Abstract
We will use García-Falset and Lloréns Fuster's paper on the AMC-property to prove that a Banach space that embeds in a subspace of a Banach space with a 1-unconditional basis has the property AMC and thus the weak fixed point property. We will apply this to some results by Cowell and Kalton to prove that every reflexive real Banach space with the property and its dual have the and that a real Banach space such that is sequentially compact and has has the .
Highlights
In 1988 Sims 1 introduced the notion of weak orthogonality WORTH and asked whether spaces with WORTH have the weak fixed point property wFPP
We will use Garcıa-Falset and Llorens Fuster’s paper on the AMC-property to prove that a Banach space X that 1 δ embeds in a subspace Xδ of a Banach space Y with a 1-unconditional basis has the property AMC and the weak fixed point property. We will apply this to some results by Cowell and Kalton to prove that every reflexive real Banach space with the property WORTH and its dual have the FPP and that a real Banach space X such that BX∗ is w∗ sequentially compact and X∗ has WORTH∗ has the wFPP
In 2008 Cowell and Kalton 6 studied properties au and au∗ in a Banach space X, where au coincides with WORTH if X is separable and au∗ in X coincides with WORTH∗ in X∗ if X is a separable Banach space. Among other things they proved that a real Banach space with au∗ embeds almost isometrically in a space with a shrinking 1unconditional basis and observed that au and au∗ are equivalent if X is reflexive
Summary
In 1988 Sims 1 introduced the notion of weak orthogonality WORTH and asked whether spaces with WORTH have the weak fixed point property wFPP. In 1993 Garcıa-Falset 2 proved that if X is uniformly nonsquare and has WORTH it has the wFPP, Mazcuna ́n Navarro in her doctoral dissertation 3 showed that uniform nonsquareness is enough In this work she showed that WORTH plus 2-UNC implies the wFPP. In 2008 Cowell and Kalton 6 studied properties au and au∗ in a Banach space X, where au coincides with WORTH if X is separable and au∗ in X coincides with WORTH∗ in X∗ if X is a separable Banach space Among other things they proved that a real Banach space with au∗ embeds almost isometrically in a space with a shrinking 1unconditional basis and observed that au and au∗ are equivalent if X is reflexive. Fixed Point Theory and Applications the wFPP Combining this with Cowell and Kalton’s results we were able to show that a reflexive real Banach space with WORTH and its dual both have FPP, giving a partial answer to Sims’ question. We showed that a separable space X such that X∗ has WORTH∗ and BX∗ is w∗ sequentially compact has the wFPP
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