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## Asymptotically isometric copies of $\ell^{1\boxplus 0}$

#### Veysel NEZİR [1]

Using James' Distortion Theorems, researchers have inquired relations between spaces containing nice copies of $c_0$ or $\ell^1$ and the failure of the fixed point property for nonexpansive mappings especially after the fact that every classical nonreflexive Banach space contains an isometric copy of either $\ell^1$ or $c_0$. For instance, finding asymptotically isometric (ai) copies of $\ell^1$ or $c_0$ inside a Banach space reveals the space's failure of the fixed point property for nonexpansive mappings. There has been many researches done using these tools developed by James and followed by Dowling, Lennard, and Turett mainly to see if a Banach space can be renormed to have the fixed point property for nonexpansive mappings when there is failure.

In this paper, we introduce the concept of Banach spaces containing ai copies of $\ell^{1\boxplus 0}$ and give alternative methods of detecting them. We show the relations
between spaces containing these copies and the failure of the fixed point property for nonexpansive mappings. Finally, we give some remarks and examples pointing our vital result: if a Banach space contains an ai copy of $\ell^{1\boxplus 0}$, then it contains an ai copy of $\ell^1$ but the converse does not hold.
Fixed point property, nonexpansive mapping, renorming, asymptotically isometric copy of $c_0$, asymptotically isometric copy of $\ell^1$
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Primary Language en Mathematics Mathematics Orcid: 0000-0001-9640-8526Author: Veysel NEZİR (Primary Author)Institution: KAFKAS UNIVERSITYCountry: Turkey Publication Date : June 2, 2020
 Bibtex @research article { hujms507488, journal = {Hacettepe Journal of Mathematics and Statistics}, issn = {2651-477X}, eissn = {2651-477X}, address = {}, publisher = {Hacettepe University}, year = {2020}, volume = {49}, pages = {984 - 997}, doi = {10.15672/hujms.507488}, title = {Asymptotically isometric copies of \$\\ell\^\{1\\boxplus 0\}\$}, key = {cite}, author = {Nezi̇r, Veysel} } APA Nezi̇r, V . (2020). Asymptotically isometric copies of $\ell^{1\boxplus 0}$ . Hacettepe Journal of Mathematics and Statistics , 49 (3) , 984-997 . DOI: 10.15672/hujms.507488 MLA Nezi̇r, V . "Asymptotically isometric copies of $\ell^{1\boxplus 0}$" . Hacettepe Journal of Mathematics and Statistics 49 (2020 ): 984-997 Chicago Nezi̇r, V . "Asymptotically isometric copies of $\ell^{1\boxplus 0}$". Hacettepe Journal of Mathematics and Statistics 49 (2020 ): 984-997 RIS TY - JOUR T1 - Asymptotically isometric copies of $\ell^{1\boxplus 0}$ AU - Veysel Nezi̇r Y1 - 2020 PY - 2020 N1 - doi: 10.15672/hujms.507488 DO - 10.15672/hujms.507488 T2 - Hacettepe Journal of Mathematics and Statistics JF - Journal JO - JOR SP - 984 EP - 997 VL - 49 IS - 3 SN - 2651-477X-2651-477X M3 - doi: 10.15672/hujms.507488 UR - https://doi.org/10.15672/hujms.507488 Y2 - 2019 ER - EndNote %0 Hacettepe Journal of Mathematics and Statistics Asymptotically isometric copies of $\ell^{1\boxplus 0}$ %A Veysel Nezi̇r %T Asymptotically isometric copies of $\ell^{1\boxplus 0}$ %D 2020 %J Hacettepe Journal of Mathematics and Statistics %P 2651-477X-2651-477X %V 49 %N 3 %R doi: 10.15672/hujms.507488 %U 10.15672/hujms.507488 ISNAD Nezi̇r, Veysel . "Asymptotically isometric copies of $\ell^{1\boxplus 0}$". Hacettepe Journal of Mathematics and Statistics 49 / 3 (June 2020): 984-997 . https://doi.org/10.15672/hujms.507488 AMA Nezi̇r V . Asymptotically isometric copies of $\ell^{1\boxplus 0}$. Hacettepe Journal of Mathematics and Statistics. 2020; 49(3): 984-997. Vancouver Nezi̇r V . Asymptotically isometric copies of $\ell^{1\boxplus 0}$. Hacettepe Journal of Mathematics and Statistics. 2020; 49(3): 984-997.

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