We construct a nonexpansive linear operator on the Gurarii space that "captures" all nonexpansive linear operators between separable Banach spaces. Some additional properties involving its restrictions to finite-dimensional subspaces describe this operator uniquely up to an isometry.Comment: 17 pages, extended subsection 2.2. Final versio

For a constant K ≥ 1 let B K be the class of pairs (X, (e n ) n∈ω ) consisting of a Banach space X and an unconditional Schauder basis (e n ) n∈ω for X, having the unconditional basic constant K u ≤ K. Such pairs are called K-based Banach spaces. A based Banach space X is rational if the unit ball of any finite-dimensional subspace spanned by finitely many basic vectors is a polyhedron whose vertices have rational coordinates in the Schauder basis of X.Using the technique of Fraïssé theory, we construct a rational K-based Banach space U K , (e n ) n∈ω which is RI K -universal in the sense that each basis preserving isometry f : Λ → U K defined on a based subspace Λ of a finitedimensional rational K-based Banach space A extends to a basis preserving isometryf : A → U K of the based Banach space A. We also prove that the K-based Banach space U K is almost FI 1 -universal in the sense that any base preserving ε-isometry f : Λ → U K defined on a based subspace Λ of a finitedimensional 1-based Banach space A extends to a base preserving ε-isometrȳ f : A → U K of the based Banach space A. On the other hand, we show that no almost FI K -universal based Banach space exists for K > 1.The Banach space U K is isomorphic to the complementably universal Banach space for the class of Banach spaces with an unconditional Schauder basis, constructed by Pe lczyński in 1969.

Abstract. We show that for each p ∈ (0, 1] there exists a separable p-Banach space