Weak$^*$ fixed point property and the space of affine functions

First we prove that if a separable Banach space X contains an isometric copy of an infinite-dimensional space A(S) of affine continuous functions on a Choquet simplex S, then its dual X∗ lacks the weak∗ fixed point property for nonexpansive mappings. Then, we show that the dual of a separable L1-predual X fails the weak∗ fixed point property for nonexpansive mappings if and only if X has a quotient isometric to some infinite-dimensional space A(S). Moreover, we provide an example showing that “quotient” cannot be replaced by “subspace”. Finally, it is worth mentioning that in our characterization the space A(S) cannot be substituted by any space C(K) of continuous functions on a compact Hausdorff K.