论文标题
小组动作在扭曲的巴拉克空间
Group actions on twisted sums of Banach spaces
论文作者
论文摘要
我们从准线图的角度研究了Banach空间的精确序列的组和Semigroups $ g $的有界动作，通过换向器估计来表征扭曲的总空间上的动作，并介绍$ G $ cum-centralizer和$ g $ equivariant地图的相关概念。我们将表明，当（a）$ g $是一个可正常的群体时，（u）目标空间以$ g $ equivariant的投影在其双重元组中进行补充，然后统一限制的操作员兼容兼容的兼容家族在扭曲的总空间上产生有界的动作；该兼容的准线性图是$ g $ centralizers的线性扰动；在（a）和（u）下，$ g $ - 中心化器的界限为$ g $ equivariant地图。先前的结果是最佳的。提供了几个示例和反例，其中涉及$ l_p（0,1）的等轴测组，在卡尔顿 - 佩克空间上的p \ neq 2 $ $ z_p $，某些不可用的三角形表示，自由组的$ f_ \ f_ \ f_ \ f_ \ f_ \ f_ \ f_ \ f_ \ f_ \ f_ \ f_ \ f_ \ f_ \ f_ \ f_ \ $ L_P $ - 空格。在最后一部分中，我们考虑了$ g $ -BANACH空间的类别，并研究其确切序列，这表明在（a）和（u）下，$ g $ splitting和通常的分裂一致。
We study bounded actions of groups and semigroups $G$ on exact sequences of Banach spaces from the point of view of quasilinear maps, characterize the actions on the twisted sum space by commutator estimates and introduce the associated notions of $G$-centralizer and $G$-equivariant map. We will show that when (A) $G$ is an amenable group and (U) the target space is complemented in its bidual by a $G$-equivariant projection, then uniformly bounded compatible families of operators generate bounded actions on the twisted sum space; that compatible quasilinear maps are linear perturbations of $G$-centralizers; and that, under (A) and (U), $G$-centralizers are bounded perturbations of $G$-equivariant maps. The previous results are optimal. Several examples and counterexamples are presented involving the action of the isometry group of $L_p(0,1), p\neq 2$ on the Kalton-Peck space $Z_p$, certain non-unitarizable triangular representations of the free group $F_\infty$ on the Hilbert space, the compatibility of complex structures on twisted sums, or bounded actions on the interpolation scale of $L_p$-spaces. In the last section we consider the category of $G$-Banach spaces and study its exact sequences, showing that, under (A) and (U), $G$-splitting and usual splitting coincide.
