On Ball Dentable Property in Banach Spaces

TL;DR

This paper introduces the concept of Ball dentable property in Banach spaces, explores its stability, and demonstrates how it can be transferred from ideals to the entire space, with specific results for spaces like $C(K,X)^*$.

Contribution

It defines the $w^*$-Ball dentable property in Banach spaces and establishes stability and lifting results for this property in the context of ideals and $M$-ideals.

Findings

The $w^*$-Ball dentable property can be lifted from an $M$-ideal to the whole Banach space.

Stability results for the $w^*$-Ball dentable property are established.

The space $C(K,X)^*$ has $w^*$-Ball dentable property when $K$ is dispersed and $X^*$ has the property.

Abstract

In this work, we introduce the notion of Ball dentable property in Banach spaces. We study certain stability results for the $w^*$-Ball dentable property leading to a discussion on Ball dentability in the context of ideals of Banach spaces. We prove that the $w^*$-Ball-dentable property can be lifted from an $M$-ideal to the whole Banach Space. We also prove similar results for strict ideals of a Banach space. We note that the space $C(K,X)^*$ has $w*$-Ball dentable property when $K$ is dispersed and $X^\ast$ has the $w^*$-Ball dentable property.