On Ball Dentable Property in Banach Spaces

In this work, we introduce the notion of Ball dentable property in Banach spaces. We study certain stability results for the $w^{\ast }$-Ball dentable property leading to a discussion on Ball dentability in the context of ideals of Banach spaces. We prove that the $w^{\ast }$-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 $C(K,X{)}^{\ast }$ has $w\ast$-Ball dentable property when $K$ is dispersed and $X^{\ast }$ has the $w^{\ast }$-Ball dentable property.