Small Diameter Properties In Ideals of Banach Spaces

TL;DR

This paper investigates the ball huskable property ($BHP$) and related properties in Banach spaces and their ideals, establishing how these properties transfer between spaces, duals, and ideals, with applications to spaces of continuous functions.

Contribution

It introduces new results on the transfer of $BHP$, $BDP$, and $BSCSP$ properties between Banach spaces, their duals, and ideals, including conditions involving $M$-ideals and almost isometric ideals.

Findings

$BHP$ transfers from a Banach space to its $M$-ideals.

$w^*$-$BHP$ in an $M$-ideal's dual implies $w^*$-$BHP$ in the dual of the ambient space.

$BHP$ and related properties can be lifted from ideals to the whole space under certain conditions.

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.