New applications of extremely regular function spaces

Trond A. Abrahamsen, Olav Nygaard and Märt Põldvere

Vol. 301 (2019), No. 2, 385–394

DOI: 10.2140/pjm.2019.301.385

Abstract

Let $L$ be an infinite locally compact Hausdorff topological space. We show that extremely regular subspaces of $C_0\left(L\right)$ have very strong diameter $2$ properties and, for every real number $\epsilon$ with $0<\epsilon <1$, contain an $\epsilon$-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 ${\ell }_1$.

Keywords

extremely regular function space, strong diameter 2 property, almost square space, octahedrality, Daugavet property

Mathematical Subject Classification 2010

Primary: 46B20, 46B22, 46B04

Milestones

Received: 2 November 2017

Revised: 25 September 2018

Accepted: 9 December 2018

Published: 24 October 2019