Weak* fixed point property and the space of affine functions

Abstract

First we prove that if a separable Banach space X X contains an isometric copy of an infinite-dimensional space A ( S ) A(S) of affine continuous functions on a Choquet simplex S S , then its dual X ∗ X^* lacks the weak ∗ ^* fixed point property for nonexpansive mappings. Then, we show that the dual of a separable L 1 L_1 -predual X X fails the weak ∗ ^* fixed point property for nonexpansive mappings if and only if X X has a quotient isometric to some infinite-dimensional space A ( S ) 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 ) A(S) cannot be substituted by any space C ( K ) \mathcal {C}(K) of continuous functions on a compact Hausdorff K K .