关于紧算子空间的某些扩张性质
On certain extension properties for the space of compact operators
摘要：
设$Z$是固定可分算子空间，$X\subset Y$是一般可分算子空间，$T:X\to Z$是完全有界映射. $Z$被称为具有完全可分扩张性质（CSEP），如果每个这样的映射允许$Y$的完全有界扩张;混合可分扩张性质（MSEP），如果每个这样的$T$允许$Y$的有界扩张。最后，如果$Z$是局部自反的，并且$T$允许$Y$的完全有界扩张，如果$Y$是局部自反的，并且$T$是完全满射同构，则称$Z$具有完全可分补性质（CSCP）。设${\bf K}$表示可分Hilbert空间上的紧算子空间，${\bf K}_0$表示${\Cal M}_n$的$c_0$和（"小紧算子"空间）.利用第二作者以前的结论${\bf K}_0$具有CSCP性质，证明了${\bf K}$具有CSCP性质.本文对E. Kirchberg），${\bf K}_0$（因此${\bf K}$）未通过CSEP。${\bf K}$是否具有MSEP仍然是一个悬而未决的问题，它被证明是等价的，${\bf K}_0$是否有这个属性。引入一个新的Banach空间概念--可拓局部自反性（ELR）来研究这一问题。进一步的补充和开放的问题进行了讨论。
英文摘要：
Let $Z$ be a fixed separable operator space, $X\subset Y$ general separable operator spaces, and $T:X\to Z$ a completely bounded map. $Z$ is said to have the Complete Separable Extension Property (CSEP) if every such map admits a completely bounded extension to $Y$; the Mixed Separable Extension Property (MSEP) if every such $T$ admits a bounded extension to $Y$. Finally, $Z$ is said to have the Complete Separable Complementation Property (CSCP) if $Z$ is locally reflexive and $T$ admits a completely bounded extension to $Y$ provided $Y$ is locally reflexive and $T$ is a complete surjective isomorphism. Let ${\bf K}$ denote the space of compact operators on separable Hilbert space and ${\bf K}_0$ the $c_0$ sum of ${\Cal M}_n$'s (the space of ``small compact operators''). It is proved that ${\bf K}$ has the CSCP, using the second author's previous result that ${\bf K}_0$ has this property. A new proof is given for the result (due to E. Kirchberg) that ${\bf K}_0$ (and hence ${\bf K}$) fails the CSEP. It remains an open question if ${\bf K}$ has the MSEP; it is proved this is equivalent to whether ${\bf K}_0$ has this property. A new Banach space concept, Extendable Local Reflexivity (ELR), is introduced to study this problem. Further complements and open problems are discussed.
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