, let A(K) denote the space of real-valued affine continuous functions, equipped with the supremum norm. For a closed subspace \(X \subset A(K)\) we give sufficient conditions, so that the weak \(^*\) closure of the set of extreme points of the dual unit ball has a decomposition in terms of ‘positive’ and ‘negative’ parts. We give several applications of these ideas to convexity and positivity. When K is a Choquet simplex, we show that the dual unit ball of such an X, inherits nice facial structure. We also use this to partly solve the open problem of exhibiting faces that are Choquet simplexes in the dual unit ball of a Banach space.

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几何约旦分解定理

摘要对于一个紧凑凸集 K，让 A(K) 表示实值仿射连续函数的空间，并配备上顶规范。对于一个封闭子空间 \(X \subset A(K)\) 我们给出了充分条件，使得对偶单位球极值点集合的弱\(^*\) 闭合具有 "正 "和 "负 "部分的分解。我们给出了这些观点在凸性和正性方面的若干应用。当 K 是一个 Choquet 单纯形时，我们证明这样一个 X 的对偶单位球继承了很好的面结构。我们还利用这一点部分地解决了在巴拿赫空间的对偶单位球中展示 Choquet 单纯形的面这一未决问题。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas

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