Weak compactness and fixed point property for affine bi-Lipschitz maps

Let X be a Banach space and let Cbe a closed convex bounded subset of X. It is proved that Cis weakly compact if, and only if, Chas the genericfixed point property (G-FPP) for the class of L-bi-Lipschitz affine mappings for every L > 1. It is also proved that if Xhas Pelczynski's property (u), then either Cis weakly compact, contains an l(1)-sequence or a c(0)-summing basic sequence. In this case, weak compactness of Cis equivalent to the G-FPP for the strengthened class of affine mappings that are uniformly bi-Lipschitz. We also introduce a generalized form of property (u), called property (su), and use it to prove that if X has property (su) then either Cis weakly compact or contains a wide-(s) sequence which is uniformly shift equivalent. In this case, weak compactness in such spaces can also be characterized in terms of the G-FPP for affine uniformly bi-Lipschitz mappings. It is also proved that every Banach space with a spreading basis has property (su), thus property (su) is stronger than property (u). These results yield a significant strengthening of an important theorem of Benavides, Japon-Pineda and Prus published in 2004. (c) 2020 Elsevier Inc. All rights reserved.