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
Full-text PDF
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.
- [1] L. Asimow, Directed Banach spaces of affine functions, Trans. Amer. Math. Soc. 143 (1969), 117-132. MR 0247419 (40:685)
- [2] -, Extensions of continuous affine functions, (to appear).
- [3] E. Bishop, A minimal boundary for function algebras, Pacific J. Math. 9 (1959), 629-642. MR 22 #191. MR 0109305 (22:191)
- [4] P. C. Curtis and A. Figá-Talamanca, Factorization theorems for Banach algebras, Proc. Internat. Sympos. Function Algebras (Tulane Univ. 1965), Scott-Foresman, Chicago, Ill., 1966, pp. 169-185. MR 34 #3350. MR 0203500 (34:3350)
- [5] R. R. Phelps, Lectures on Choquet's theorem, Van Nostrand, Princeton, N. J., 1966. MR 33 #1690. MR 0193470 (33:1690)
Retrieve articles in Proceedings of the American Mathematical Society with MSC: 46.55
Retrieve articles in all journals with MSC: 46.55
Additional Information
DOI:
https://doi.org/10.1090/S0002-9939-1970-0259607-8
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