关于Lindenstrauss-Pejeczynski空间
On Lindenstrauss-Pełczyński spaces
摘要：
在这项工作中，我们将关注的一些稳定性方面的经典问题的扩展$C(K)$值的运营商。本文引入Lindenstrauss-Petersczynski型巴拿赫空间类$\mathscr{LP}$，使得从$c_0$的子空间到该空间中的每个算子都可以扩张到$c_0$ .我们证明了所有的$\mathscr{LP}$ -空间都是$\mathcal L_\infty$型的，但不是相反的.此外，$\mathcal L_\infty$ -空间将被刻画为这样的空间$E$，使得来自$l_1$的$w^*(l_1,c_0)$ -闭子空间的$E$ -值算子扩展到$l_1$ . $C(K)$的可补子空间和可分内射空间是$\mathscr{LP}$ -空间的子类，并且证明了前者不包含后者。证明了不含$l_1$的$\mathcal L_\infty$ -空间是$\mathscr{LP}$ -空间的同分空间，不含$c_0$的$\mathcal L_\infty$ -空间、$\mathscr{LP}$ -空间被可分内射空间的同分空间以及$\mathscr{LP}$ -空间的扭和是$\mathscr{LP}$ -空间.
英文摘要：
In this work we shall be concerned with some stability aspects of the classical problem of extension of $C(K)$-valued operators. We introduce the class $\mathscr{LP}$ of Banach spaces of Lindenstrauss-Pełczyński type as those such that every operator from a subspace of $c_0$ into them can be extended to $c_0$. We show that all $\mathscr{LP}$-spaces are of type $\mathcal L_\infty$ but not the converse. Moreover, $\mathcal L_\infty$-spaces will be characterized as those spaces $E$ such that $E$-valued operators from $w^*(l_1,c_0)$-closed subspaces of $l_1$ extend to $l_1$. Complemented subspaces of $C(K)$ and separably injective spaces are subclasses of $\mathscr{LP}$-spaces and we show that the former does not contain the latter. It is established that $\mathcal L_\infty$-spaces not containing $l_1$ are quotients of $\mathscr{LP}$-spaces, while $\mathcal L_\infty$-spaces not containing $c_0$, quotients of an $\mathscr{LP}$-space by a separably injective space and twisted sums of $\mathscr{LP}$-spaces are $\mathscr{LP}$-spaces.
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