Extreme points and the diameter norm
Let \(X\) be an infinite compact Hausdorff space. Let \(A \subset C(X)\) be a closed, point separating subspace containing constants \(C\) (real scalars). It is well-known that \(A\) can be identified with the space of affine continuous functions on its state space. If one equips the quotient space \(A/C\) with the norm \(\| [f]\| = \text{diam}(f(X))\), the reviewer and \textit{A. K. Roy} [J. Aust. Math. Soc. 70, 323--335 (2001; Zbl 1027.46009)] have given a complete description of the extreme points of the dual unit ball of \(A/C\) under the assumption that every extreme point of the state space is a split face. In this paper, the authors show that the same description holds when \(A\) satisfies the unique decomposition property. When the state space is a Choquet simplex, \(A\) satisfies both the conditions described above.
scientific article; zbMATH DE number 2143229