arXiv subjects

Michael A. Rincón-Villamizar

Publications and source records attributed to Michael A. Rincón-Villamizar.

5 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.

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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.

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arXiv2023

Almost Fraïssé Banach spaces

Continuing with the study of Approximately ultrahomogeneous and Fraïssé Banach spaces introduced by V. Ferenczi, J. López-Abad, B. Mbombo and S. Todorcevic, we define formally weaker and in some aspects more natural properties of Banach spaces which we call Almost ultrahomogeneity and the Almost Fraïssé Property. We obtain relations between these different homogeneity properties of a space $E$ and relate them to certain pseudometrics on the class $\mathrm{Age}(E)$ of finite dimensional subspaces of $E$. We prove that ultrapowers of an almost Fraïssé Banach space are ultrahomogeneous. We also study two properties called finitely isometrically extensible and almost finitely isometrically extensible, respectively, and prove that approximately ultrahomogeneous Banach spaces are finitely isometrically extensible.

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