arXiv subjects

Juan Guerrero-Viu

Publications and source records attributed to Juan Guerrero-Viu.

7 recordsLinked to original sources

arXiv2026

Banach spaces with the weak diametral diameter two property

We introduce and systematically study the weak diametral diameter two property (weak-DD2P), a new geometric property that lies strictly between the diametral diameter two property and both the diameter two property and the convex diametral local diameter two property. A necessary condition for the weak-DD2P is obtained through the structure of the extreme points of the dual unit ball, which leads to characterisations for several classical classes of spaces, including $C(K)$ spaces, $L_1$-preduals, unital uniform algebras, and (vector-valued) function algebras. We establish stability results under standard constructions such as absolute sums, Köthe--Bochner spaces, and projective (symmetric) tensor products. Moreover, we provide complete descriptions of the weak-DD2P for vector-valued spaces of the form $L_1(μ,X)$, $L_\infty(μ,X)$, and $C(K,X)$. These results yield a wide range of new examples and show that the weak-DD2P exhibits a behaviour genuinely different from that of other diameter two properties.

math.FA↗

arXiv2026

Almost preserved extreme points

In this paper we introduce the notion of an almost preserved extreme point (APEP) of a set as a weakening of the concept of preserved extreme points, and we systematically study such points. As a main result, we prove that a Banach space $X$ has the Radon-Nikodým property (RNP) if and only if every closed, convex, and bounded subset of the space has an APEP. Similarly, we prove that $X$ has the RNP if and only if the unit ball of every equivalent renorming has an APEP. We further investigate APEPs of the unit ball of classical Banach spaces, absolute sums, Lipschitz-free spaces, and projective tensor products. In the latter setting, our work also describes the preserved extreme points in the unit ball under the assumption that every bounded operator is compact, thereby partially solving an open problem.

math.FA↗

arXiv2026

On super Delta-points and the convex-DLD2P in absolute sums

We partially answer two open questions concerning diameter two properties in absolute sums. First, we identify the conditions that a super $Δ$-point in an absolute sum of Banach spaces imposes on the coordinates. Secondly, we show that the convex diametral local diameter two property (convex-DLD2P) passes from an absolute sum $X\oplus_N Y$ to its factors whenever $N$ is not the $\ell_\infty$-norm.

math.FA↗

arXiv2026

Functions in $L_1(μ,Y)$ with optimal tensor representations

We study the existence and characterization of optimal tensor representations of elements in the space $L_1(μ,Y)$ of Bochner integrable functions. We completely describe the set of norm-attaining elements in two settings. First, when the Banach space $Y$ is strictly convex, and second, when $Y=L_1(ν)$ and $\mathbb K=\mathbb R$. In both situations, our analysis yields the existence of non-norm-attaining tensors whenever the underlying measures are not purely atomic. Finally, we introduce a geometric property over $Y$ ensuring that every element in $L_1(μ, Y)$ admits an optimal representation. In particular, this holds for Lipschitz-free spaces over complete scattered metric spaces, for $C(K)$ spaces when $K$ is a compact Hausdorff totally disconnected space, and for $c_0(Γ)$ where $Γ$ is any index set. As a byproduct, we settle two open questions regarding projective norm-attainment.

math.FA↗

arXiv2026

Composition operators for holomorphic Lipschitz functions

We study composition operators on spaces of holomorphic Lipschitz functions defined on the open unit ball of a complex Banach space. Our approach is based on the linearization of the symbol through the holomorphic Lipschitz-free spaces, which allow composition operators to be realized as adjoints of linear operators. For spaces with the bounded approximation property, we characterize composition operators between spaces of holomorphic Lipschitz functions vanishing at the origin and describe when composition operators are onto isomorphisms. We further investigate compactness and weak compactness properties of composition operators. In the finite-dimensional setting, compactness and weak compactness are shown to coincide, and a complete characterization is obtained in terms of the symbol. Finally, we analyze the asymptotic behavior of the iterates of composition operators, proving convergence to zero whenever the supremum norm of the symbol is less than one, and we extend several results to the case not vanishing at 0.

math.FA↗

arXiv2026

Integral representations of projective norm-attaining tensors

We introduce a Bochner integral approach to projective norm attainment in tensor products of Banach spaces by defining the class of integral projective norm-attaining tensors. This framework provides a broader, measure-theoretic approach to the study of projective norm attainment in tensor products of Banach spaces. We show that every integral norm-attaining tensor can be approximated in norm by norm-attaining tensors with finite representations. As a consequence, the Bishop-Phelps type density problem for classical norm-attaining tensors is equivalent to the corresponding density problem for integral norm-attaining tensors. Moreover, we prove that if an integral projective norm-attaining tensor represented by a Radon measure is an extreme point, then it must be an elementary tensor. We further investigate weaker topological versions of integral norm-attainment, including weak and weak$^*$ integral representations, providing sufficient conditions for the existence of Bochner representations. Finally, we extend known constructions of projective tensor products containing non-norm-attaining tensors to the integral setting. We show, for instance, that $L_1\widehat{\otimes}_πL_p$ and the real $c_0\widehat{\otimes}_πL_p$ contain non-norm-attaining tensors for $1<p<\infty$.

math.FA↗

arXiv2024

Projective tensor products where every element is norm-attaining

In this paper we analyse when every element of $X\widehat{\otimes}_πY$ attains its projective norm. We prove that this is the case if $X$ is the dual of a subspace of a predual of an $\ell_1(I)$ space and $Y$ is $1$-complemented in its bidual under approximation properties assumptions. This result allows us to provide some new examples where $X$ is a Lipschitz-free space. We also prove that the set of norm-attaining elements is dense in $X\widehat{\otimes}_πY$ if, for instance, $X=L_1(μ)$ and $Y$ is any Banach space, or if $X$ has the metric $π$-property and $Y$ is a dual space with the RNP.

math.FA↗