Polynomials and injections of Banach spaces into superreflexive spaces

Abstract

We construct a nonseparable analogue of the Tsirelson space, namely a space that contains no subspace of the same density character admitting an injection into some superreflexive space. We relate the existence of injections to the behavior of polynomials on subspaces. The original construction of Tsirelson space T* [11] yields a separable reflexive space which does not contain any superreflexive subspace, in particular lp, 1 1, whenever X admits a polynomial that separates the origin from the unit sphere of X. In nonseparable case, we often replace the question of existence of a subspace isomorphic to lp (F) of a space X, by the question of existence of injections from subspaces of X into some Ip (F). The purpose of this note is to show certain nonseparable analogues to the results of Tsirelson and Deville. The results of this kind are also interesting in the context of smooth optimization on nonseparable spaces. For example, it is an open problem whether the space c o (F) admits a C2-smooth function that attains its minimum at exactly one point, if F is uncountable. Before passing to our main results, we present a related separable statement. This result was also recently proved independently by R. Gonzalo and J. Jaramillo [7] by a different technique and a slightly different version of this was also recently shown by J. Farmer [6], also by a different proof.