Universal decomposed Banach spaces

AbstractLet$${\mathcal {B}}$$Bbe a class of finite-dimensional Banach spaces. A$${\mathcal {B}}$$B-decomposed Banach spaceis a Banach spaceXendowed with a family$${\mathcal {B}}_X\subset {\mathcal {B}}$$BX⊂Bof subspaces ofXsuch that each$$x\in X$$x∈Xcan be uniquely written as the sum of an unconditionally convergent series$$\sum _{B\in {\mathcal {B}}_X}x_B$$∑B∈BXxBfor 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∈BXlet$$\mathrm {pr}_B:X\rightarrow B$$prB:X→Bdenote 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. TheC-decomposition constant$$K_C$$KCof a$${\mathcal {B}}$$B-decomposed Banach space$$(X,{\mathcal {B}}_X)$$(X,BX)is the smallest number$$K_C$$KCsuch that for every function$$\alpha :{\mathcal {F}}\rightarrow C$$α:F→Cfrom a finite subset$${\mathcal {F}}\subset {\mathcal {B}}_X$$F⊂BXthe operator$$T_\alpha =\sum _{B\in {\mathcal {F}}}\alpha (B)\cdot \mathrm {pr}_B$$Tα=∑B∈Fα(B)·prBhas norm$$\Vert T_\alpha \Vert \le K_C$$‖Tα‖≤KC. By$$\varvec{{\mathcal {B}}}_C$$BCwe denote the class of$${\mathcal {B}}$$B-decomposed Banach spaces withC-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∈BCwhich contains an almost isometric copy of each$${\mathcal {B}}$$B-decomposed Banach space$$X\in \varvec{{\mathcal {B}}}_C$$X∈BC. If$${\mathcal {B}}$$Bis the class of all 1-dimensional (resp. finite-dimensional) Banach spaces, then$$\mathbb {U}_{C}$$UCis isomorphic to the complementably universal Banach space for the class of Banach spaces with an unconditional (f.d.) basis, constructed by Pełczyński (and Wojtaszczyk).