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Isometric embedding of $ \ell_1$ into Lipschitz-free spaces and $ \ell_\infty$ into their duals


Authors: Marek Cúth and Michal Johanis
Journal: Proc. Amer. Math. Soc. 145 (2017), 3409-3421
MSC (2010): Primary 46B03, 54E35
DOI: https://doi.org/10.1090/proc/13590
Published electronically: April 12, 2017
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Abstract: We show that the dual of every infinite-dimensional Lipschitz-free Banach space contains an isometric copy of $ \ell _\infty $ and that it is often the case that a Lipschitz-free Banach space contains a $ 1$-complemented subspace isometric to $ \ell _1$. Even though we do not know whether the latter is true for every infinite-dimensional Lipschitz-free Banach space, we show that the space is never rotund. In the last section we survey the relations between isometric embeddability of  $ \ell _\infty $ into $ X^*$ and containment of a good copy of $ \ell _1$ in $ X$ for a general Banach space $ X$.


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Additional Information

Marek Cúth
Affiliation: Department of Mathematical Analysis, Charles University, Sokolovská 83, 186 75 Praha 8, Czech Republic
Email: cuth@karlin.mff.cuni.cz

Michal Johanis
Affiliation: Department of Mathematical Analysis, Charles University, Sokolovská 83, 186 75 Praha 8, Czech Republic
Email: johanis@karlin.mff.cuni.cz

DOI: https://doi.org/10.1090/proc/13590
Keywords: Lipschitz-free spaces, isometric embedding of $\ell_1$, isometric embedding of $\ell_\infty$
Received by editor(s): May 23, 2016
Received by editor(s) in revised form: September 1, 2016
Published electronically: April 12, 2017
Additional Notes: The first author is a junior researcher in the University Centre for Mathematical Modelling, Applied Analysis and Computational Mathematics (MathMAC) and was supported by grant P201/12/0290
The second author was supported by GAČR 16-07378S
Communicated by: Thomas Schlumprecht
Article copyright: © Copyright 2017 American Mathematical Society