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On the monotone σ-envelope of A(K)
Published online by Cambridge University Press: 24 October 2008
Extract
Let K be a compact convex set regularly embedded in a locally convex Hausdorff topological vector space E. Let Ab(K) be the GM-space (6) with unit 1K consisting of all real valued bounded of affine functions on K and let A(K) be the GM-subspace of Ab(K) consisting of continuous functions. For properties of K and A(K), see (2). The monotone σ-envelope A(K)μ of A(K) is the smallest subset of Ab(K) containing A(K) and such that the pointwise limit of every uniformly bounded monotone sequence (an) in A(K)μ also lies in A(K)μ. In (1), Alfsen shows that when K is a metrizable simplex A(K)μ is a vector lattice. In this note it is shown conversely that if K is metrizable and A(K)μ is a vector lattice, then K is a simplex.
- Type
- Research Article
- Information
- Mathematical Proceedings of the Cambridge Philosophical Society , Volume 78 , Issue 2 , September 1975 , pp. 315 - 316
- Copyright
- Copyright © Cambridge Philosophical Society 1975
References
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