Well located Banach spaces - A combined approach to L_1- preduals and Hilbert space

Abstract

In the description of L_1- preduals on the one hand and of Hilbert spaces, on the other, one encounters the surprising appearance of similar classes of operators in terms of their extendibility properties in the setting of these two classes of spaces. It also turns out that the extension properties are closely tied to how the two classes of spaces are positioned inside the ambient spaces. It is well known that in the case of all subspaces being positioned inside a larger space Z as a complemented subspace leads to Z being (isomorphically) a Hilbert space whereas in the case of a Banach space X which is complemented inside all its superspaces, it turns out that X is an injective Banach space. In this paper, we shall be concerned mostly with such properties as demanding that X sit inside the ambient space as a locally complemented subspace (and sometimes as a Lipchitz retract). It turns out that the class of Banach spaces X determined by the indicated properties are precisely the class of L_1- preduals in the former case and of Hilbert spaces, in the latter. As an important consequence of this approach, we are led to several new characterisations of Hilbert spaces in terms of the extendibility of bounded linear maps coming from well-known operator (nonlinear operator) ideals.