Stability Results Of Small Diameter Properties In Banach Spaces

TL;DR

This paper introduces the Ball Huskable Property ($BHP$) for Banach spaces, compares it with related properties, and studies its stability and transferability under various conditions, enriching the geometric understanding of Banach space structures.

Contribution

The paper defines the new $BHP$ property, compares it with $BSCSP$ and $BDP$, and establishes stability results and ideal-related transfer properties for these geometric properties.

Findings

$BDP$ implies $BHP$, which implies $BSCSP$; no reverse implications.

All properties are stable under $l_p$ sums for $1 \\leq p \\leq \\infty$.

Properties can be lifted from M-Ideals and strict ideals to the whole Banach space.

Abstract

The geometric notion of huskability initiated and developed in [B3], [BR] ,[EW], [GM] was subsequently extensively studied in the context of dentability and Radon Nikodym Property in [GGMS]. In this work, we introduce a new geometric property of Banach space, the Ball Huskable Property ($BHP$), namely, the unit ball has relatively weakly open subsets of arbitrarily small diameter. We compare this property to two related geometric properties, $BSCSP$ namely, the unit ball has convex combination of slices of arbitrarily small diameter and $BDP$ namely, the closed unit ball has slices of arbitrarily small diameter. We show $BDP$ implies $BHP$ which in turn implies $BSCSP$ and none of the implications can be reversed. We prove similar results for the $w^*$-versions. We prove that all these properties are stable under $l_p$ sum for $1\leq p \leq \infty$. These stability results lead to a discussion in the context of ideals of Banach spaces. We prove that $BSCSP$ (respectively $BHP$, $BDP$) can be lifted from an M-Ideal to the whole space. We also show similar results for strict ideals. We note that the space $C(K,X)^*$ has $w^*$-$BSCSP$ (respectively $w^*$-$BHP$, $w^*$-$BDP$) when K is dispersed and $X^*$has the $w^*$-$BSCSP$ (respectivley $w^*$-$BHP$, $w^*$-$BDP$).