Nonlinear classification of Banach spaces

Abstract

We study the geometric classi&#64257;cation of Banach spaces via Lipschitz, uniformly continuous, and coarse mappings. We prove that a Banach space which is uniformly homeomorphic to a linear quotient of lp is itself a linear quotient of lp when p<2. We show that a Banach space which is Lipschitz universal for all separable metric spaces cannot be asymptotically uniformly convex. Next we consider coarse embedding maps as de&#64257;ned by Gromov, and show that lp cannot coarsely embed into a Hilbert space when p> 2. We then build upon the method of this proof to show that a quasi-Banach space coarsely embeds into a Hilbert space if and only if it is isomorphic to a subspace of L0(??) for some probability space (&#937;,B,??).