Decomposable compact convex sets and peak sets for function spaces.
Author:
Leonard Asimow
Journal:
Proc. Amer. Math. Soc. 25 (1970), 75-79
MSC:
Primary 46.55
DOI:
https://doi.org/10.1090/S0002-9939-1970-0259607-8
MathSciNet review:
0259607
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Abstract | References | Similar Articles | Additional Information
Abstract: Geometric conditions are known under which a closed face of a compact convex set is a peak set with respect to the space of continuous affine (real-valued) functions. The purpose of this note is to give an application of this “abstract-geometric” set-up to the problem of finding peak sets (or points) in a compact Hausdorff space with respect to a closed subspace of continuous complex-valued functions. In this fashion we obtain the strong hull criteria of Curtis and Figá-Talamanca and in particular the Bishop peak point theorem for function algebras.
- Leonard Asimow, Directed Banach spaces of affine functions, Trans. Amer. Math. Soc. 143 (1969), 117–132. MR 247419, DOI https://doi.org/10.1090/S0002-9947-1969-0247419-7 ---, Extensions of continuous affine functions, (to appear).
- Errett Bishop, A minimal boundary for function algebras, Pacific J. Math. 9 (1959), 629–642. MR 109305
- Philip C. Curtis Jr. and Alessandro Figà-Talamanca, Factorization theorems for Banach algebras, Function Algebras (Proc. Internat. Sympos. on Function Algebras, Tulane Univ., 1965) Scott-Foresman, Chicago, Ill., 1966, pp. 169–185. MR 0203500
- Robert R. Phelps, Lectures on Choquet’s theorem, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1966. MR 0193470
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Additional Information
Keywords:
Compact convex sets,
Functions spaces,
affine functions,
extreme points,
functions algebras,
peak sets,
peak faces,
strong hull
Article copyright:
© Copyright 1970
American Mathematical Society