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

First we prove that if a separable Banach space X contains an isometric\r\ncopy of an infinite-dimensional space A(S) of affine continuous functions\r\non a Choquet simplex S, then its dual X∗ lacks the weak∗ fixed point property\r\nfor nonexpansive mappings. Then, we show that the dual of a separable\r\nL1-predual X fails the weak∗ fixed point property for nonexpansive mappings\r\nif and only if X has a quotient isometric to some infinite-dimensional space\r\nA(S). Moreover, we provide an example showing that “quotient” cannot be\r\nreplaced by “subspace”. Finally, it is worth mentioning that in our characterization\r\nthe space A(S) cannot be substituted by any space C(K) of continuous\r\nfunctions on a compact Hausdorff K.