The Fixed Point Property in c0 with an Equivalent Norm

Abstract EN

We study the fixed point property (FPP) in the Banach space c0 with the equivalent norm ‖⋅‖D.

The space c0 with this norm has the weak fixed point property.

We prove that every infinite-dimensional subspace of (c0,‖⋅‖D) contains a complemented asymptotically isometric copy of c0, and thus does not have the FPP, but there exist nonempty closed convex and bounded subsets of (c0,‖⋅‖D) which are not ω-compact and do not contain asymptotically isometric c0—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 (c0,‖⋅‖D), and we give some of its properties.

We also prove that the dual space of (c0,‖⋅‖D) over the reals is the Bynum space l1∞ and that every infinite-dimensional subspace of l1∞ does not have the fixed point property.