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The Fixed Point Property in c0 with an Equivalent Norm

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  • Berta Gamboa de Buen
  • Fernando Núñez-Medina

Abstract

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.

Suggested Citation

Handle: RePEc:wly:jnlaaa:v:2011:y:2011:i:1:n:574614
DOI: 10.1155/2011/574614
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