Into isometries that preserve finite dimensional structure of the range

In this paper we study linear into isometries of non-reflexive spaces (embeddings) that preserve finite dimensional structure of the range space. We consider this for various aspects of the finite dimensional structure, covering the recent notion of an almost isometric ideals introduced by Abrahamsen et.al., the well studied notions of a M -ideal and that of an ideal. We show that if a separable non-reflexive Banach space X, in all embeddings into its bidual X∗∗, is an almost isometric ideal and if X∗ is isometric to L1(μ), for some positive measure μ, then X is the Gurariy space. For a fixed infinite compact Hausdorff space K, if every embedding of a separable space X into C(K) is an almost isometric ideal and X∗ is a non-separable space, then again X is the Gurariy space. We show that if a separable Banach space contains an isometric copy of c0 and if it is a M -ideal in its bidual in the canonical embedding, then there is another embedding of the space in its bidual, in which it is not a M -ideal.