Open Access
Uniqueness of unconditional basis of ℓ2⊕T(2)
(American Mathematical Society, 2022) Albiac Alesanco, Fernando José; Ansorena, José L.; Estadística, Informática y Matemáticas; Estatistika, Informatika eta Matematika
We provide a new extension of Pitt’s theorem for compact operators between quasi-Banach lattices which permits to describe unconditional bases of finite direct sums of Banach spaces X1 · · · Xn as direct sums of unconditional bases of their summands. The general splitting principle we obtain yields, in particular, that if each Xi has a unique unconditional basis (up to equivalence and permutation), then X1 · · · Xn has a unique unconditional basis too. Among the novel applications of our techniques to the structure of Banach and quasi-Banach spaces we have that the space ℓ2⊕T(2) has a unique unconditional basis.
Open Access
Structure of the Lipschitz free p-spaces Fp(Zd) and Fp(Rd) for 0 < p ≤ 1
(Springer, 2021) Albiac Alesanco, Fernando José; Ansorena, José L.; Cúth, Marek; Doucha, Michal; Estatistika, Informatika eta Matematika; Institute for Advanced Materials and Mathematics - INAMAT2; Estadística, Informática y Matemáticas
Our aim in this article is to contribute to the theory of Lipschitz free p-spaces for 0 < p ≤ 1 over the Euclidean spaces Rd and Zd. To that end, on one hand we show that Fp(Rd) admits a Schauder basis for every p ∈ 2 (0, 1], thus generalizing the corresponding result for the case p = 1 by H_ajek and Perneck_a [20, Theorem 3.1] and answering in the positive a question that was raised in [3]. Explicit formulas for the bases of both Fp(Rd) and its isomorphic space Fp([0, 1]d) are given. On the other hand we show that the well-known fact that F(Z) is isomorphic to l1 does not extend to the case when p < 1, that is, Fp(Z) is not isomorphic to lp when 0 < p < 1.
Open Access
Uniqueness of unconditional basis of infinite direct sums of quasi-Banach spaces
(Kluwer Academic Publishers, 2022) Albiac Alesanco, Fernando José; Ansorena, José L.; Estadística, Informática y Matemáticas; Estatistika, Informatika eta Matematika
This paper is devoted to providing a unifying approach to the study of the uniqueness of unconditional bases, up to equivalence and permutation, of infinite direct sums of quasi-Banach spaces. Our new approach to this type of problem permits to show that a wide class of vector-valued sequence spaces have a unique unconditional basis up to a permutation. In particular, solving a problem from Albiac and Leránoz (J Math Anal Appl 374(2):394-401, 2011. https://doi.org/10.1016/j.jmaa.2010.09.048) we show that if X is quasi-Banach space with a strongly absolute unconditional basis then the infinite direct sum -1(X) has a unique unconditional basis up to a permutation, even without knowing whether X has a unique unconditional basis or not. Applications to the uniqueness of unconditional structure of infinite direct sums of non-locally convex Orlicz and Lorentz sequence spaces, among other classical spaces, are also obtained as a by-product of our work.
Open Access
Embeddability of ℓp and bases in Lipschitz free p-spaces for 0 < p ≤ 1
(Elsevier, 2020) Albiac Alesanco, Fernando José; Ansorena, José L.; Cúth, Marek; Doucha, Michal; Estatistika, Informatika eta Matematika; Institute for Advanced Materials and Mathematics - INAMAT2; Estadística, Informática y Matemáticas
Our goal in this paper is to continue the study initiated by the authors in of the geometry of the Lipschitz free p-spaces over quasimetric spaces for 0 < p ≤ 1, denoted Fp(M). Here we develop new techniques to show that, by analogy with the case p = 1, the space p embeds isomorphically in Fp(M) for 0 < p < 1. Going further we see that despite the fact that, unlike the case p = 1, this embedding need not be complemented in general, complementability of p in a Lipschitz free p-space can still be attained by imposing certain natural restrictions to M. As a by-product of our discussion on bases in Fp([0, 1]), we obtain examples of p-Banach spaces for p < 1 that are not based on a trivial modification of Banach spaces, which possess a basis but fail to have an unconditional basis.
Open Access
Lipschitz free spaces isomorphic to their infinite sums and geometric applications
(American Mathematical Society, 2021) Albiac Alesanco, Fernando José; Ansorena, José L.; Cúth, Marek; Doucha, Michal; Estatistika, Informatika eta Matematika; Institute for Advanced Materials and Mathematics - INAMAT2; Estadística, Informática y Matemáticas
We find general conditions under which Lipschitz-free spaces over metric spaces are isomorphic to their infinite direct _1-sum and exhibit several applications. As examples of such applications we have that Lipschitz-free spaces over balls and spheres of the same finite dimensions are isomorphic, that the Lipschitz-free space over Zd is isomorphic to its _1-sum, or that the Lipschitz-free space over any snowflake of a doubling metric space is isomorphic to l1. Moreover, following new ideas of Bruè et al. from [J. Funct. Anal. 280 (2021), pp. 108868, 21] we provide an elementary self-contained proof that Lipschitz-free spaces over doubling metric spaces are complemented in Lipschitz-free spaces over their superspaces and they have BAP. Everything, including the results about doubling metric spaces, is explored in the more comprehensive setting of p-Banach spaces, which allows us to appreciate the similarities and differences of the theory between the cases p < 1 and p = 1.
Open Access
Addendum to "uniqueness of unconditional basis of infinite direct sums of quasi-Banach spaces"
(Springer, 2022) Albiac Alesanco, Fernando José; Ansorena, José L.; Estadística, Informática y Matemáticas; Estatistika, Informatika eta Matematika; Institute for Advanced Materials and Mathematics - INAMAT2; Universidad Pública de Navarra / Nafarroako Unibertsitate Publikoa
After [Uniqueness of unconditional basis of infinite direct sums of quasi-Banach spaces, Positivity 26 (2022), Paper no. 35] was published, we realized that Theorem 4.2 therein, when combined with work of Casazza and Kalton (Israel J. Math. 103:141-175, 1998) , solves the long-standing problem whether there exists a quasi-Banach space with a unique unconditional basis whose Banach envelope does not have a unique unconditional basis. Here we give examples to prove that the answer is positive. We also use auxiliary results in the aforementioned paper to give a negative answer to the question of Bourgain et al. (Mem Am Math Soc 54:iv+111, 1985)*Problem 1.11 whether the infinite direct sum l(1)(X) of a Banach space X has a unique unconditional basis whenever X does.
Open Access
On a 'philosophical' question about Banach envelopes
(Springer, 2021) Albiac Alesanco, Fernando José; Ansorena, José L.; Wojtaszczyk, Przemyslaw; Estatistika, Informatika eta Matematika; Institute for Advanced Materials and Mathematics - INAMAT2; Estadística, Informática y Matemáticas
We show how to construct non-locally convex quasi-Banach spaces X whose dual separates the points of a dense subspace of X but does not separate the points of X. Our examples connect with a question raised by Pietsch (Rev Mat Complut 22(1):209-226, 2009) and shed light into the unexplored class of quasi-Banach spaces with nontrivial dual which do not have sufficiently many functionals to separate the points of the space.
Open Access
Projections and unconditional bases in direct sums of ℓp SPACES, 0
(Wiley, 2021) Albiac Alesanco, Fernando José; Ansorena, José L.; Estadística, Informática y Matemáticas; Estatistika, Informatika eta Matematika
We show that every unconditional basis in a finite direct sum ⊕p∈Aℓp , with A ⊂ (0,∞], splits into unconditional bases of each summand. This settles a 40 years old question raised in 'A. Ortyński, Unconditional bases in ℓp ⊕ ℓq, 0< p < q <1, Math. Nachr. 103 (1981), 109–116'. As an application we obtain that for any A ⊂ (0,1] finite, the spaces Z = ⊕p∈A ℓp,Z ⊕ ℓ2, and Z ⊕ c0 have a unique unconditional basis up to permutation.