Universal decomposed Banach spaces

Abstract

AbstractLet $${\mathcal {B}}$$B be a class of finite-dimensional Banach spaces. A $${\mathcal {B}}$$B-decomposed Banach space is a Banach space X endowed with a family $${\mathcal {B}}_X\subset {\mathcal {B}}$$BX⊂B of subspaces of X such that each $$x\in X$$x∈X can be uniquely written as the sum of an unconditionally convergent series $$\sum _{B\in {\mathcal {B}}_X}x_B$$∑B∈BXxB for some $$(x_B)_{B\in {\mathcal {B}}_X}\in \prod _{B\in {\mathcal {B}}_X}B$$(xB)B∈BX∈∏B∈BXB. For every $$B\in {\mathcal {B}}_X$$B∈BX let $$\mathrm {pr}_B:X\rightarrow B$$prB:X→B denote the coordinate projection. Let $$C\subset [-1,1]$$C⊂[-1,1] be a closed convex set with $$C\cdot C\subset C$$C·C⊂C. The C-decomposition constant $$K_C$$KC of a $${\mathcal {B}}$$B-decomposed Banach space $$(X,{\mathcal {B}}_X)$$(X,BX) is the smallest number $$K_C$$KC such that for every function $$\alpha :{\mathcal {F}}\rightarrow C$$α:F→C from a finite subset $${\mathcal {F}}\subset {\mathcal {B}}_X$$F⊂BX the operator $$T_\alpha =\sum _{B\in {\mathcal {F}}}\alpha (B)\cdot \mathrm {pr}_B$$Tα=∑B∈Fα(B)·prB has norm $$\Vert T_\alpha \Vert \le K_C$$‖Tα‖≤KC. By $$\varvec{{\mathcal {B}}}_C$$BC we denote the class of $${\mathcal {B}}$$B-decomposed Banach spaces with C-decomposition constant $$K_C\le 1$$KC≤1. Using the technique of Fraïssé theory, we construct a rational $${\mathcal {B}}$$B-decomposed Banach space $$\mathbb {U}_C\in \varvec{{\mathcal {B}}}_C$$UC∈BC which contains an almost isometric copy of each $${\mathcal {B}}$$B-decomposed Banach space $$X\in \varvec{{\mathcal {B}}}_C$$X∈BC. If $${\mathcal {B}}$$B is the class of all 1-dimensional (resp. finite-dimensional) Banach spaces, then $$\mathbb {U}_{C}$$UC 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).