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arXiv · 2509.22328

Duality of Lipschitz-free spaces over ultrametric spaces

Trond A. Abrahamsen·Vegard Lima·Andre Ostrak
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Abstract

We give a metric characterisation of when the Lipschitz-free space over a separable ultrametric space is a dual Banach space. In the case where the Lipschitz-free space has a predual, we show that this predual is M-embedded if and only if the metric space is proper. We show that for ultrametric spaces, the little Lipschitz space is always an M-ideal in the corresponding space of Lipschitz functions, and we show that this is not the case for metric spaces in general, thus answering a question posed by Werner in the negative. Finally, we show that the space of Lipschitz functions of an ultrametric space contains a strongly extreme point.

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Trond A. Abrahamsen, Vegard Lima, Andre Ostrak. 2025-09-26. Duality of Lipschitz-free spaces over ultrametric spaces. https://arxiv.org/abs/2509.22328

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