Asymptotically isometric copies of c(0) and renormings of Banach spaces

If a Banach space X contains an asymptotically isometric copy of c(0), then X fails to have weak normal structure. Consequently, if X is an infinite-d imensional subspace of (c(0), parallel to . parallel to(infinity)), then X fails to have weak normal structure. Also, every equivalent renorming of c( 0)(Gamma), for Gamma uncountable, fails to have weak normal structure. Ever y separable Banach space can be equivalently renormed so as not to contain an asymptotically isometric copy of c(0). Every Banach space with the gener alized Gossez-Lami Dozo (GGLD) property fails to contain a subspace isomorp hic to c(0). (C) 1998 Academic Press.