Extension of smooth subspaces in Lindenstrauss spaces
It follows from [FW] that in the Gurariy space G every finite dimensional smooth subspace is contained in a bigger smooth subspace. We show that this property does not characterise the Gurariy space among Lindenstrauss spaces and we provide various examples to show that C(K) spaces do not have this property. The starting point of this paper is the following observation which easily follows from [FW, Theorem 1.2] (see the proof of the Observation below). Observation. Let L ⊂ G be a finite-dimensional smooth subspace of the Gurariy spaceG. Then there is a smooth subspaceM ⊂ G withM ⊃ L andM 6= L. Reacall that a point x ∈ SX of the unit sphere SX of a Banach space X is called a smooth point of SX if there is the unique linear functional f ∈ SX∗ such that f(x) = 1. A subspace X ⊂ Y of a Banach space Y is called smooth if any point x ∈ SX is a smooth point of SX . A separable Banach space G is called a Gurariy space if given ε > 0 and an isometric embedding T : L → G of a finitedimensional normed spaceL intoG, for any finite-dimensional spaceM ⊃ L there is an extension T̃ : M → G with ||T̃ ||||T̃−1|| ≤ 1 + ε. We say that the pair L ⊂ M of normed spaces has the unique Hahn-Banach extension property (UHB in short) if for any functional f ∈ L∗ there is a unique extension f̂ ∈ M∗ with ||f̂ || = ||f ||. For instance if M is smooth and L ⊂ M,dimL <∞, then this pair has UHB. In the proof of Observation we use the following theorem which is the main results of the paper [FW] Theorem 0.1. Let X be a separable Banach space. TFAE (a) X = G. (b) Let L ⊂ M, dimL < ∞, codimML = 1, be a pair with property UHB and let T : L→ X be an isometric embedding of L into X. Then there is an isometric extension T̃ : M → X of T . Proof of the Observation. Put M1 = L⊕R and define in M1 the norm as follows ||(x, t)|| = (||x|| + t), x ∈ L, t ∈ R. Since L is smooth it easily follows that M1 is also smooth, and hence the pair L ⊂ M1 has UHB. By (b), Theorem 0.1 (T = Id) there is an isometric extension T̃ : M1 → G of T . Put M = T̃ (M1) and finish the proof. First named author was partially supported by the Israel Science Foundation, Grant 209/09 and by European Grant Spade 2. Second named author was partially supported by the Center for Advanced Studies in Mathematics of the Ben-Gurion University of the Negev, the “HPC Infrastructure for Grand Challenges of Science and Engineering Project, co-financed by the European Regional Development Fund under the Innovative Economy Operational Programme” and Polish NCN grant DEC2011/03/B/ST1/04902.