On Property (H) of c_0 and Lipschitz Problems of Gromov and Johnson

Abstract

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 $\psi:S_V\to S_{\ell_2}$ such that the restrictions $\psi|_{S_{V_j}}$, $j=1,2,\ldots$, 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 $\mu_n$ of the Lipschitz constants of all nonzero-degree maps from $S_{\ell_\infty^n}$ to $S_{\ell_2^n}$ satisfies $\lim_{n\to\infty} \mu_n/\log n=1/2$. (2) Let $\lambda_n$ 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 $\lim_{n\to\infty} \lambda_n/\log n=\lim_{n\to\infty} C_n/\log n=1/2$. 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.