The coarse geometry of Tsirelson’s space and applications
The coarse geometry of Tsirelson’s space and applications
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DOI：
10.1090/jams/899
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发表时间：
2017-05
影响因子：
3.9
通讯作者：
F. Baudier;G. Lancien;T. Schlumprecht
中科院分区：
文献类型：
--
作者：
F. Baudier;G. Lancien;T. Schlumprecht
本文的主要结果是关于粗嵌入到Tsirelson原空间T^*中的Banach空间的扩张模型结构的刚性结果。每个可粗嵌入到$T^*$中的Banach空间一定是自反的，并且它的所有扩张模型都必须同构于$c_0$。我们的僵化结果带来了几个重要的后果。我们得到了Tsirelson的一个有影响力的定理的粗略版本：$T^*$对于[1，\inty)$中的$p，不包含$c_0$或$\ell_p$。我们证明了不存在粗略嵌入到每个无限维Banach空间中的无限维Banach空间。特别地，我们驳斥了可分无限维Hilbert空间粗嵌入到每个无限维Banach空间的猜想。刚性结果源于取值于$T^*$的无限Hamming图上的Lipschitz映射的一个新的浓度不等式，以及无限Hamming图在允许不同构于$c0$的扩展模型的Banach空间中的可嵌入性.此外，还得到了有限维的纯度量刻画。
The main result of this article is a rigidity result pertaining to the spreading model structure for Banach spaces coarsely embeddable into Tsirelson's original space $T^*$. Every Banach space that is coarsely embeddable into $T^*$ must be reflexive and all its spreading models must be isomorphic to $c_0$. Several important consequences follow from our rigidity result. We obtain a coarse version of an influential theorem of Tsirelson: $T^*$ does not coarsely contain $c_0$ nor $\ell_p$ for $p\in[1,\infty)$. We show that there is no infinite dimensional Banach space that coarsely embeds into every infinite dimensional Banach space. In particular, we disprove the conjecture that the separable infinite dimensional Hilbert space coarsely embeds into every infinite dimensional Banach space. The rigidity result follows from a new concentration inequality for Lipschitz maps on the infinite Hamming graphs and taking values in $T^*$, and from the embeddability of the infinite Hamming graphs into Banach spaces that admit spreading models not isomorphic to $c_0$. Also, a purely metric characterization of finite dimensionality is obtained.