扭曲非交换$L_p$空间
Twisting non-commutative $L_p$ spaces
摘要：
本文的第一个步骤的研究扩展（“扭曲和”）的非交换$L^p$ -空间被视为Banach模的基础冯诺依曼代数$\mathcal M$。我们的方法结合了Kalton对中心化扩展的描述（这些是某些映射，一般来说，既不是线性的，也不是有界的）与一般原则，由于Rochberg和Weiss说，每当一个人发现一个给定的巴拿赫空间$Y$作为一个中间空间在一个（复杂的）插值尺度，一个自动得到一个自扩展$ 0\longrightarrow Y\longrightarrow X\longrightarrow Y \longrightarrow 0. $对于半有限代数，将$L^p=L^p(\mathcal M,τ)$看作是$\mathcal M$与它的预对偶$\mathcal M_*$之间的插值空间，得到了$L^p$的某种自扩张，它是一种非交换的Kalton-Peck空间，具有自然双模结构。这些空间的一些有趣的性质。对于一般代数，包括III型代数，插值机制产生两个（而不是一个）单侧模的扩张，一个是左模，另一个是右模。是否可以找到（非平凡的）自扩张的双模在所有情况下是开放的。
英文摘要：
The paper makes the first steps into the study of extensions ("twisted sums") of noncommutative $L^p$-spaces regarded as Banach modules over the underlying von Neumann algebra $\mathcal M$. Our approach combines Kalton's description of extensions by centralizers (these are certain maps which are, in general, neither linear nor bounded) with a general principle, due to Rochberg and Weiss saying that whenever one finds a given Banach space $Y$ as an intermediate space in a (complex) interpolation scale, one automatically gets a self-extension $ 0\longrightarrow Y\longrightarrow X\longrightarrow Y \longrightarrow 0. $ For semifinite algebras, considering $L^p=L^p(\mathcal M,τ)$ as an interpolation space between $\mathcal M$ and its predual $\mathcal M_*$ one arrives at a certain self-extension of $L^p$ that is a kind of noncommutative Kalton-Peck space and carries a natural bimodule structure. Some interesting properties of these spaces are presented. For general algebras, including those of type III, the interpolation mechanism produces two (rather than one) extensions of one sided modules, one of left-modules and the other of right-modules. Whether or not one can find (nontrivial) self-extensions of bimodules in all cases is left open.
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