arXiv subjects

Carlos Uzcátegui Aylwin

Publications and source records attributed to Carlos Uzcátegui Aylwin.

4 recordsLinked to original sources

arXiv2026

Structure properties of Banach spaces of I-null sequences

We investigate structural properties of the ideal-null sequence spaces c_{0,I} and c_{0,I}(X), and of the quotient l_infty/c_{0,I}, emphasizing the interaction between Banach space theory and combinatorial, topological, and measure-theoretic properties of the underlying ideal. We study classical problems related to c_0, including the dual space, compactness criteria, bounded and compact c_{0,I}-valued operators, Sobczyk-type extension phenomena, and complemented copies of c_0. We identify c_{0,I}^* isometrically with a space of bounded finitely additive measures on I and obtain a canonical atomic-singular decomposition, with l_1 as an isometrically complemented summand. A Dini principle for ideal convergence yields characterizations of relatively compact subsets of c_{0,I} and of compact operators with range in c_{0,I}. We prove a Sobczyk theorem for c_{0,I} and show that its separable-injectivity constant is exactly 2 for every proper ideal. We also construct, in ZFC, a class of statistical ideals for which c_{0,I} is not complemented in l_infty, and characterize complemented copies of c_0 in c_{0,I}(X) via the corresponding properties of c_{0,I} and X. Finally, we describe quotients associated with direct and Frolik sums of ideals, characterize Banach-lattice copies of c_0(kappa) in l_infty/c_{0,I} through I-almost disjoint families, identify ad(I) with the cellularity of the associated Stone space, and determine its behavior under Fubini products.

math.FA↗

arXiv2026

On the complementation of spaces of $\mathcal I$-null sequences

We study the complementation (in $\ell_\infty$) of the Banach space $c_{0,\mathcal{I}}$, consisting of all bounded sequences $(x_n)$ that $\mathcal{I}$-converge to $0$, endowed with the supremum norm, where $\mathcal{I}$ is an ideal of subsets of $\mathbb{N}$. We show that the complementation of these spaces is related to a condition requiring that the ideal is the intersection of at most a countable family of maximal ideals, which we refer to as at most $ω$-maximal ideals. We prove that $\mathcal{I}$ is at most $ω$-maximal exactly when $c_{0,\mathcal{I}}$ is the kernel of an operator from $\ell_\infty$ to itself satisfying a certain property. In addition, we show that the existence of a Banach lattice isomorphism from the quotient $\ell_\infty/c_{0,\mathcal{I}}$ onto a closed sublattice of $\ell_\infty$ is equivalent to $\mathcal{I}$ being an at most $ω$-maximal ideal. Moreover, $\ell_\infty/c_{0,\mathcal I}$ is Banach lattice isomorphic to $\ell_\infty$ if and only if $\mathcal I$ is the intersection of a certain countably infinite family of maximal ideals; we refer to such ideals as strongly $ω$-maximal. Finally, for two ideals $\mathcal I\subsetneq\mathcal J$, we characterize when the quotient space $c_{0,\mathcal{J}} / c_{0,\mathcal{I}}$ is finite-dimensional.

math.FA↗

arXiv2026

Grothendieck ideals of $\ell_\infty$

We answer several questions in the literature concerning the Grothendieck property of ideals of the Banach lattice $\ell_\infty$ that contain $c_0$. Any such ideal can be represented as a space $c_{0,\mathcal I}$ for $\mathcal I$ an ideal over the natural numbers. We provide a characterization of when $c_{0,\mathcal I}$ is a Grothendieck space in terms of finitely additive measures over $\mathcal P(ω)$ and elements of $\mathcal I$. Using this characterization we show that there are analytic ideals $\mathcal I$ such that $c_{0,\mathcal I}$ is Grothendieck. In the opposite direction we show that for any AD family $\mathcal A$, $c_{0,\mathcal I(\mathcal A)}$ and $C(K_\mathcal A)$ are not Grothendieck spaces, and that for most of the Borel ideals present in the literature, $c_{0,\mathcal I}$ is not Grothendieck. In particular, the family of ideals that do not have the Grothendieck property is cofinal in the Rudin-Keisler order, so the Grothendieck property is not downward closed in the Katětov order. Continuing the work in \cite{Sobota-Zuchowski, Zuchowski}, we also provide similar results on the Nikodym property of the Boolean subalgebras of $\mathcal P(ω)$ generated by the ideal $\mathcal I$.

math.FA↗

arXiv2023

Banach spaces of $\mathcal I$-convergent sequences

We study the space $c_{0,\mathcal{I}}$ of all bounded sequences $(x_n)$ that $\mathcal{I}$-converge to $0$, endowed with the sup norm, where $\mathcal{I}$ is an ideal of subsets of $\mathbb{N}$. We show that two such spaces, $c_{0,\mathcal{I}}$ and $c_{0,\mathcal{J}}$, are isometric exactly when the ideals $\mathcal{I}$ and $\mathcal{J}$ are isomorphic. Additionally, we analyze the connection of the well-known Katětov pre-order $\leq_K$ on ideals with some properties of the space $c_{0,\mathcal{I}}$. For instance, we show that $\mathcal{I}\leq_K\mathcal{J}$ exactly when there is a (not necessarily onto) Banach lattice isometry from $c_{0,\mathcal{I}}$ to $c_{0,\mathcal{J}}$, satisfying some additional conditions. We present some lattice-theoretic properties of $c_{0,\mathcal{I}}$, particularly demonstrating that every closed ideal of $\ell_\infty$ is equal to $c_{0,\mathcal{I}}$ for some ideal $\mathcal{I}$ on $\mathbb{N}$. We also show that certain classical Banach spaces are isometric to $c_{0,\mathcal{I}}$ for some ideal $\mathcal{I}$, such as the spaces $\ell_\infty(c_0)$ and $c_0(\ell_\infty)$. Finally, we provide several examples of ideals for which $c_{0,\mathcal{I}}$ is not a Grothendieck space.

math.FA↗