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On the L-embeddability of Lipschitz-free spaces in their biduals
Abstract
A Banach space X is L-embedded in its bidual if there is a projection P: X**→ X such that |x|=|Px|+|x−Px| for all x∈ X**. We show that, as long as X has dimension at least $2$, its Lipschitz-free space, denoted by ℱ(X), is not L-embedded in its bidual. Our methods have applications to the problem of when the L-embeddability of ℱ(M) passes to ℱ(A) for a metric space M and A⊆ M. This is the case when A is compact or when A is geodesically closed.
Related
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BibTeX
@misc{braga2026lembeddability,
title = {{On the $L$-embeddability of Lipschitz-free spaces in their biduals}},
author = {Bruno de Mendonça Braga and Chris Gartland and Gilles Lancien and Pavlos Motakis and Eva Pernecká and Thomas Schlumprecht},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.08616},
eprint = {2610.08616},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.08616},
}
RIS
TY - UNPB
TI - On the L-embeddability of Lipschitz-free spaces in their biduals
AU - Bruno de Mendonça Braga
AU - Chris Gartland
AU - Gilles Lancien
AU - Pavlos Motakis
AU - Eva Pernecká
AU - Thomas Schlumprecht
PY - 2026
DA - 2026/10/06
PB - arXiv
DO - 10.48550/arxiv.2610.08616
UR - https://arxiv.org/abs/2610.08616
AB - A Banach space X is L-embedded in its bidual if there is a projection P: X^**→ X such that |x|=|Px|+|x−Px| for all x∈ X^**. We show that, as long as X has dimension at least $2$, its Lipschitz-free space, denoted by ℱ(X), is not L-embedded in its bidual. Our methods have applications to the problem of when the L-embeddability of ℱ(M) passes to ℱ(A) for a metric space M and A⊆ M. This is the case when A is compact or when A is geodesically closed.
ER -