New asymptotically isometric properties that imply the failure of the fixed point property in copies of <mml:math display="inline"><mml:msup><mml:mrow><mml:mi>ℓ</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:math>

Abstract

In this study, we introduce three new notions which may occur for some Banach spaces. We call these new properties AAI1, AAI2 and AAI3 where AAI stands for “alternative asymptotically isometric”. We prove that if a Banach space has any of them, then it fails to have the fixed point property for nonexpansive mappings. We provide alternative ways of detecting if a Banach space has any of these properties. We show that AAI1 is an equivalent property for a Banach space to have an asymptotically isometric copy of ℓ1ℓ1ℓ1