Examples of uniformly homeomorphic Banach spaces
Several examples of separable Banach spaces which are linearly non-isomorphic but uniformly homeomorphic are given. The main tool used is a reworking of \textit{M. Ribe}'s construction [Isr. J. Math. 48, 139--147 (1984; Zbl 0577.46015)]. It is shown that for every \(1 < p \neq 2 < \infty\) there are two uniformly homeomorphic subspaces (respectively, quotients) of \(\ell_p\) which are not linearly isomorphic; similarly, \(c_0\) has two uniformly homeomorphic subspaces which are not isomorphic, but it is not known if we can replace subspaces by quotients in the statement for \(c_0\). An example is also given of two non-isomorphic separable \(\mathcal{L}_\infty\)-spaces which are coarsely homeomorphic (i.e., have Lipschitz equivalent nets). The author also demonstrates the limitations on the preservation of asymptotic structure by uniform homeomorphisms. Let \((G_n)_{n=1}^\infty\) be a sequence of finite-dimensional normed spaces dense for the Banach-Mazur distance in all finite-dimensional normed spaces. The space \(C_p = (\sum_{n=1}^\infty G_n)_{\ell_p}\) has asymptotic moduli of convexity and smoothness satisfying \(\bar{\delta}_{C_p}(t) \approx \bar{\rho}_{C_p}(t) \approx t^p\). But \(C_p\) is shown to be uniformly homeomorphic to \[ X = \left( (\sum_{n=1}^\infty G_n)_{T_p}\right)^2 \oplus \left( (\sum_{n=1}^\infty G_n)_{T^*_{p'}}\right)^2 \] which cannot be renormed to have either \(\bar{\rho}_X(t) \leq C t^p\) or \(\bar{\delta}_X(t) \geq c t^p\), where \(0 < c,C < \infty\) are constants. Here, \(T_p\), \(1 < p < \infty\), denotes the \(p\)-convexification of the Tsirelson space. For positive results in this direction, see the paper [\textit{N. J. Kalton}, Trans. Am. Math. Soc. 365, No. 2, 1051--1079 (2013; Zbl 1275.46011)].
scientific article; zbMATH DE number 6176541
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