Lipschitz-free spaces over properly metrisable spaces and approximation properties

2023/08/27 by R. Jeffrey Smith, Smith, Richard J., Filip Talimdjioski +1 · 1 citation
Mathematics · #Advanced Topology and Set Theory #Advanced Banach Space Theory #Mathematical and Theoretical Analysis

paper · pdf · doi:10.48550/arxiv.2308.14121

Abstract

Let T be a topological space admitting a compatible proper metric, that is, a locally compact, separable and metrisable space. Let M^T be the non-empty set of all proper metrics d on T compatible with its topology, and equip M^T with the topology of uniform convergence, where the metrics are regarded as functions on T². We prove that the set A^T,1 of metrics d\inM^T for which the Lipschitz-free space F(T,d) has the metric approximation property is a dense set in M^T, and is furthermore residual in M^T when T is zero-dimensional. We also prove that if T is uncountable then the set A^T_f of metrics d\inM^T for which F(T,d) fails the approximation property is dense in M^T. Combining the last statement with a result of Dalet, we conclude that for any `properly metrisable' space T, A^T_f is either empty or dense in M^T.

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