关于$c_0$的性质(H)以及Gromov和Johnson的球面问题
On Property (H) of c_0 and Lipschitz Problems of Gromov and Johnson
- Xiamen University（厦门大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文证明零序列Banach空间$c_0$不具有有理性质(H)，从而解决了Kasparov-Yu问题的一个关键部分，并同时解决了Gromov和Johnson提出的一个相关问题。
AI中文摘要：
设$X$为赋范空间，$S_X$为$X$的单位球面。我们称$X$具有性质(H)，如果存在两个递增的有限维子空间序列$V_j\subset X$、$H_j\subset \ell_2, j=1,2,\cdots$，使得并集$V\equiv\bigcup_j V_j$在$X$中稠密，并且存在一个一致连续映射$\psi: S_V\rightarrow S_{\ell_2}$，使得限制$\psi|_{S_{V_j}}, j=1,2,\cdots$是从$S_{V_j}$到$S_{H_j}$的同构。Kasparov-Yu问题是：所有零序列构成的Banach空间$c_0$是否具有性质(H)，或者更弱地，是否具有有理性质(H)。在本文中，我们证明$c_0$不具有有理性质(H)。同时，我们证明了以下结果，这解决了Gromov和Johnson的一个问题。
英文摘要：
Let $X$ be a normed space and $S_X$ be the unit sphere of $X$. We say that $X$ has Property (H) if there exist two increasing sequences $V_j\subset X$ and $H_j\subset \ell_2$, $j=1,2,\ldots$, of finite dimensional subspaces such that $V=\bigcup_j V_j$ is dense in $X$, and there is a uniformly continuous mapping $ψ:S_V\to S_{\ell_2}$ such that the restrictions $ψ|_{S_{V_j}}$, $j=1,2,\ldots$, are isomorphisms from $S_{V_j}$ onto $S_{H_j}$. The Kasparov-Yu problem asks whether the Banach space $c_0$ of all null sequences admits Property (H), or rational Property (H). In this paper, we prove that $c_0$ admits neither Property (H) nor rational Property (H). We also obtain the following results. (1) The infimum $μ_n$ of the Lipschitz constants of all nonzero-degree maps from $S_{\ell_\infty^n}$ to $S_{\ell_2^n}$ satisfies $\lim_{n\to\infty} μ_n/\log n=1/2$. (2) Let $λ_n$ and $C_n$ denote the infima of the Lipschitz constants of homeomorphisms from $S_{\ell_\infty^n}$ onto $S_{\ell_2^n}$ and from $B_{\ell_\infty^n}$ onto $B_{\ell_2^n}$, respectively. Then $\lim_{n\to\infty} λ_n/\log n=\lim_{n\to\infty} C_n/\log n=1/2$. The first result solves a problem of Gromov on nonzero-degree maps between finite-dimensional spheres, while the second settles Johnson's problem on the Lipschitz distortion of finite-dimensional unit balls.
