On the duality of the symmetric strong diameter $2$ property in Lipschitz spaces

TL;DR

This paper characterizes the weak* symmetric strong diameter 2 property in Lipschitz spaces through a new property called decomposable octahedrality, exploring its duality with the symmetric strong diameter 2 property.

Contribution

It introduces decomposable octahedrality as a new property of Lipschitz-free spaces and studies its duality with the symmetric strong diameter 2 property.

Findings

Decomposable octahedrality is sufficient for dual spaces to have the weak* symmetric strong diameter 2 property.

The duality between decomposable octahedrality and the symmetric strong diameter 2 property is established.

The necessity of decomposable octahedrality for the property remains an open question.

Abstract

We characterise the weak$^*$ symmetric strong diameter $2$ property in Lipschitz function spaces by a property of its predual, the Lipschitz-free space. We call this new property decomposable octahedrality and study its duality with the symmetric strong diameter $2$ property in general. For a Banach space to be decomposably octahedral it is sufficient that its dual space has the weak$^*$ symmetric strong diameter $2$ property. Whether it is also a necessary condition remains open.