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Two Banach spaces X and Y are symmetrically complemented in each other if there exists a supplement of Y in X which is isomorphic to some supplement of X in Y. In 1996, W. T. Gowers solved the Schroeder-Bernstein (or Cantor-Bernstein) Problem for Banach spaces by constructing two non-isomorphic Banach spaces which are symmetrically complemented in each other. In this paper, we show how to modify such a symmetry in order to ensure that X is isomorphic to Y. To do this, first we introduce the notion...

Suppose that X and Y are Banach spaces that embed complementably into each other. Are X and Y necessarily isomorphic? In this generality, the answer is no, as proved by W. T. Gowers in 1996. However, if X contains a complemented copy of its square X², then X is isomorphic to Y whenever there exists p ∈ ℕ such that $X^p$ can be decomposed into a direct sum of $X^{p-1}$ and Y. Motivated by this fact, we introduce the concept of (p,q,r) widely complemented subspaces in Banach spaces, where p,q and r ∈ ℕ. Then,...

We classify, up to isomorphism, the spaces of compact operators (E,F), where E and F are the Banach spaces of all continuous functions defined on the compact spaces $2^{}\times [0,\alpha ]$, the topological products of Cantor cubes $2^{}$ and intervals of ordinal numbers [0,α].

We establish the existence of Banach spaces E and F isomorphic to complemented subspaces of each other but with $E^m\oplus Fⁿ$ isomorphic to $E^p\oplus F^q$, m, n, p, q ∈ ℕ, if and only if m = p and n = q.

In relation to some Banach spaces recently constructed by W. T. Gowers and B. Maurey, we introduce the notion of Schroeder-Bernstein index SBi(X) for every Banach space X. This index is related to complemented subspaces of X which contain some complemented copy of X. Then we establish the existence of a Banach space E which is not isomorphic to Eⁿ for every n ∈ ℕ, n ≥ 2, but has a complemented subspace isomorphic to E². In particular, SBi(E) is uncountable. The construction of E follows the approach...

Let X,Y,A and B be Banach spaces such that X is isomorphic to Y ⊕ A and Y is isomorphic to X ⊕ B. In 1996, W. T. Gowers solved the Schroeder-Bernstein problem for Banach spaces by showing that X is not necessarily isomorphic to Y. In the present paper, we give a necessary and sufficient condition on sextuples (p,q,r,s,u,v) in ℕ with p + q ≥ 2, r + s ≥ 1 and u, v ∈ ℕ* for X to be isomorphic to Y whenever these spaces satisfy the following decomposition scheme: ⎧ $X^u\sim X^p\oplus Y^q$, ⎨ ⎩ $Y^v\sim A^r\oplus B^s$. Namely, Ω = (p-u)(s-r-v)...

We provide a complete isomorphic classification of the Banach spaces of continuous functions on the compact spaces $2^{}\oplus [0,\alpha ]$, the topological sums of Cantor cubes $2^{}$, with smaller than the first sequential cardinal, and intervals of ordinal numbers [0,α]. In particular, we prove that it is relatively consistent with ZFC that the only isomorphism classes of $C\left(2^{}\oplus [0,\alpha ]\right)$ spaces with ≥ ℵ₀ and α ≥ ω₁ are the trivial ones. This result leads to some elementary questions on large cardinals.

Let X and Y be two Banach spaces, each isomorphic to a complemented subspace of the other. In 1996, W. T. Gowers solved the Schroeder-Bernstein Problem for Banach spaces by showing that X is not necessarily isomorphic to Y. In this paper, we obtain necessary and sufficient conditions on the quintuples (p,q,r,s,t) in ℕ for X to be isomorphic to Y whenever ⎧$X{X}^p\oplus Y^q$, ⎨ ⎩ $Y^t{X}^r\oplus Y^s$. Such quintuples are called Schroeder-Bernstein quintuples for Banach spaces and they yield a unification of the known decomposition...

Inspired by Pełczyński's decomposition method in Banach spaces, we introduce the notion of Schroeder-Bernstein quadruples for Banach spaces. Then we use some Banach spaces constructed by W. T. Gowers and B. Maurey in 1997 to characterize them.

We present an extension of the classical isomorphic classification of the Banach spaces C([0,α]) of all real continuous functions defined on the nondenumerable intervals of ordinals [0,α]. As an application, we establish the isomorphic classification of the Banach spaces $C\left(2^{}\times [0,\alpha ]\right)$ of all real continuous functions defined on the compact spaces $2^{}\times [0,\alpha ]$, the topological product of the Cantor cubes $2^{}$ with smaller than the first sequential cardinal, and intervals of ordinal numbers [0,α]. Consequently, it is relatively...

Let us denote by C(α) the classical Banach space C(K) when K is the interval of ordinals [1,α] endowed with the order topology. In the present paper, we give an answer to a 1960 Bessaga and Pełczyński question by providing tight bounds for the Banach-Mazur distance between C(ω) and any other C(K) space which is isomorphic to it. More precisely, we obtain lower bounds L(n,k) and upper bounds U(n,k) on d(C(ω),C(ωⁿk)) such that U(n,k) - L(n,k) < 2 for all 1 ≤ n, k < ω.

A classical result of Cembranos and Freniche states that the C(K,X) space contains a complemented copy of c₀ whenever K is an infinite compact Hausdorff space and X is an infinite-dimensional Banach space. This paper takes this result as a starting point and begins a study of conditions under which the spaces C(α), α < ω₁, are quotients of or complemented in C(K,X). In contrast to the c₀ result, we prove that if C(βℕ ×[1,ω],X) contains a complemented copy of $C\left({\omega }^{\omega }\right)$ then X contains a copy of c₀. Moreover,...

We completely determine the ${\ell }_q$ and C(K) spaces which are isomorphic to a subspace of ${\ell }_p\otimes {̂}_{\pi }C\left(\alpha \right)$, the projective tensor product of the classical ${\ell }_p$ space, 1 ≤ p < ∞, and the space C(α) of all scalar valued continuous functions defined on the interval of ordinal numbers [1,α], α < ω₁. In order to do this, we extend a result of A. Tong concerning diagonal block matrices representing operators from ${\ell }_p$ to ℓ₁, 1 ≤ p < ∞. The first main theorem is an extension of a result of E. Oja and states that the only...

In the spirit of the classical Banach-Stone theorem, we prove that if K and S are intervals of ordinals and X is a Banach space having non-trivial cotype, then the existence of an isomorphism T from C(K, X) onto C(S,X) with distortion $\left|\right|T\left|\right|\left|\right|T^{-1}\left|\right|$ strictly less than 3 implies that some finite topological sum of K is homeomorphic to some finite topological sum of S. Moreover, if Xⁿ contains no subspace isomorphic to $X^{n+1}$ for every n ∈ ℕ, then K is homeomorphic to S. In other words, we obtain a vector-valued Banach-Stone...

We prove the following quasi-dichotomy involving the Banach spaces C(α,X) of all X-valued continuous functions defined on the interval [0,α] of ordinals and endowed with the supremum norm. Suppose that X and Y are arbitrary Banach spaces of finite cotype. Then at least one of the following statements is true. (1) There exists a finite ordinal n such that either C(n,X) contains a copy of Y, or C(n,Y) contains a copy of X. (2) For any infinite countable...

In this short note we give a negative answer to a question of Argyros, Castillo, Granero, Jiménez and Moreno concerning Banach spaces which contain complemented and uncomplemented subspaces isomorphic to c.

For a locally compact Hausdorff space K and a Banach space X we denote by C₀(K,X) the space of X-valued continuous functions on K which vanish at infinity, provided with the supremum norm. Let n be a positive integer, Γ an infinite set with the discrete topology, and X a Banach space having non-trivial cotype. We first prove that if the nth derived set of K is not empty, then the Banach-Mazur distance between C₀(Γ,X) and C₀(K,X) is greater than or equal to 2n + 1. We also show that the Banach-Mazur...