New applications of extremely regular function spaces

TL;DR

This paper investigates the geometric properties of extremely regular subspaces of $C_0(L)$ spaces, revealing strong diameter 2 properties, the presence of $ ext{ extcyr{c}}_0$ copies, and the Daugavet property under certain conditions.

Contribution

It establishes new geometric and structural properties of extremely regular subspaces of $C_0(L)$, including diameter 2, $ ext{ extcyr{c}}_0$ embeddings, and the Daugavet property.

Findings

Extremely regular subspaces have strong diameter 2 properties.

They contain $ ext{ extcyr{c}}_0$ copies for every $ ext{ extepsilon}$ in (0,1).

If $L$ has no isolated points, these subspaces have the Daugavet property.

Abstract

Let $L$ be an infinite locally compact Hausdorff topological space. We show that extremely regular subspaces of $C_0(L)$ have very strong diameter $2$ properties and, for every real number $\varepsilon$ with $0<\varepsilon<1$, contain an $\varepsilon$-isometric copy of $c_0$. If $L$ does not contain isolated points they even have the Daugavet property, and thus contain an asymptotically isometric copy of $\ell_1$.