Some topological and metric properties of the space of $\ell _1$-predual hyperplanes in $c$

Abstract

We study topological and metric properties of the space (H, d), where H is the set of all 1-predual hyperplanes in the space c of convergent sequences, d denotes the Banach-Mazur distance and we identify hyperplanes that are almost isometric. The space c and its subspace c0 of sequences converging to 0 are the simplest examples of elements in H. First we prove that (H, d) is homeomorphic to (K, d 1 ) with K = {x ∈ 1 : x ≤ 1 and x(i) ≥ x(i + 1) ≥ 0 for all i ∈ N}. We provide optimal lower bounds for the distortions T T −1 of isomorphic embeddings T from an arbitrarily chosen element of H into an infinite-dimensional L1-predual X such that (ext BX * ) ⊂ rBX * for some r ∈ [0, 1). Finally, we apply the above results to construct a homotopy contracting H to c0 along the shortest paths in H and we calculate their lengths. For instance, we show that the shortest path in H joining c and c0 has length ln 4.

A note on the Banach–Mazur distances between c0 and other ℓ1 -preduals

“…Now observe that, by Theorem 2.3, there exist a subspace Z of c 0 and an isomorphism K : c 0 / ker πT −1 → Z such that K K −1 < 1 + ε. Hence, applying Theorem 4.1 in [9], we obtain…”