The Fixed Point Property in c0 with an Equivalent Norm

We study the fixed point property (FPP) in the Banach space c(0) with the equivalent norm parallel to . parallel to(D). The space c(0) with this norm has the weak fixed point property. We prove that every infinite-dimensional subspace of (c(0), parallel to . parallel to(D)) contains a complemented asymptotically isometric copy of c(0), and thus does not have the FPP, but there exist nonempty closed convex and bounded subsets of (c(0), parallel to . parallel to(D)) which are not omega-compact and do not contain asymptotically isometric c(0)-summing basis sequences. Then we define a family of sequences which are asymptotically isometric to different bases equivalent to the summing basis in the space (c(0), parallel to . parallel to(D)), and we give some of its properties. We also prove that the dual space of (c(0), parallel to . parallel to(D)) over the reals is the Bynum space l(1 infinity) and that every infinite-dimensional subspace of l(1 infinity) does not have the fixed point property.