BUSCADOR RECOLECTA

The list of known Banach spaces whose linear geometry determines the (nonlinear) democracy functions of their quasi-greedy bases to the extent that they end up being democratic, reduces to c0, ℓ2, and all separable L1-spaces. Oddly enough, these are the only Banach spaces that, when they have an unconditional basis, it is unique. Our aim in this paper is to study the connection between quasi-greediness and democracy of bases in non-locally convex spaces. We prove that all quasi-greedy bases in ℓp for 0<p<1 (which also has a unique unconditional basis) are democratic with fundamental function of the same order as (m1/p)∞m=1. The methods we develop allow us to obtain even more, namely that the same occurs in any separable Lp-space, 0<p<1, with the bounded approximation property., F. Albiac acknowledges the support of the Spanish Ministry for Science and Innovation under Grant PID2019-107701GB-I00 for Operators, lattices, and structure of Banach spaces. F. Albiac and J. L. Ansorena acknowledge the support of the Spanish Ministry for Science, Innovation, and Universities under Grant PGC2018-095366-B-I00 for Análisis Vectorial, Multilineal y Aproximación. P. Wojtaszczyk was supported by National Science Centre, Poland grant UMO-2016/21/B/ST1/00241.

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., Fernando Albiac acknowledges the support of the Spanish Ministry for Economy and Competitivity under Grant MTM2016-76808-P and the Spanish Ministry for Science and Innovation under Grant PID2019-107701GB-I00 for Operators, lattices, and structure of Banach spaces. Fernando Albiac and Jose L. Ansorena acknowledge the support of the Spanish Ministry for Science, Innovation, and Universities under Grant PGC2018-095366-B-I00 for Analisis Vectorial, Multilineal y Aproximacion. P. Wojtaszczyk was supported by National Science Centre, Poland Grant UMO-2016/21/B/ST1/00241.

One of the main results of the article Gelfand theory for real Banach algebras, recently published in [Rev R Acad Cienc Exactas Fís Nat Ser A Mat RACSAM 114(4):163, 2020] is Proposition 4.1, which establishes that the norm inequality ||a2||≤ ||a2+ b2|| for a, b∈ A is sufficient for a commutative real Banach algebra A with a unit to be isomorphic to the space CR(K) of continuous real-valued functions on a compact Hausdorff space K. Moreover, in this proposition is also shown that if the above condition (which involves all the operations of the algebra) holds, then the real-algebra isomorphism given by the Gelfand transform preserves the norm of squares. A very natural question springing from the above-mentioned result is whether an isomorphism of A onto CR(K) is always norm-preserving of squares. This note is devoted to providing a negative answer to this problem. To that end, we construct algebra norms on spaces CR(K) which are (1 + ϵ) -equivalent to the sup-norm and with the norm of the identity function equal to 1, where the norm of every nonconstant function is different from the standard sup-norm. We also provide examples of two-dimensional normed real algebras A where ||a2|| ≤ k|| a2+ b2|| for all a, b∈ A, for some k>1 , but the inequality fails for k= 1., The first two authors acknowledge the support from the Spanish Ministry for Science, Innovation, and Universities under Grant PGC2018-095366-B-I00 for Análisis Vectorial, Multilineal y Aproximación. F. Albiac was also supported by the Spanish Ministry for Science and Innovation under Grant PID2019-107701GB-I00 for Operators, lattices, and structure of Banach spaces.

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., F. Albiac acknowledges the support of the Spanish Ministry for Science and Innovation under Grant PID2019-107701GB-I00 for Operators, lattices, and structure of Banach spaces. F. Albiac and J. L. Ansorena acknowledge the support of the Spanish Ministry for Science, Innovation, and Universities under Grant PGC2018-095366-B-I00 for Análisis Vectorial, Multilineal y Aproximación. M. Cúth has been supported by Charles University Research program No. UNCE/SCI/023. M. Doucha was supported by the GAČR project EXPRO 20-31529X and RVO: 67985840.

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., The first author acknowledges the support of the Spanish Ministry for Science and Innovation under Grant PID2019-107701GB-I00 for Operators, lattices, and structure of Banach spaces and the support of the Spanish Ministry for Science, Innovation, and Universities under Grant PGC2018-095366-B-I00 for Análisis Vectorial, Multilineal y Aproximación. The second author acknowledges the support of the Spanish Ministry for Science, Innovation, and Universities under Grant PGC2018-095366-B-I00 for Análisis Vectorial, Multilineal y Aproximación. The third author was supported by Charles University Research program No. UNCE/SCI/023. The fourth author was supported by the GACˇR project 19-05271Y and RVO: 67985840.

The general problem addressed in this work is the development of a systematic study of the thresholding greedy algorithm for general biorthogonal systems in quasi-Banach spaces from a functional-analytic point of view. If (Formula Presented) is a biorthogonal system in X then for each x ∈ X we have a formal expansion (Formula Presented). The thresholding greedy algorithm (with threshold ε > 0) applied to x is formally defined as (Formula Presented). The properties of this operator give rise to the different classes of greedy-type bases. We revisit the concepts of greedy, quasi-greedy, and almost greedy bases in this comprehensive framework and provide the (non-trivial) extensions of the corresponding characterizations of those types of bases. As a by-product of our work, new properties arise, and the relations among them are carefully discussed., F. Albiac acknowledges the support of the Spanish Ministry for Science and Innovation under Grant PID2019-107701GB-I00 for Operators, lattices, and structure of Banach spaces. F. Albiac and J. L. Ansorena acknowledge the support of the Spanish Ministry for Science, Innovation, and Universities Grant PGC2018-095366-B-I00 for Análisis Vectorial, Multilineal y Aplicaciones. P. M. Berná acknowledges the support of the Spanish Ministry for Economy and Competitivity Grants MTM-2016-76566-P and PID2019-105599GB-100 (Agencia Estatal de Investigación). P. M. Berná was also supported by Grant 20906/PI/18 from Fundación Séneca (Región de Murcia, Spain). P. Wojtaszczyk was partially supported by National Science Centre, Poland, grant UMO-2016/21/B/ST1/00241. This work was supported by EPSRC grant number EP/R014604/1.

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., F. Albiac acknowledges the support of the Spanish Ministry for Science and Innovation under Grant PID2019-107701GB-I00 for Operators, lattices, and structure of Banach spaces. F. Albiac and J. L. Ansorena acknowledge the support of the Spanish Ministry for Science, Innovation, and Universities under Grant PGC2018- 095366-B-I00 for Análisis Vectorial, Multilineal y Aproximación.

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., Both authors were supported by the Spanish Ministry for Science, Innovation, and Universities, Grant PGC2018-095366-B-I00 for ‘Análisis Vectorial, Multilineal y Approximación’. The first-named author also acknowledges the support from the Spanish Ministry for Science and Innovation, Grant PID2019-107701GB-I00 for ‘Operators, Lattices, and Structure of Banach spaces.

Addendum en https://hdl.handle.net/2454/45133, 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., F. Albiac acknowledges the support of the Spanish Ministry for Science and Innovation under Grant PID2019-107701GB-I00 for Operators, lattices, and structure of Banach spaces. F. Albiac and J. L. Ansorena acknowledge the support of the Spanish Ministry for Science, Innovation, and Universities under Grant PGC2018-095366-B-I00 for Análisis Vectorial, Multilineal y Aproximación.

We investigate a way to turn an arbitrary (usually, unbounded) metric space M into a bounded metric space B in such a way that the corresponding Lipschitz-free spaces F(M) and F(B) are isomorphic. The construction we provide is functorial in a weak sense and has the advantage of being explicit. Apart from its intrinsic theoretical interest, it has many applications in that it allows to transfer many arguments valid for Lipschitz-free spaces over bounded spaces to Lipschitz-free spaces over unbounded spaces. Furthermore, we show that with a slightly modified pointwise multiplication, the space Lip(0)(M) of scalar-valued Lipschitz functions vanishing at zero over any (unbounded) pointed metric space is a Banach algebra with its canonical Lipschitz norm., This work was supported by the Spanish Ministry for Science and Innovation [PID2019-107701GBI00 for Operators, lattices, and structure of Banach spaces to F.A.]; the Spanish Ministry for Science, Innovation, and Universities [PGC2018-095366-B-I00 for Analisis Vectorial, Multilineal, y Aproximacion to F.A. and J.L.A.]; the Charles University Research program [UNCE/SCI/023 to M.C.]; and the GAC. R project [EXPRO 20-31529X and RVO: 67985840 to M.D.].