论文标题
纯原子$ l_1 $ preduals的单位球中弱拓扑的表征
A characterization of the weak topology in the unit ball of purely atomic $L_1$ preduals
论文作者
论文摘要
我们研究的Banach空间具有弱稳定的单位球，即Banach空间，其单位球中相对较弱的开放子集的每个凸组合再次是其单位球中相对较弱的开放子集。事实证明，$ l_1 $ $稳定的单位球的级别与纯原子的$ l_1 $预期相一致，对于某些$ quamγ$而言，这是$ \ ell_1（γ）$的预期，以这种方式获得了完全纯粹的$ l_1 $ $ l_1 $的完全几何表征。结果，我们证明了其他作者先前研究的不同属性的$ l_1 $的等价性，该属性在较弱的稳定性方面。另外，每当$ k $是hausdorff且零散的本地紧凑空间时，我们就会获得$ c_0（k，x）$的单位球的稳定性弱，并且$ x $具有稳定且弱稳定的单位球，这使$ C_0（k，x）的单位球的稳定性较弱（k，x）$ x $ x $ x $ x $ x $ x $ x $ x $ x $ x $ $ k $ af of the and af of yat af yat af yat af af af af yat af af af yat。最后，我们证明，稳定单位球弱的Banach空间满足了直径两种属性的非常强大的新版本。
We study Banach spaces with a weak stable unit ball, that is Banach spaces where every convex combination of relatively weakly open subsets in its unit ball is again a relatively weakly open subset in its unit ball. It is proved that the class of $L_1$ preduals with a weak stable unit ball agree with those $L_1$ preduals which are purely atomic, that is preduals of $\ell_1(Γ)$ for some set $Γ$, getting in this way a complete geometrical characterization of purely atomic preduals of $L_1$, which answers a setting problem. As a consequence, we prove the equivalence for $L_1$ preduals of different properties previously studied by other authors, in terms of slices around weak stability. Also we get the weak stability of the unit ball of $C_0(K,X)$ whenever $K$ is a Hausdorff and scattered locally compact space and $X$ has a norm stable and weak stable unit ball, which gives the weak stability of the unit ball in $C_0(K,X)$ for finite-dimensional $X$ with a stable unit ball and $K$ as above. Finally we prove that Banach spaces with a weak stable unit ball satisfy a very strong new version of diameter two property.
