MMN-4892
New asymptotically isometric properties that imply the failure of the fixed point property in copies of ℓ¹
Shilpa Das; Veysel Nezir; Aysun Güven;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 [fpp(ne)]. 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 (ai) copy of $\ell^1$. That is, a Banach space contains an ai copy of $\ell^1$ if and only if it has the property AAI1. In fact, we obtain generalized version of property AAI1, which we call property AAI3, obtained as a conclusion of property AAI2. We prove that all properties AAI1, AAI2 and AAI3 are equivalent. We also support our ideas with some examples and remarks for our equivalent properties to contain ai copy of $\ell^1$.
Vol. 26 (2025), No. 1, pp. 181-193