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 $${\mathcal {B}}_X\subset {\mathcal {B}}$$ of subspaces of X such that each $$x\in X$$ can be uniquely written as the sum of an unconditionally convergent series $$\sum _{B\in {\mathcal {B}}_X}x_B$$ for some $$(x_B)_{B\in {\mathcal {B}}_X}\in \prod _{B\in {\mathcal {B}}_X}B$$. For every $$B\in {\mathcal {B}}_X$$ let $$\mathrm {pr}_B:X\rightarrow B$$ denote the coordinate projection. Let $$C\subset [-1,1]$$ be a closed convex set with $$C\cdot C\subset C$$. 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 $$T_\alpha =\sum _{B\in {\mathcal {F}}}\alpha (B)\cdot \mathrm {pr}_B$$ 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 $$K_C\le 1$$. Using the technique of Fraisse theory, we construct a rational $${\mathcal {B}}$$-decomposed Banach space $$\mathbb {U}_C\in \varvec{{\mathcal {B}}}_C$$ which contains an almost isometric copy of each $${\mathcal {B}}$$-decomposed Banach space $$X\in \varvec{{\mathcal {B}}}_C$$. 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 Pelczynski (and Wojtaszczyk).

中文翻译：

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泛分解 Banach 空间

令 $${\mathcal {B}}$$ 是一类有限维 Banach 空间。$${\mathcal {B}}$$-decomposed Banach 空间是一个 Banach 空间 X，它被赋予了 X 的子空间的族 $${\mathcal {B}}_X\subset {\mathcal {B}}$$使得每个 $$x\in X$$ 可以唯一地写成一个无条件收敛级数的和 $$\sum _{B\in {\mathcal {B}}_X}x_B$$ 对于某些 $$(x_B )_{B\in {\mathcal {B}}_X}\in \prod _{B\in {\mathcal {B}}_X}B$$。对于每个 $$B\in {\mathcal {B}}_X$$，让 $$\mathrm {pr}_B:X\rightarrow B$$ 表示坐标投影。令 $$C\subset [-1,1]$$ 是具有 $$C\cdot C\subset C$$ 的闭凸集。$${\mathcal {B}}$$-decomposed Banach space $$(X,{\mathcal {B}}_X)$$的C-分解常数$$K_C$$是最小的数$$K_C $$ 这样对于每个函数 $$\alpha ：{\mathcal {F}}\rightarrow C$$ 来自有限子集 $${\mathcal {F}}\subset {\mathcal {B}}_X$$ 运算符 $$T_\alpha =\sum _{B \in {\mathcal {F}}}\alpha (B)\cdot \mathrm {pr}_B$$ 有范数 $$\Vert T_\alpha \Vert \le K_C$$。我们用 $$\varvec{{\mathcal {B}}}_C$$ 表示具有 C-分解常数 $$K_C\le 1$$ 的 $${\mathcal {B}}$$-decomposed Banach 空间的类. 使用 Fraisse 理论的技术，我们构造了一个有理的 $${\mathcal {B}}$$-decomposed Banach 空间 $$\mathbb {U}_C\in \varvec{{\mathcal {B}}}_C$$其中包含每个 $${\mathcal {B}}$$ 分解的 Banach 空间 $$X\in \varvec{{\mathcal {B}}}_C$$ 的几乎等距副本。如果 $${\mathcal {B}}$$ 是所有一维（或有限维）Banach 空间的类，