Extreme points and the diameter norm
by
Manuel Sanchis
Departament de Matemàtiques, Universitat Jaume I (Spain)
Coauthors: J. Font

Let X be a (infinite) compact Hausdorff space. As usual, C(X) stands for the space of all continuous real-valued functions on X endowed with the supremum norm ||·||_\infty. The diameter (semi)norm of any function f in C(X) is defined to be the diameter of the range of f. This seminorm was first studied by Györy and Molnár in [4] when dealing with linear bijections of C(X) (X compact and first countable) which leave the diameter of the range of every function invariant, which is to say, diameter preserving mappings. Since then, considerable attention has been given to these mappings and to the diameter norm. Namely, González and Uspenskij [5] and, independently, Cabello [1] removed the hypothesis of first countability on X and characterized the extreme points of the closed unit ball of the dual of C(X) endowed with the diameter semnorm (see also [3] for related results on certain subspaces of continuous functions). Rao and Roy [6] have obtained analogue characterizations in the context of spaces of affine functions and vector-valued continuous functions. Recently, in [2], the authors have proved that the diameter norm is maximal on C₀(X), where X is a connected non-compact manifold.

Let A be a point-separating closed linear subspace of (C(X), ||·||_\infty) which contains the constant functions. Let C denote the constant functions on X. In this talk we study the extreme points of the closed unit ball of the dual of the quotient A/C endowed with the diameter norm. We prove the following results (here Ch(A) stands for the Choquet boundary of A and di{x} for the Dirac delta at the point x):

Theorem 1. Let A be a point-separating closed linear subspace of C(X) with the constants. Then the set of extreme points of the closed unit ball of (A_d) ^* is included in {di{x}{y}:x, y in Ch(A), x =/= y}.

Theorem 2. Let A be a point-separating closed linear subspace of C(X) with the constants. If A satisfies the unique decomposition property, then the set of extreme points of the closed unit ball of (A_d) ^* consists exactly of {di{x}{y}:x, y in Ch(A), x =/= y}.

We give an example showing that theinclusion in Theorem 1 can be strict. As examples of usual closed linear subspaces of C(X) enjoying the unique descomposition property we have simplicial function spaces and the space A(K) of all continuous affine functions on a compact convex set K if either every norm-exposed face of K is projective or K satisfies an adequate intersection property.

References

[1] F. Cabello Sánchez, Diameter preserving linear maps and isometries, Archiv Math. (Basel) 73 (1999), 373-379.

[2] A. Cabello Sánchez and F. Cabello Sánchez, Maximal norms on Banach spaces of continuous functions. Corrigendum to: Örthonormal systems in Banach spaces and their applications" [Math. Proc. Cambridge Philos. Soc. 79 (1976), no. 3, 493-510] by N. J. Kalton and G. V. Wood. Math. Proc. Cambridge Philos. Soc. 129 (2000), no. 2, 325-330.

[3] J.J. Font and M. Sanchis, A characterization of locally compact spaces with homeomorphic one-point compactifications, Top. and its Appl. 121 (2002), no. 1-2, 91-104.

[4] M. Györy and L. Molnár, Diameter preserving bijections of C(X), Arch. Math. 71 (1998), 301-310.

[5] F. González and V.V. Uspenskij, On homomorphisms of groups of integer-valued functions, Extracta Math. 14 (1) (1999), 19-29.

[6]T.S.S.R.K. Rao and A.K. Roy, Diameter-preserving linear bijections of function spaces, J. Australian Math. Soc. 70 (2001) (3), 323-335.

Date received: May 15, 2003