Group Actions on Twisted Sums of Banach Spaces
Group Actions on Twisted Sums of Banach Spaces
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Banach 空间扭曲和的群作用
DOI：
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发表时间：
2020
影响因子：
1.2
通讯作者：
V. Ferenczi
中科院分区：
文献类型：
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作者：
J. Castillo;V. Ferenczi
从（广义）拟线性映射的角度研究了群和半群G在Banach空间正合序列上的有界作用，利用交换子估计刻画了扭和空间上的作用，并引入了G -中心化子和G -等变映射的概念.我们将证明：当（A）G是一个顺从群且（U）目标空间在其双对偶中被一个G -等变投影补时，一致有界相容算子族在扭和空间上生成有界作用;相容拟线性映射是G -中心化子的线性扰动;在（A）和（U）下，G -中心化子是G -等变映射的有界扰动.前面的结果是最优的。给出了几个例子和反例，包括L_p（0，1），p\ne 2 L_p（0，1），p\ne 2 L_p（0，1）在Kalton-Peck空间Z_p上的作用，自由群F ∞在Hilbert空间上的某些不可幺正三角表示，扭和上的复结构的相容性，或L_p L_p -空间内插尺度上的有界作用。在倒数第二节中，我们考虑了G-Banach空间范畴，并研究了它的正合列，证明了在（A）和（U）下，G -分裂与通常的分裂是一致的。最后一节的目的是提出一些应用，表明以前的几个结果是最佳的，并建议进一步的开放路线的研究。
We study bounded actions of groups and semigroups G on exact sequences of Banach spaces from the point of view of (generalized) 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 $$L_p(0,1), p\ne 2$$ L p ( 0 , 1 ) , p ≠ 2 on the Kalton–Peck space $$Z_p$$ Z p , certain non-unitarizable triangular representations of the free group $${\mathbb {F}}_\infty $$ F ∞ on the Hilbert space, the compatibility of complex structures on twisted sums, or bounded actions on the interpolation scale of $$L_p$$ L p -spaces. In the penultimate 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. The purpose of the final section is to present some applications, showing that several previous result are optimal and to suggest further open lines of research.
Banach空间中的三表示问题
影响因子：
0.8
作者：
Kuchment, P.
通讯作者：
Kuchment, P.
Banach 空间同构的轻群和不变 LUR 重整
影响因子：
0.6
作者：
Antunes, Leandro;Ferenczi, Valentin;Grivaux, Sophie;Rosendal, Christian
通讯作者：
Rosendal, Christian