论文标题
巴拉克空间中小直径特性的稳定性结果
Stability Results Of Small Diameter Properties In Banach Spaces
论文作者
论文摘要
随后在[GGMS]中的可义能力和radon Nikodym属性的背景下，对[b3]，[br]，[ew]，[gm]启动和开发的Huskability的几何概念进行了广泛的研究。在这项工作中，我们引入了Banach Space的新几何特性，Ballable属性（$ bhp $），即，单位球具有相对较弱的任意小直径的开放子集。我们将该属性与两个相关的几何属性进行了比较，即$ bscsp $，即，单位球具有任意小直径和$ bdp $的切片组合，即，封闭的单位球具有任意小直径的切片。我们显示$ bdp $意味着$ bhp $，这又意味着$ bscsp $，而没有任何含义可以逆转。我们证明了$ w^*$ - 版本的相似结果。我们证明，所有这些属性在$ l_p $ sum in $ 1 \ leq p \ leq \ infty $下都是稳定的。这些稳定性结果导致在Banach空间理想的背景下进行讨论。我们证明，可以将$ bscsp $（分别$ bhp $，$ bdp $）从M-Ideal提起到整个空间。对于严格的理想，我们还显示出类似的结果。我们注意到，当k $ w^*$ - $ bscsp $（分别$ w^*$ - $ bhp $，$ w^*$ - $ bdp $）时，空间$ c（k，x）^*$具有$ w^*$ - $ bdp $）
The geometric notion of huskability initiated and developed in [B3], [BR] ,[EW], [GM] was subsequently extensively studied in the context of dentability and Radon Nikodym Property in [GGMS]. In this work, we introduce a new geometric property of Banach space, the Ball Huskable Property ($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 $w^*$-$BHP$, $w^*$-$BDP$).
