Involutions on Banach Spaces and Reflexivity

spaces. E is said to be finitely representable in F if, given e> 0 and a finite dimensional subspace E0 of E, there exists a subspace F0 of F such that d(E0, F0) _< 1 + e, where d(E0, F0) = inf { T T-1 ß T is an isomorphism from E0 onto F0} denotes the Banach-Mazur distance coefficient. E is said to be super-reflexive if every Banach space which is finitely representable in E is re-flexive. Super-reflexivity has been characterized in terms of the notion of J-convexity: suppose that n> _ 1 and that e> 0; E is said to be J(n, e)-convex if, for all Xl,..., x, • in the unit ball of E, we have inf Xl q-'' ' q- Xk-- Xk+l..... Xn I-- • 7 /-- e. l<k<n--1 The "if " part of the following theorem was proved in [12] and [5], and the "only if " part was proved in [10]. TnEOaEM A. E is super-ret•exive //'and only iœE is J(n, e)-convex for some n> 1 ande> 0