可分Banach空间的对偶作为Calkin代数与泛型理想商

中文摘要 AI 辅助

对于$\mathbb{K}=\mathbb{R}$或$\mathbb{C}$以及每个可分Banach空间$V$，我们使用一个依赖于oracle的双类型有限扩张构造，其中oracle记录$V$的局部有理结构，来构造一个可分Banach空间$X_V$，使得作为Banach代数有$$ \operatorname{Cal}(X_V)=\mathcal{B}(X_V)/\mathcal{K}(X_V)\simeq \begin{pmatrix} \mathbb{K}&0\\\\ V^*&\mathbb{K} \end{pmatrix} $$。$V^*$与Jacobson根的等距同构是等距的。因此，每个具有可分预对偶的非零对偶Banach空间，在等价重赋范后，允许一个单Banach代数结构同构于某个可分Banach空间的Calkin代数。此外，存在一个可分Banach空间$X$，使得每个可分Banach空间都等距于$\mathcal{J}/\mathcal{K}(X)$，其中$\mathcal{J}$是$\mathcal{B}(X)$的闭双边理想，并且子空间-理想对应保持典范序。

英文摘要

For $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$ and every separable Banach space $V$, we use an oracle-relative two-sorted finite-extension construction, whose oracle records the local rational structure of $V$, to construct a separable Banach space $X_V$ such that $$ \operatorname{Cal}(X_V)=\mathcal{B}(X_V)/\mathcal{K}(X_V)\simeq \begin{pmatrix} \mathbb{K}&0\\ V^*&\mathbb{K} \end{pmatrix} $$ as Banach algebras. The identification of $V^*$ with the Jacobson radical is isometric. Consequently, every nonzero dual Banach space with a separable predual admits, after an equivalent renorming, a unital Banach-algebra structure isomorphic to the Calkin algebra of a separable Banach space. Moreover, there is a separable Banach space $X$ such that every separable Banach space is isometric to $\mathcal{J}/\mathcal{K}(X)$ for a closed two-sided ideal $\mathcal{J}$ of $\mathcal{B}(X)$, and the subspace--ideal correspondence preserves the canonical order.

发表机构

• School of Mathematical Sciences and LPMC, Nankai University（南开大学数学科学学院）

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