Universal decomposed Banach spaces

Abstract

Let {mathcal {B}} be a class of finite-dimensional Banach spaces. A {mathcal {B}}-decomposed Banach space is a Banach space X endowed with a family {mathcal {B}}_Xsubset {mathcal {B}} of subspaces of X such that each xin X can be uniquely written as the sum of an unconditionally convergent series sum _{Bin {mathcal {B}}_X}x_B for some (x_B)_{Bin {mathcal {B}}_X}in prod _{Bin {mathcal {B}}_X}B. For every Bin {mathcal {B}}_X let mathrm {pr}_B:Xrightarrow B denote the coordinate projection. Let Csubset [-1,1] be a closed convex set with Ccdot Csubset C. The C-decomposition constant K_C of a {mathcal {B}}-decomposed Banach space (X,{mathcal {B}}_X) is the smallest number K_C such that for every function alpha :{mathcal {F}}rightarrow C from a finite subset {mathcal {F}}subset {mathcal {B}}_X the operator T_alpha =sum _{Bin {mathcal {F}}}alpha (B)cdot mathrm {pr}_B has norm Vert T_alpha Vert le K_C. By varvec{{mathcal {B}}}_C we denote the class of {mathcal {B}}-decomposed Banach spaces with C-decomposition constant K_Cle 1. Using the technique of Fraïssé theory, we construct a rational {mathcal {B}}-decomposed Banach space mathbb {U}_Cin varvec{{mathcal {B}}}_C which contains an almost isometric copy of each {mathcal {B}}-decomposed Banach space Xin varvec{{mathcal {B}}}_C. If {mathcal {B}} is the class of all 1-dimensional (resp. finite-dimensional) Banach spaces, then mathbb {U}_{C} is isomorphic to the complementably universal Banach space for the class of Banach spaces with an unconditional (f.d.) basis, constructed by Pełczyński (and Wojtaszczyk).