Universal decomposed Banach spaces

Let B\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {B}}$$\end{document} be a class of finite-dimensional Banach spaces. A B\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {B}}$$\end{document}-decomposed Banach space is a Banach space X endowed with a family BX⊂B\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {B}}_X\subset {\mathcal {B}}$$\end{document} of subspaces of X such that each x∈X\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x\in X$$\end{document} can be uniquely written as the sum of an unconditionally convergent series ∑B∈BXxB\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sum _{B\in {\mathcal {B}}_X}x_B$$\end{document} for some (xB)B∈BX∈∏B∈BXB\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(x_B)_{B\in {\mathcal {B}}_X}\in \prod _{B\in {\mathcal {B}}_X}B$$\end{document}. For every B∈BX\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B\in {\mathcal {B}}_X$$\end{document} let prB:X→B\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm {pr}_B:X\rightarrow B$$\end{document} denote the coordinate projection. Let C⊂[-1,1]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C\subset [-1,1]$$\end{document} be a closed convex set with C·C⊂C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C\cdot C\subset C$$\end{document}. The C-decomposition constant KC\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K_C$$\end{document} of a B\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {B}}$$\end{document}-decomposed Banach space (X,BX)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(X,{\mathcal {B}}_X)$$\end{document} is the smallest number KC\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K_C$$\end{document} such that for every function α:F→C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha :{\mathcal {F}}\rightarrow C$$\end{document} from a finite subset F⊂BX\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {F}}\subset {\mathcal {B}}_X$$\end{document} the operator Tα=∑B∈Fα(B)·prB\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_\alpha =\sum _{B\in {\mathcal {F}}}\alpha (B)\cdot \mathrm {pr}_B$$\end{document} has norm ‖Tα‖≤KC\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert T_\alpha \Vert \le K_C$$\end{document}. By BC\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{{\mathcal {B}}}_C$$\end{document} we denote the class of B\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {B}}$$\end{document}-decomposed Banach spaces with C-decomposition constant KC≤1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K_C\le 1$$\end{document}. Using the technique of Fraïssé theory, we construct a rational B\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {B}}$$\end{document}-decomposed Banach space UC∈BC\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {U}_C\in \varvec{{\mathcal {B}}}_C$$\end{document} which contains an almost isometric copy of each B\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {B}}$$\end{document}-decomposed Banach space X∈BC\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X\in \varvec{{\mathcal {B}}}_C$$\end{document}. If B\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {B}}$$\end{document} is the class of all 1-dimensional (resp. finite-dimensional) Banach spaces, then UC\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {U}_{C}$$\end{document} 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).


Introduction
A Banach space X is (complementably) universal for a given class of Banach spaces if X belongs to this class and every space from the class is isomorphic to a (complemented) subspace of X .
In 1969 Pełczyński [15] constructed a complementably universal Banach space for the class of Banach spaces with a Schauder basis. In 1971 Kadec [10] constructed a complementably universal Banach space for the class of spaces with the bounded approximation property (BAP). In the same year Pełczyński [13] showed that every Banach space with BAP is isomorphic to a complemented subspace of a Banach space with a basis. Pełczyński and Wojtaszczyk [16] constructed in 1971 a (complementably) universal Banach space for the class of spaces with a finite-dimensional decomposition. A simple construction of the Pełczyński universal space was suggested by Schechtman [17]. Applying Pełczyński's decomposition argument [14], one immediately concludes that all three universal spaces are isomorphic. It is worth mentioning a negative result of Johnson and Szankowski [9] saying that no separable Banach space can be complementably universal for the class of all separable Banach spaces.
Lindenstrauss and Tzafriri showed in [12, 1.g.5] that a Banach space with an unconditional finite-dimensional decomposition is isomorphic to a subspace of a space with an unconditional basis.
In [7] the second author constructed an isometric version of the Kadec-Pełczyński-Wojtaszczyk space. 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. Also the notion of a decomposed Banach space generalizes the notions of a Banach space with unconditional base and of a Banach space with uncounditional f.d. base, which was considered in [16].
For a set X by [X ] <ω we denote the family of all finite subsets of X . The sets of rational and real numbers are denoted by Q and R, respectively. All Banach spaces considered in this paper are separable and are over the field R of real numbers.
Let (x α ) α∈A be a sequence of points of a Banach space X , indexed by elements of a set A. We say that a series α∈A x α converges (more precisely, unconditionally converges) to a point x ∈ X if for every ε > 0 there exists a finite subset F ⊂ A such that x − α∈B x α < ε for any finite subset B ⊂ A that contains F.

Decomposed Banach spaces
Let B be a class of finite-dimensional Banach spaces such that for any linear bijective isometry f : X → Y between Banach spaces the inclusion X ∈ B implies Y ∈ B. Banach spaces from the class B will be called basic Banach spaces. For example, for B we can take the class of all 1-dimensional Banach spaces (or all finite-dimensional Banach spaces).
By a B-decomposed Banach space we understand a pair (X , B X ) consisting of a Banach space X and a family B X ⊂ B of finite-dimensional subspaces of X such that every x ∈ X can be uniquely written as the sum of an unconditionally convergent series B∈B X x B consisting of points x B ∈ B for B ∈ B X . The convergence x = B∈B X x B means that for every ε > 0 there exists a finite subfamily F ⊂ B X such that x − B∈E x B < ε for any finite subfamily E ⊂ B X that contains F. In this case the family B X is called the B-decomposition of X . If the class B is clear from the context, then we shall speak about decomposed Banach spaces instead of B-decomposed Banach spaces.
Fix a class B of finite-dimensional Banach spaces. For every B ∈ B X let pr B : X → B denote the coordinate projector. It assigns to each point x ∈ X the B-th term x B of the unique sequence (x E ) E∈B X with x = E∈B X x E . The Open Mapping Principle implies that the coordinate projector pr B : X → B ⊂ X is bounded (see [4, p. 33] for more details).
For every finite subfamily F ⊂ B X , consider the projector For every x ∈ X the unconditional convergence of the series B∈B X pr B (x) implies the set { pr F (x) : F ∈ [B X ] <ω } is bounded and then by the Banach-Steinhaus Uniform Boundedness Principle, the real number is well-defined. This number will be called the suppression constant of the decomposed Banach space (X , B X ).

The decomposition constants of a decomposed Banach space
Let (X , B X ) be a decomposed Banach space with the suppression constant K s . For every function α : F → R defined on a finite subfamily F ⊂ B X , consider the operator For a bounded subset C ⊂ R let The number 6. K tC = |t| · K C for any real number t.
Proof 1. To prove that K C ≥ K s · sup c∈C |c|, assume that this inequality does not hold. Then there exists c ∈ C and a finite subfamily F ⊂ B X such that K C < |c| · pr F . Now consider the constant function α : F → {c} ⊂ C and observe that T α = c · pr F and hence which is a desired contradiction completing the proof. 2. To prove that K C ≤ K s · c∈C |c|, observe that for every function α : F → C defined on a finite subset F ⊂ B X we have 3. The inequality K C ≤ K C∪{0} follows from the inclusion C ⊂ C ∪ {0}; the inequality K C∪{0} ≤ K C follows from the observation that for every function α ∈ (C ∪ {0}) <B X the operator T α is equal to the operator T β for the function β = α α −1 (C) ∈ C <B X . 4. The inequality K C ≤ K conv(C∪{0}) follows from the inclusion C ⊂ conv(C ∪ {0}).
To prove the reverse inequality, we need to prove that T α ≤ K C for any function α : F → conv(C ∪ {0}) defined on a finite set F ⊂ B X . Let C 0 = C ∪{0} and for every n ∈ ω let C n+1 := { 1 2 x + 1 2 y : x, y ∈ C n }. It is easy to see that the union C ω = n∈ω C n is a dense subset of conv(C ∪ {0}). By induction we shall show that for every n ∈ ω and every function α : F → C n the operator T α has norm T α ≤ K C . For n = 0 this follows from the equality K C 0 = K C∪{0} = K C , which was proved in the preceding statement.
Assume that for some n ∈ ω we proved that for every α : F → C n the operator T α has norm T α ≤ K C . Given any function α : F → C n+1 , find two functions β, γ : Now we see that for any function α : F → C ω the operator T α has norm T α ≤ K C . The density of C ω in conv(C ∪ {0}) and the continuity of the projectors pr B for B ∈ F imply that for every function α : By the statements (4) and (2), we obtain 6. For any t ∈ R, the definition of the constants K C and K tC ensure that

Rational decomposed Banach spaces
For a Banach space X by · X we denote the norm of X and by the closed unit ball of X . A finite-dimensional decomposed Banach space (X , B X ) is rational if its unit ball is a convex polyhedron spanned by finitely many points that have rational coordinates in some basis {e 1 , . . . , e n } ⊂ B∈B X B of X . It is easy to see that any decomposed subspace of a rational finite-dimensional decomposed Banach space is rational.

Categories
Let K be a category. For two objects A, B of the category K, by K(A, B) we denote the set of all K-morphisms from A to B. A subcategory of K is a category L such that each object of L is an object of K and each morphism of L is a morphism of K. Morphisms and isomorphisms of a category K will be called K-morphisms and K-isomorphisms, respectively.
A subcategory L of a category K is full if each K-morphism between objects of the category L is an L-morphism.
A category L is cofinal in K if for every object A of K there exists an object B of L such that the set K(A, B) is nonempty. A category K has the amalgamation property if for every objects A, B, C of K and for every morphisms In this paper we shall work in the category B, whose objects are B-decomposed To shorten notation, we shall often denote decomposed Banach spaces (X , B X ) by X .
A morphism T : X → Y of the category B is called an isometry (or else an isometry morphism) if T (x) Y = x X for any x ∈ X . By BI we denote the category whose objects are decomposed Banach spaces and morphisms are isometry morphisms of decomposed Banach spaces. The category BI is a subcategory of the category B.
By fB (resp. fBI) we denote the full subcategory of B (resp. BI), whose objects are finite-dimensional decomposed Banach spaces, and by rB (resp. rBI) the full subcategory of fB (resp. fBI) whose objects are rational finite-dimensional decomposed Banach spaces.

B-universal decomposed Banach spaces
Applying the decomposition method of Pełczyński [15], we can prove the following uniqueness result.
Theorem 1 Any two B-universal decomposed Banach spaces are B-isomorphic.

The categories B C and BI C
Given a closed convex set C ⊂ [−1, 1], let B C (resp. BI C ) be the full subcategory of B (resp. BI) whose objects are decomposed Banach spaces with C-decomposition Also consider the full subcategory fB C = fB ∩ B C of the category B and the full subcagories fBI C = fBI ∩ BI C and rBI C = rBI ∩ BI C of the category BI.
From now on we assume that C ⊂ [−1, 1] is a fixed closed convex set such that where C · C = {x y : x, y ∈ C}.

Amalgamation
In this section we prove that the categories fBI C and rBI C have the amalgamation property. Proof We shall prove this lemma in the special case when the isometries i, j are identity inclusions; the general case is analogous but has more complicated notation. Our We define linear operators i : Let us show i and j are isometries. Indeed, for every On the other hand, for every z ∈ Z Similarly, we can show that j is an isometry. We shall identify X and Y with their images i (X ) and j (Y ) in W . In this case we can consider the union B W := B X ∪ B Y and can show that B W is a decomposition of the (finite-dimensional) Banach space W .
Let us show that the decomposed Banach space (W , B W ) has C-decomposition constant K C ≤ 1. Given any function α : B W → C, we should prove the upper bound for every w ∈ W . Since 0 is a non-isolated point of the convex set C, we lose no generality assuming that for every B ∈ B W the number α B := α(B) is not equal to zero.
Taking into account that the decomposed Banach spaces (X , B X ) and (Y , B Y ) have C-unconditional constant ≤ 1, we obtain: If the finite-dimensional based Banach spaces X and Y are rational, then so is their sum X ⊕ Y and so is the quotient space W of X ⊕ Y .

rBI C -universal decomposed Banach spaces
Definition 2 A decomposed Banach space X is called rBI C -universal if -each finite-dimensional decomposed subspace of X is an object of the category rBI C , and -for any object A of the category rBI C , any BI-morphism f : Λ → X defined on a decomposed subspace Λ of A can be extended to an BI-morphismf : A → X .
We recall that rBI C denotes the full subcategory of BI C whose objects are rational finite-dimensional decomposed Banach spaces with C-decomposition constant K C ≤ 1. Obviously, up to BI-isomorphisms the category rBI C contains countably many objects. By Lemma 2, the category rBI C has the amalgamation property. We now use the concepts from [11] for constructing a "generic" sequence in rBI C . A sequence (X n ) n∈ω of objects of the category rBI C is called a chain if each decomposed Banach space X n is a subspace of the decomposed Banach space X n+1 .

Definition 3
A chain (U n ) n∈ω of objects of the category rBI C is Fraïssé if for any n ∈ ω and BI-morphism f : U n → Y ∈ rBI C there exist m > n and an BI- Definition 3 implies that the Fraïssé sequence {U n } n∈ω is cofinal in the category rBI C in the sense that each object A of the category rBI C admits a BI-morphism A → U n for some n ∈ ω. In this case the category rBI C is countably cofinal.
The name "Fraïssé sequence", as in [11], is motivated by the model-theoretic theory of Fraïssé limits developed by Roland Fraïssé [6]. One of the results in [11] is that every countably cofinal category with amalgamation has a Fraïssé sequence. Applying this general result to our category rBI C we get: Theorem 2 [11] The category rBI C has a Fraïssé sequence.
From now on, we fix a Fraïssé sequence (U n ) n∈ω in rBI C . 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].
is an rBI C -universal decomposed Banach space.
To shorten notation, the rBI C -universal decomposed Banach space (U C , B U C ) will be denoted by U C . 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].

Theorem 4
Any two rBI C -universal decomposed Banach spaces X , Y are BIisomorphic, which means that there exists a linear bijective isometry X → Y preserving the decompositions B X and B Y .

Corollary 1 Any rBI C -universal decomposed Banach space X is BI-isomorphic
to the rBI C -universal decomposed Banach space U C .
The following universality property of the space U C can be proved by analogy with Theorem 5.5 in [1].

Almost fBI C -universality
By analogy with an rBI C -universal decomposed Banach space, one can try to introduce an fBI C -universal decomposed Banach space. However such notion is vacuous as each decomposed Banach space has only countably many finite-dimensional decomposed subspaces whereas in general the category fBI C contains continuum many pairwise non BI-isomorphic 2-dimensional decomposed Banach spaces. A "right" definition is that of an almost fBI C -universal decomposed Banach space, introduced with the help of ε-isometries.
For a positive real number ε, a linear operator f : X → Y between Banach spaces X and Y is called an ε-isometry if for every x ∈ X \{0}. This definition implies that each ε-isometry is an injective linear operator.

Definition 4 A decomposed
Banach space X called almost fBI C -universal if for any ε > 0 and finite-dimensional decomposed Banach space A ∈ fBI C , any ε-isometry B-morphism f : Λ → X defined on a decomposed subspace Λ of A can be extended to an ε-isometry B-morphismf : A → X .

Theorem 6 Any rBI C -universal decomposed Banach space X is almost fBI Cuniversal.
Proof We shall use the fact, that the norm of any finite-dimensional decomposed Banach space can be approximated by a rational norm.
To prove that X is almost fBI C -universal, take any ε > 0, any finite-dimensional decomposed Banach space A ∈ fBI C and an ε-isometry B-morphism f : Λ → X defined on a decomposed subspace Λ of A. For every function α : By · A and · Λ we denote the norms of the Banach spaces A and Λ. The morphism f determines a new norm · Λ on Λ, defined by a Λ = f (a) X for a ∈ Λ. Taking into account that the decomposed Banach space X is rational with C-unconditional constant K C ≤ 1, we conclude that · Λ is a rational norm on Λ such that T α (a) Λ ≤ a Λ for every a ∈ Λ and every function α : B Λ → C.
Since f is an ε-isometry, (1 + ε) −1 < a Λ < (1 + ε) for every a ∈ Λ with a Λ = 1. By the compactness of the unit sphere in Λ, there exists a positive δ < ε such that (1 + δ) −1 < a Λ < (1 + δ) for every a ∈ Λ with a Λ = 1. This inequality implies 1 and B Λ = {a ∈ Λ : a Λ ≤ 1} are the closed unit balls of Λ in the norms · Λ and · Λ . Choose positive real numbers δ , ε such that δ < δ < ε < ε. Next, Since the norm · Λ is rational, there exists a basis {e 1 , . . . , e n } ⊂ ∪B A of the Banach space A such that the ball B Λ is a convex polyhedron whose vertices have rational coordinates in the basis e 1 , . . . , e n . Such a polyhedron in A will be called rational.
Let B A = {a ∈ A : x A ≤ 1} be the closed unit ball of the Banach space A. Choose a rational polyhedron P in A such that P = −P and Next consider the convex hull B A := conv(P ) of the set Repeating the proof of Lemma 1(4), one can show that B A coincides with the convex hull of the set which implies that B A is a rational polyhedron in the decomposed Banach space A.
Taking into account that [c, c ] ⊂ 1+δ 1+δ C, P ⊂ 1 1+δ B A , and A is a decomposed Banach space with C-decomposition constant ≤ 1, we conclude that Then The convex symmetric set B A := conv(P ) determines a rational norm · A on A whose unit ball coincides with B A . By A we denoted the decomposed Banach space A endowed with the norm · A . We claim that the decomposition B A = B A of the Banach space A has C-unconditional constant K C ≤ 1. Indeed, for any function α : which means that the operator T α : A → A has norm T α ≤ 1 and hence the decomposed Banach space A has C-unconditional constant K C ≤ 1. It remains to check that a A = a Λ for each a ∈ Λ, which is equivalent to the equality holding for all a ∈ A\{0}. Let Λ and A be the decomposed Banach spaces Λ and A endowed with the new rational norms · Λ and · A , respectively. It is clear that Λ ⊂ A . The definition of the norm · Λ ensures that f : Λ → X is a BI-morphism. Using the rBI C -universality of X , extend the BI-morphism f : Λ → X to a BI-morphismf : A → X . The inequalities (2) ensure thatf : A → X is an ε-isometry B-morphism from A, extending the ε-isometry f . This completes the proof of the almost fBI C -universality of X . Theorem 7 Let X and Y be almost fBI C -universal decomposed Banach spaces and ε > 0. Each ε-isometry B-morphism f : X 0 → Y defined on a finite-dimensional decomposed subspace X 0 of the decomposed Banach space X ∈ B C can be extended to an ε-isometry B-isomorphismf : X → Y .
Theorem 7 can be proved by analogy with Theorem 5.3 in [1].

Corollary 3 Any almost fBI C -universal decomposed Banach space is B-isomorphic
to the rBI C -universal decomposed Banach space U C . Theorem 8 Let U be an almost fBI C -universal decomposed Banach space. For any ε > 0 and any decomposed Banach space X ∈ B C , there exists an ε-isometry Bmorphism f : X → U .
Theorem 8 can be proved by analogy with Theorem 5.5 in [1]. Applying Theorems 6 and 8 to the rBI C -universal space U C we obtain its B C -universality.

Corollary 4
For any ε > 0 and any decomposed Banach space X ∈ B C , there exists an ε-isometry B-morphism f : X → U C .

Corollary 5 Each almost fBI C -universal decomposed Banach space U is B-universal and hence is B-isomorphic to the rBI
Proof Given a B-decomposed Banach space X , we need to prove that X is Bisomorphic to a subspace of the decomposed space U . Denote by X 1 the decomposed Banach space X endowed with the equivalent norm Using the inclusion C ·C ⊂ C, it is easy to check that X 1 has C-decompostion constant ≤ 1 and hence X 1 ∈ B C . By Theorem 8, for ε = 1 2 there exists an ε-isometry Bmorphism f : X 1 → U . Then f is a B-isomorphism between X and the decomposed (and hence complemented) subspace f (X ) = f (X 1 ) of the decomposed Banach space U .
Therefore, the decomposed Banach space U is B-universal. By Theorem 1, U is B-isomorphic to any other B-universal decomposed Banach space, in particular, to U [0,1] .

Some special cases
In this section we survey some implications of the results obtained in the preceding sections for some special choices of the class B of basic Banach spaces and the convex set C ⊂ [−1, 1].

B-Decomposed Banach spaces with bounds on suppression or unconditional constants
Let us recall that for a decomposed Banach space X its suppression constant K s is equal to K

Universal Banach spaces with unconditional bases
If B is the class of all 1-dimensional Banach spaces, then B-decomposed Banach spaces can be identified with based Banach spaces, considered in [1,2]. We recall that a based Banach space is a pair (X , B X ) consisting of a Banach space X and a subset B X ⊂ X such that every x ∈ X can be uniquely written as the sum of an unconditionally convergent series b∈B X x b · b for some sequence of real numbers (x b ) b∈B X . The set B X is called the unconditional basis of X . Each unconditional basis B X induces a 1-dimensional decomposition B X := {R · b} b∈B X ⊂ B of the Banach space X . Conversely, for any decomposition B X ⊂ B of a Banach space X , in each 1-dimensional subspace B ∈ B X we can choose an element e B ∈ B of norm e B = 1 and obtain an unconditional basis B X := {e B } B∈B X for X . The suppression and unconditional constants of the obtained base B X coincides with the suppression and decomposition constants of the decomposition B X .
A based Banach space (X , B X ) is called a subspace of a based Banach space It is clear that each base-preserving operator is decomposition-preserving. In [1] (resp. [2]) for every real number K ≥ 1 we have constructed a rational based Banach space (U K , B U K ) with suppression (resp. unconditional) constant ≤ K such that for any rational based Banach space (A, B A ) with suppression (resp. unconditional) constant ≤ K , any base-preserving linear isometry f : Λ → U K defined on a finite-dimensional based subspace (Λ, B Λ ) of (A, B A ) can be extended to a base-preserving linear isometry f : A → U K . The based space (U K , B U K ) will be called the universal rational based Banach space with suppression (resp. unconditional) constant K . In [1] (resp. [2]) it was shown that the universal space U K is isomorphic to the universal Banach space U with unconditional basis, constructed by Pełczyński [15].
By [1,3] (resp. [2]), for K = 1 and any ε > 0 the universal based Banach space (U K , B U K ) has the following ε-universality property: for any based Banach space (A, B A ) with suppression (resp. unconditional) constant ≤ K , any basepreserving linear ε-isometry f : Λ → U K defined on a finite-dimensional based subspace (Λ, B Λ ) of (A, B A ) can be extended to a base-preserving linear εisometry f : A → U K . By [2,3], the latter ε-extension property of U K does not hold for K > 1. However, Theorem 6 implies the following weaker ε-universality property of (U K , B K ), holding for every K > 1.

Corollary 10
Let K ≥ 1 and (U K , B U K ) be a rational universal based Banach space with suppression (resp. unconditional) constant K . For every ε > 0 and any based Banach space (A, B A ) with suppression (resp. unconditional) constant ≤ K , any decomposition-preserving linear ε-isometry f : Λ → U K defined on a finite-dimensional based subspace (Λ, B Λ ) of (A, B A ) can be extended to a decomposition-preserving linear ε-isometry f : A → U K .
This corollary follows from Theorem 6 and the observation that the universal rational based Banach sppace (U K , B U K ) endowed with the 1-dimensional decomposition generated by its basis B U K is a rational rBI C -universal B-decomposed Banach space for C = 0, 1 K resp. − 1 K , 1 K .

Conclusions
In this paper we consider the notion of a decomposed Banach space, which is more flexible comparing to a more restrictive notion of a based Banach space, exploited in [1,2]. In particular, it allowed us to introduce a true notion of an almost universal space and establish an almost universality of the rBI C -universal based Banach spaces, constructed in [1,2]. Moreover, using the C-unconditional constants K C allowed us to generalize the results of the papers [1][2][3] related to Banach spaces with some restrictions on suppression or unconditional of unconditional bases.