论文标题
本地几乎是方形的Banach格子
Locally almost square Banach lattices
论文作者
论文摘要
如果在其单位球体中的每一个$ y $中，Banach空间几乎都是正方形的，那么在其单位球体中存在一个序列$（x_n）$，以至于$ \ lim \ | y \ y \ pm x_n \ | = 1 $。如果我们要求序列$（x_n）$弱null，那么Banach空间几乎是弱的。众所周知，这两个属性是不同的，因此我们旨在研究局部几乎几乎平方，是否通过用网代替序列来暗示后一种属性的较弱版本。为了实现这一结果，我们将自己限制在Banach晶格中，并通过要求该序列在晶格的正锥中引入局部几乎平方。作为这种表征的应用，我们证明了局部几乎方形的这种正变体意味着单位球中每个相对较弱的开放式均具有直径2，也就是说，Banach空间具有所谓的直径两个属性。这特别使我们还可以生成享受直径两个特性的Banach空间的新示例。
A Banach space is locally almost square if, for every $y$ in its unit sphere, there exists a sequence $(x_n)$ in its unit sphere such that $\lim\|y\pm x_n\|=1$. A Banach space is weakly almost square if, in addition, we require the sequence $(x_n)$ to be weakly null. It is known that these two properties are distinct, so we aim to investigate if local almost squareness implies a weaker version of the latter property by replacing the sequence with a net. In order to achieve this result, we restrict ourselves to Banach lattices and introduce a strengthening of local almost squareness by requiring that the sequence is in the positive cone of the lattice. As an application of such characterization, we prove that this positive variant of local almost squareness implies that every relatively weakly open set in the unit ball has diameter 2, that is, the Banach space has the so called diameter two property. This in particular allows us also to generate new examples of Banach spaces enjoying the diameter two property.
