超交替达戈维特性质、无条件基与 SCD 几何
The super Alternative Daugavet property, unconditional bases and SCD geometry
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中文总结 AI 辅助
回答关于具 1 - 无条件基且满足超交替达戈维特性质的无限维巴拿赫空间存在性问题，处理相关弱拓扑结构等问题，证明具特定基的巴拿赫空间子集性质及存在单位球非切片可数确定的巴拿赫空间。
中文摘要 AI 辅助
我们对 Langemets、Lõo、Martín、Perreau 和 Rueda Zoca 提出的关于存在具有 1 - 无条件基且满足超交替达戈维特性质的无限维巴拿赫空间的问题 6.4 给出否定回答。我们还处理了 Lõo 和 Perreau 最近提出的关于具有无条件基的巴拿赫空间中弱拓扑结构和切片可数确定现象的两个问题。具体而言，我们证明了具有收缩或有界完备绍德尔基的巴拿赫空间的每个有界凸子集都有一个可数弱 $\pi$ - 基，对问题 5.1 给出部分肯定回答。最后，我们证明对于每个 $k > 1$，存在一个具有 $k$ - 无条件基的巴拿赫空间，其单位球不是切片可数确定的，从而对问题 5.4 给出肯定回答。
英文摘要
We answer negatively the 1-unconditional part of Question 6.4, by Langemets, Lõo, Martín, Perreau and Rueda Zoca, and, more generally, prove that no infinite-dimensional Banach space with a $K$-unconditional basis, for $1\leq K<3/2$, can satisfy the super Alternative Daugavet property. We also address two recent questions posed by Lõo and Perreau concerning weak topological structures and slicely countably determined phenomena in Banach spaces with unconditional bases. More precisely, we prove that the weak unit ball of every Banach space with a 1-unconditional basis admits a countable $π$-base, thereby giving a positive answer to Question 5.1. We further show that every bounded convex subset of a Banach space with a Schauder basis that is shrinking or boundedly complete admits a countable $π$-base for its relative weak topology. We complement this result with two permanence principles: one for shrinking Schauder decompositions whose finite partial sums have countable weak $π$-bases, and another for unconditional sums over boundedly complete Banach sequence spaces. Finally, we prove that, for every $k>1$, there exists a Banach space with a $k$-unconditional basis whose unit ball is not slicely countably determined, thereby giving a positive answer to Question 5.4.
