PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 139, Number 3, March 2011, Pages 1099–1104 S 0002-9939(2010)10534-8 Article electronically published on August 10, 2010 ISOMORPHISMS OF SPACES OF CONTINUOUS AFFINE FUNCTIONS ON COMPACT CONVEX SETS WITH LINDEL ¨ OF BOUNDARIES PAVEL LUDV ´ IK AND JI ˇ R ´ I SPURN ´ Y (Communicated by Nigel J. Kalton) Abstract. Let X, Y be compact convex sets such that every extreme point of X and Y is a weak peak point and both ext X and ext Y are Lindel¨of spaces. We prove that if there exists an isomorphism T : A c (X) → A c (Y ) with ‖T ‖·‖T −1 ‖ < 2, then ext X is homeomorphic to ext Y . This generalizes results of C. H. Chu and H. B. Cohen. 1. Introduction If X is a compact convex set in a real locally convex space, let A c (X) stand for the space of all continuous affine functions, A b (X) for the space of all bounded affine functions on X, and ext X for the set of extreme points. We refer the reader to [5, pp. 72, 73, 75] for notions of the theory of compact convex sets. We just mention that X can be embedded to (A c (X)) ∗ via the eval- uation mapping φ : X → (A c (X)) ∗ defined as φ(x)(f )= f (x), f ∈ A c (X), x ∈ X. The dual unit ball B (A c (X)) ∗ equals the convex hull co (X ∪−X), and (A c (X)) ∗ coincides with span X, the linear span of X. Further, any function f ∈ A b (X) has a unique extension to span X, and this provides an identification of (A c (X)) ∗∗ with A b (X). For a set F ⊂ X, the complementary set F cs is defined as the union of all faces of X disjoint from F . A face F of X is said to be a split face if its complementary set F cs is convex (and hence a face; see [1, p.132]) and every point in X \ (F ∪ F cs ) can be uniquely represented as a convex combination of a point in F and a point in F cs . We call x ∈ ext X a weak peak point if given ε ∈ (0, 1) and an open neighborhood U of x, there exists h ∈ A c (X) such that ‖h‖≤ 1, h(x) > 1 − ε and |h| <ε on ext X \ U . Received by the editors January 7, 2010 and, in revised form, April 9, 2010. 2010 Mathematics Subject Classification. Primary 46A55, 46E15, 54D20. Key words and phrases. Compact convex set, extreme point, weak peak point, Lindel¨of space, continuous affine function. The first author was supported by grant GA ˇ CR 401/09/H007. The second author was supported in part by the grants GAAV IAA 100190901 and GA ˇ CR 201/07/0388, and in part by the Research Project MSM 0021620839 from the Czech Ministry of Education. c 2010 American Mathematical Society Reverts to public domain 28 years from publication 1099 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use