Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-12-05T02:12:03.043Z Has data issue: false hasContentIssue false
  • English
  • Français

Distances between convex subsets of state spaces

Published online by Cambridge University Press:  17 April 2009

A.J. Ellis
A.J. Ellis
Affiliation:
Department of Mathematics, Institute of Advanced Studies, Australian National Univerity, GPO Box 4, Canberra, ACT 2601, Australia.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let L be a closed linear space of continuous real-valued functions, containing constants, on a compact Hausdorff space Ω. This paper gives some new criteria for a closed subset E of Ω to be an L-interpolation set, or more generally for L|E to be uniformly closed or simplicial, in terms of distances between certain compact convex subsets of the state space of L. These criteria involve the facial structure of the state space and hence are of a geometric nature. The results sharpen some standard results of Glicksberg.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 32 , Issue 1 , August 1985 , pp. 109 - 117
Copyright
Copyright © Australian Mathematical Society 1985

References

[1]Alfsen, E.M., Compact convex sets and boundary integrals (Ergebnisse der Mathematik und ihrer Grenzgebiete, 57. Springer-Verlag, Berlin, Heidelberg, New York, 1971).CrossRefGoogle Scholar
[2]Asimow, L. and Ellis, A.J., Convexity theory and its applications in functional analysis (Academic Press, London, New York, 1980).Google Scholar
[3]Batty, C.J.K., “Some properties of maximal measures on compact convex sets”, Math. Proc. Cambridge Philos. Soc. 94 (1983), 297–305.CrossRefGoogle Scholar
[4]Bishop, E. and Phelps, R.R., “The support functionals of a convex set”, Convexity, – (Proc. Sympos. in Pure Math. 7. American Mathematical Society, Providence, Rhode Island, 1963).CrossRefGoogle Scholar
[5]Briem, E., “Peak sets for the real part of a function algebra”, Proc. Amer. Math. Soc. 85 (1983), 77–78.CrossRefGoogle Scholar
[6]Effros, E.G., “Structure in simplexes II”, J. Funct. Anal. 1 (1967), 361–391.CrossRefGoogle Scholar
[7]Ellis, A.J. and Roy, A.K., “Dilated sets and characterizations of simplexes”, Invent. Math. 56 (1980), 101–108.CrossRefGoogle Scholar
[8]Glicksberg, I., “Measures orthogonal to algebras and sets of anti-symmetry”, Trans. Amer. Math. Soc. 105 (1962), 415–435.CrossRefGoogle Scholar
[9]Hoffman, K. and Wermer, J., “A characterization of C(X)”, Pacific J. Math. 12 (1962), 941–944.CrossRefGoogle Scholar
[10]McDonald, J.N., “Compact convex sets with the equal support property”, Pacific J. Math. 37 (1971), 429–443.CrossRefGoogle Scholar
[11]Teleman, S., “On the regularity of the boundary measures” (INCREST preprint, no. 30/1981).Google Scholar