On the duality between Diameter Two Properties and Ideals in Banach Spaces

TL;DR

This paper investigates the relationship between diameter two properties and ideals in Banach spaces, showing that duals of M ideals inherit these properties but not vice versa, with applications to C(K) spaces.

Contribution

It establishes that the dual of an M ideal inherits diameter two properties and provides counterexamples, advancing understanding of these properties in Banach space theory.

Findings

Duals of M ideals inherit diameter two properties.

Counterexamples show the converse does not hold.

Applications to C(K) spaces and their duals.

Abstract

We prove that the dual of an M ideal of a Banach space inherits all the versions of $w^*$ diameter two properties. We give a counter example to show that the converse is not true. We use these results to explore these properties in $C(K)$ spaces and its duals.