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

Abstract

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.

Keywords

• Fixed point property,nonexpansive mapping,renorming,asymptotically isometric copy of $c_0$,asymptotically isometric copy of $\ell^1$

References

1. [1] J. Diestel, Sequences and series in Banach spaces, Springer Science & Business Media, 2012.
2. [2] S.J. Dilworth, M. Girardi and J. Hagler, Dual Banach spaces which contain an iso- metric copy of L1, Bull. Polish Acad. Sci. Math. 48, 1–12, 2000.
3. [3] P.N. Dowling and C.J. Lennard, Every nonreflexive subspace of $L_1[0, 1]$ fails the fixed point property, Proc. Amer. Math. Soc. 125, 443–446, 1997.
4. [4] P.N. Dowling, C.J. Lennard and B. Turett, Asymptotically isometric copies of $c_0$ in Banach Spaces, J. Math. Anal. Appl. 219, 377–391, 1998.
5. [5] P.N. Dowling, C.J. Lennard and B. Turett, Renormings of $\ell^1$ and $c_0$ and fixed point properties, in: Handbook of Metric Fixed Point Theory, Springer, Netherlands, 269– 297, 2001.
6. [6] P.N. Dowling, W.B. Johnson, C.J. Lennard and B. Turett, The optimality of James’s distortion theorems, Proc. Amer. Math. Soc. 125, 167–174, 1997.
7. [7] R.C. James, Uniformly non-square Banach spaces, Ann. of Math. 80 (3), 542–550, 1964.
8. [8] C.J. Lennard, Personal communication, 2017.

9. [9] P.K. Lin, There is an equivalent norm on $\ell^1$ that has the fixed point property, Nonlinear Anal. 68, 2303–2308, 2008.
10. [10] J. Lindenstrauss and L. Tzafriri, Classical Banach spaces I: Sequence Spaces, in: Ergebnisse der Mathematik und ihrer Grenzgebiete 92, Springer-Verlag, 1977.
11. [11] J. Lindenstrauss and L. Tzafriri, Classical Banach spaces II: Function Spaces, in: Ergebnisse der Mathematik und ihrer Grenzgebiete 97, Springer-Verlag, 1979.
12. [12] G.G. Lorentz, Some new functional spaces, Ann. Math. 51 (1), 37–55, 1950.
13. [13] V. Nezir, Fixed point properties for $c_0$-like spaces, Ph.D., University of Pittsburgh, Pittsburgh, PA, USA, 2012.
14. [14] V. Nezir, Fixed point properties for a degenerate Lorentz-Marcinkiewicz space, Turkish J. Math. 43, 1919-1939, 2019.
15. [15] V. Nezir and N. Mustafa, On the fixed point property for a degenerate Lorentz- Marcinkiewicz space, in: Proceedings of the 5th International Conference on Recent Advances in Pure and Applied Mathematics (icrapam 2018), Karadeniz Technical University, Trabzon, 23–27 July 2018.