A universal Banach space with a $K$-unconditional basis

Abstract

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.

Universal decomposed Banach spaces

“…Let U C be the completion of the union n∈ω U n and B U C = n∈ω B U n . The proof of the following theorem literally repeats the proof of Theorem 4.4 in [1].…”

“…The following theorem shows that such space is unique up to BI C -isomorphism. It can be proved by analogy with Theorem 4.5 in [1].…”

“…The following universality property of the space U C can be proved by analogy with Theorem 5.5 in [1].…”

“…The universal Banach space from [7] was constructed using the general categorical technique of Fraïssé limits [11]. In [1,2] the authors constructed some isomorphic copies of the Pełczyński universal spaces, which are isometrically universal for the class of rational based Banach with some restrictions on the suppression or unconditional basis constants. However, as was discovered in [2], the language of based Banach spaces and base preserving morphisms does not work well for based Banach spaces with unconditional constant > 1.…”

“…In this paper we propose an alternative language of decomposable Banach spaces which is free of this drawback and allow us to establish nice almost universality properties of rational Banach spaces constructed in [1,2]. Our approach also allows to consider simultaneously the suppression and unconditional constants, which are just special cases of the C-decomposition constant K C , for C equal to {1} and {−1, 1}, respectively.…”