Small Diameter Properties In Ideals of Banach Spaces

Abstract

A Banach space has the ball huskable property ($BHP$) if the closed unit ball has weakly open sets of arbitrarily small diameter. We can analogously define $w^*$-$BHP$ in the dual space. In this short note, we study these properties in the context of ideals in Banach spaces. The notion of an ideal, was introduced by Godefroy, Kalton and Saphar. We show that if a Banach space $X$ has $BHP,$ then any $M$-ideal of $X$ also has $BHP.$ We further show that if $Y$ is an $M$-ideal of $X,$ then $Y^*$ has $w^*$-$BHP$ implies $X^*$ has $w^{*}$-$BHP.$ We use this result to prove that for a compact Hausdorff space $K$ which has an isolated point, $X$ has $BHP$ whenever $C(K,X)$ has $BHP$ and $X^*$ has $w^{*}$-$BHP$ implies $C(K,X)^*$ has $w^{*}$-$BHP.$ We also prove that $w^*$-$BHP$ can be lifted from $Y^*$ to $X^*$ provided $Y$ is a strict ideal of $X$. Lastly, we show that if $Y$ is an almost isometric ideal of $X,$ then $BHP$ can be lifted from $Y$ to $X.$ We obtain similar results for ball dentable property ($BDP$) and ball small combination of slices Property ($BSCSP$) as well.