Towards the fixed point property for superreflexive spaces (Q2777719)

This article deals with a class of Banach spaces \(X\) in which the following \((S_m)\)-property holds: every metrically convex set \(A\subset S_X\) \((S_X\) is the unit sphere of \(X)\) with diameter not greater than 1 can be (weakly) separated from zero by a functional \(F\in X^*\). In the article it is proved that all separable and all strictly convex Banach spaces have this property; furthermore, it is shown that all uniformly noncreasy spaces, a large class of superreflexive spaces introduced by S. Prus, also have the property \((S_m)\); in particular, all uniformly convex and all uniformly smooth Banach spaces have the property \((S_m)\). The main result is the following: if \(X\) is a superreflexive space and \((X)_U\) (the quotient space of \(\ell_\infty (X)\) by the subspace \(\{(x_n)\in \ell_\infty (X):\lim_U \|x_n\|=0\}\) of \(\ell_\infty (X))\) has the property \((S_m)\) for some free ultrafilter \(U\) on \(\mathbb{N}\), then \(X\) has the fixed point property. In particular, all uniformly noncreasy Banach spaces have the fixed point property.