Equivalent norms with the property $(\beta)$ of Rolewicz

TL;DR

This paper extends characterizations of Banach spaces with the property $(eta)$ of Rolewicz to non-separable spaces, involving Szlenk index and smoothness/convexity properties, thus broadening the understanding of their geometry.

Contribution

It generalizes existing characterizations of Banach spaces with property $(eta)$ to the non-separable setting using Szlenk index and asymptotic properties.

Findings

Extended characterizations to non-separable Banach spaces.

Connected property $(eta)$ with Szlenk index and asymptotic smoothness.

Facilitated application of recent nonlinear geometry results.

Abstract

We extend to the non separable setting many characterizations of the Banach spaces admitting an equivalent norm with the property $(\beta)$ of Rolewicz. These characterizations involve in particular the Szlenk index and asymptotically uniformly smooth or convex norms. This allows to extend easily to the non separable case some recent results from the non linear geometry of Banach spaces.