On the geometry of Banach spaces of the form $\mathrm{Lip}_0(C(K))$

TL;DR

This paper explores the geometric structure of Banach spaces formed by Lipschitz functions on compact spaces of continuous functions, providing conditions for their classification and isomorphism to known spaces.

Contribution

It establishes sufficient conditions under which these Lipschitz Banach spaces are isomorphic to spaces over uncountable index sets, advancing the classification theory.

Findings

Identifies conditions for isomorphism to $ ext{Lip}_0(c_0( extGamma))$

Provides a framework for classifying $ ext{Lip}_0(C(K))$ spaces

Enhances understanding of the geometry of Lipschitz Banach spaces

Abstract

We investigate the problem of classifying the Banach spaces $\mathrm{Lip}_0(C(K))$ for Hausdorff compacta $K$. In particular, sufficient conditions are established for a space $\mathrm{Lip}_0(C(K))$ to be isomorphic to $\mathrm{Lip}_0(c_0(\varGamma))$ for some uncountable set $\varGamma$.