DOI: 10.24193/fpt-ro.2024.2.11

Authors: Dawid Kapitan and Łukasz Piasecki

Abstract: We present a unified approach to describe a possibly wide class of separable Banach spaces which are extremal with respect to the minimal displacement of k-Lipschitz self-maps of the closed unit ball. The prominent member of this class, which plays a central role in our considerations, is the Banach space c₀ of real sequences converging to 0, provided with the maximum norm. Indeed, we show that if a separable Banach space X contains an isomorphic (resp. isometric) copy of c₀, then X as well as all subspaces of X of finite codimension are extremal (resp. strictly extremal). Our result encompasses and significantly extends a collection of all known examples of separable Banach spaces which are extremal (resp. strictly extremal).

Key Words and Phrases: Minimal displacement, Lipschitz map, space c₀.

2010 Mathematics Subject Classification: 47H09, 47H10, 46B20, 46B25.

Published on-line: June 15th, 2024.

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