Representation in $C(K)$ by Lipschitz functions

TL;DR

This paper investigates how Banach spaces can be represented within spaces of Lipschitz functions on compact spaces, highlighting restrictions imposed by Lipschitz regularity on the structure of both the space and the compact set.

Contribution

It provides a systematic analysis of Lipschitz function representations of Banach spaces in $C(K)$, improving upon previous results and clarifying the impact of Lipschitz regularity.

Findings

Lipschitz representations impose restrictions on the structure of Banach spaces.

Regularity conditions affect the existence of isometric embeddings.

The paper extends previous results on Lipschitz function representations.

Abstract

The isometric universality of the spaces $C(K)$ for $K$ a non scattered Hausdorff compact does not take into account the ``quality'' of the representation. Indeed, the existence of an isometric copy of a separable Banach space $X$ into $C(K)$ made of regular enough functions, say Lipschitz with respect to a lower semicontinuous metric defined on $K$, imposes severe restrictions to both $X$ and $K$. In this paper, we present a systematic treatment of the representation of Banach spaces into $C(K)$ by Lipschitz functions improving previous results of the author.