The minimum principle for affine functions and isomorphisms of continuous affine function spaces

Abstract

Let X be a compact convex set and let ext X stand for the set of extreme points of X. We show that if f : X → R is an affine function with the point of continuity property such that f ≤ 0 on ext X, then f ≤ 0 on X.As a corollary of this minimum principle we obtain a generalization of a theorem by H.B. Cohen and C.H. Chu by proving the following result. Let X, Y be compact convex sets such that every extreme point of X and Y is a weak peak point and let T : A c (X) → A c (Y ) be an isomorphism such that T · T −1 < 2. Then ext X is homeomorphic to ext Y .2010 Mathematics Subject Classification. 46A55; 46B03.

On the Banach-Mazur distance between continuous function spaces with scattered boundaries

“…Starting in [11], and continuing in [18], [13], and [22], the theorem of Amir and Cambern was extended to the context of subspaces. The final result for subspaces of scalar functions (see [22,Theorem 1.1]), reads as follows.…”

Isomorphisms of spaces of affine continuous complex functions

“…In [19] the same result is proved without the assumption of the Lindelöf property and paper [37] provides an analogous result for the case of complex functions. It turns out that this result is in a sense optimal since the bound 2 cannot be improved (see [15], where a pair of nonhomeomorphic compact spaces K 1 , K 2 for which there exists an isomorphism T : C(K 1 , R) → C(K 2 , R) with T · T −1 = 2 is constructed) and the assumption on weak peak points cannot be omitted (see [26], where the author constructs for each ε ∈ (0, 1) a pair of simplices X 1 , X 2 such that ext X 1 is not homeomorphic to ext X 2 but there is an isomorphism T :…”

An Amir-Cambern theorem for subspaces of Banach lattice-valued continuous functions

“…In [20], we were able to extend the theorem of Amir and Cambern by showing that for closed subspaces H i ⊂ C 0 (K i , R) for i = 1, 2, their Choquet boundaries are homeomorphic provided points in the Choquet boundaries are weak peak points and there exists an isomorphism T : H 1 → H 2 with T • T −1 < 2. Before that, there were proved analougous results for spaces of affine real continuous functions on compact convex sets ( [6], [16] and [10]), and spaces of affine complex continuous functions [21].…”