Baire classes of Banach spaces and strongly affine functions

Abstract

We construct a metrizable simplex X X and a Baire–two function f f on X X satisfying the barycentric formula such that f f is not of affine class two; i.e., there is no bounded sequence of affine Baire–one functions on X X converging to f f . This provides an example of a Banach L ∞ \mathcal {L}_\infty –space E E such that E 2 ∗ ∗ ≠ E B 2 ∗ ∗ E_{2}^{**}\neq E_{\mathcal {B}_2}^{**} .