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Leandro Candido

Publications and source records attributed to Leandro Candido.

16 recordsLinked to original sources
arXiv2026

Derivatives of Tensor Products and Applications to Spaces $C(K^n,X)$

In this paper we develop an abstract theory of derivatives for Banach spaces based on objects that we call \emph{bidual assignments}. This framework encompasses both the Semadeni derivative and the recently introduced Semadeni--Pe{\l}czy\'nski derivative. More generally, suitable ideals of subsets of the dual space give rise to a broad family of derivatives within this setting. We establish addition and product formulas for these derivatives, showing that they behave naturally with respect to direct sums and injective tensor products. As an application, we compute iterated derivatives for a number of spaces $C(K)$ associated with scattered compacta, including scattered compact lines and compact trees. As a further application, we establish classification results for spaces of the form $C(K^n,X)$. In particular, for uncountable ordinals $\alpha$ and $\beta$, an integer $n\geq 1$, and Banach spaces $X$ satisfying suitable rigidity assumptions, we prove that \[ C([0,\alpha]^n,X)\sim C([0,\beta]^n,X) \quad\text{if and only if}\quad C([0,\alpha])\sim C([0,\beta]). \] This extends Kislyakov's classification of spaces $C([0,\alpha])$ and its vector-valued extension due to Galego to finite powers of ordinal intervals.

math.FA↗
arXiv2026

A lifting theorem for operators between Lipschitz spaces

We prove that every bounded linear operator between Lipschitz spaces admits a lifting along the de Leeuw embedding. More precisely, given pointed metric spaces $M$ and $N$ and $\epsilon>0$, every bounded linear operator $S:\mathrm{Lip}_0(M)\to \mathrm{Lip}_0(N)$ admits a lifting $\mathfrak{S}:C(\beta \widetilde{M})\to C(\beta \widetilde{N})$ such that $\|\mathfrak{S}\|\leq \|S\|+\epsilon$ and $\mathfrak{S}(\varPhi_M(f))=\varPhi_N(S(f))$ for every $f\in \mathrm{Lip}_0(M)$. Moreover, compact operators admit compact liftings.

math.FA↗
arXiv2026

On measurability of Kurzweil--Stieltjes integrable functions on compact lines

We continue the study on Kurzweil--Stieltjes integration on compact lines initiated in [doi:10.1007/s11117-025-01161-9]. Given a real valued function $G$ on a compact line, the presented integral is called the Kurzweil--Stieltjes integral with respect to $G$, or simply the $G$-integral. %Given a compact line $K$ and a right-continuous function $G:K\to\mathbb{R}$ of bounded variation, we consider the Radon measure $\mu_G$ naturally induced by $G$. Our main results concern the relationship between $G$-integrability and measurability. We prove that, whenever $G$ is nondecreasing, every $G$-integrable function is $\mu_G$-measurable, where $\mu_G$ is the natural Radon measure induced by $G$. We also show that, for an arbitrary $G$ of bounded variation, every bounded $G$-integrable function is $\mu_G$-measurable. %, where $|\mu_G|$ denotes the total variation measure of $\mu_G$. As an application, we provide a full characterization of Lebesgue integrablility with respect to Radon measures in terms of the $G$-integral, and demonstrate that the $G$-integral represents an extension of the Lebesgue integral with respect to $\mu_G$ for suitable $G$. In addition, we establish a version of Hake's theorem for the $G$-integral in this setting.

math.FA↗
arXiv2025

Few Bilinear Operators on Spaces of Continuous Functions

Motivated by recent work exhibiting a locally compact scattered space $L$ constructed under Ostaszewski's $\clubsuit$-principle, which yielded a complete classification of linear operators on $C_0(L\times L)$, we extend the analysis to the bilinear setting. We show that, for this space $L$, every bilinear operator $G:C_0(L)\times C_0(L)\to C_0(L)$ admits a unique decomposition into the sum of trivially predictable components. This establishes a bilinear analogue of the few operators phenomenon.

math.FA↗
arXiv2025

Kurzweil--Stieltjes integration on compact lines

We develop a version of the Kurzweil--Stieltjes integral on compact lines and establish its fundamental properties. For sufficiently regular integrators, we obtain convergence theorems and show that the presented integration process generalizes Lebesgue integration with respect to positive Radon measures. Additionally, we introduce a notion of derivation on compact lines which, when paired with the proposed integral, yields a formulation of the Fundamental Theorem of Calculus in this context.

math.FA↗
arXiv2024

Few operators on Banach spaces $C_0(L\times L)$

Using Ostaszewski's $\clubsuit$-principle, we construct a non-metrizable, locally compact, scattered space $L$ in which the operators on the Banach space $C_0(L \times L)$ exhibit a remarkably simple structure. We provide a detailed analysis and, through a series of decomposition steps, offer an explicit characterization of all operators on $C_0(L \times L)$.

math.FA↗
arXiv2024

On the weak$^*$ separability of the space of Lipschitz functions

We conjecture that whenever $M$ is a metric space of density at most continuum, then the space of Lipschitz functions is $w^*$-separable. We prove the conjecture for several classes of metric spaces including all the Banach spaces with a projectional skeleton, Banach spaces with a $w^*$-separable dual unit ball and locally separable complete metric spaces.

math.FA↗
arXiv2023

Note on almost isometric ideals and local retracts in Banach and metric spaces

We exhibit a new approach to the proofs of the existence of a large family of almost isometric ideals in nonseparable Banach spaces and existence of a large family of almost isometric local retracts in metric spaces. Our approach also implies the existence of a large family of nontrivial projections on every dual of a nonseparable Banach space. We prove three possible formulations of our results are equivalent. Some applications are mentioned which witness the usefulness of our novel approach.

math.FA↗
arXiv2022

On large $\ell_1$-sums of Lipschitz-free spaces and applications

We prove that the Lipschitz-free space over a Banach space $X$ of density $κ$, denoted by $\mathcal{F}(X)$, is linearly isomorphic to its $\ell_1$-sum $\left(\bigoplus_κ\mathcal{F}(X)\right)_{\ell_1}$. This provides an extension of a previous result from Kaufmann in the context of non-separable Banach spaces. Further, we obtain a complete classification of the spaces of real-valued Lipschitz functions that vanish at $0$ over a $\mathcal{L}_p$-space. More precisely, we establish that, for every $1\leq p\leq \infty$, if $X$ is a $\mathcal{L}_p$-space of density $κ$, then $\mathrm{Lip}_0(X)$ is either isomorphic to $\mathrm{Lip}_0(\ell_p(κ))$ if $p<\infty$, or $\mathrm{Lip}_0(c_0(κ))$ if $p=\infty$.

math.FA↗
arXiv2021

Complementations in $C(K,X)$ and $\ell_\infty(X)$

We investigate the geometry of $C(K,X)$ and $\ell_{\infty}(X)$ spaces through complemented subspaces of the form $\left(\bigoplus_{i\in \varGamma}X_i\right)_{c_0}$. Concerning the geometry of $C(K,X)$ spaces we extend some results of D. Alspach and E. M. Galego from \cite{AlspachGalego}. On $\ell_{\infty}$-sums of Banach spaces we prove that if $\ell_{\infty}(X)$ has a complemented subspace isomorphic to $c_0(Y)$, then, for some $n \in \mathbb{N}$, $X^n$ has a subspace isomorphic to $c_0(Y)$. We further prove the following: (1) If $C(K)\sim c_0(C(K))$ and $C(L)\sim c_0(C(L))$ and $\ell_{\infty}(C(K))\sim \ell_{\infty}(C(L))$, then $K$ and $L$ have the same cardinality. (2) If $K_1$ and $K_2$ are infinite metric compacta, then $\ell_{\infty}(C(K_1))\sim \ell_{\infty}(C(K_2))$ if and only if $C(K_1)$ is isomorphic to $C(K_2)$.

math.FA↗
arXiv2020

On the geometry of Banach spaces of the form $\mathrm{Lip}_0(C(K))$

We investigate the problem of classifying the Banach spaces $\mathrm{Lip}_0(C(K))$ for Hausdorff compacta $K$. In particular, sufficient conditions are established for a space $\mathrm{Lip}_0(C(K))$ to be isomorphic to $\mathrm{Lip}_0(c_0(\varGamma))$ for some uncountable set $\varGamma$.

math.FA↗
arXiv2020

On operators on $C_0(α\times L)$ under the Ostaszewski's $\clubsuit$-principle

For an exotic locally compact Hausdorff space $L$, constructed under the assumption of the Ostaszewski's $\clubsuit$-principle, and a countable ordinal space $α$, we prove that all operators defined on $C_0(α\times L)$ are as simple as possible. We also investigate the geometry of such space $C_0(α\times L)$ and we classify up to isomorphisms all its complemented subspaces.

math.FA↗
arXiv2019

Isomorphisms between spaces of Lipschitz functions

We develop tools for proving isomorphisms of normed spaces of Lipschitz functions over various doubling metric spaces and Banach spaces. In particular, we show that $\operatorname{Lip}_0(\mathbb{Z}^d)\simeq\operatorname{Lip}_0(\mathbb{R}^d)$, for all $d\in\mathbb{N}$. More generally, we e.g. show that $\operatorname{Lip}_0(Γ)\simeq \operatorname{Lip}_0(G)$, where $Γ$ is from a large class of finitely generated nilpotent groups and $G$ is its Mal'cev closure; or that $\operatorname{Lip}_0(\ell_p)\simeq\operatorname{Lip}_0(L_p)$, for all $1\leq p<\infty$. We leave a large area for further possible research.

math.FA↗
arXiv2015

On complemented copies of $c_0(ω_1)$ in $C(K^n)$ spaces

Given a compact Hausdorff space $K$ we consider the Banach space of real continuous functions $C(K^n)$ or equivalently the $n$-fold injective tensor product $\hat\bigotimes_{\varepsilon}C(K)$ or the Banach space of vector valued continuous functions $C(K, C(K, C(K ..., C(K)...)$. We address the question of the existence of complemented copies of $c_0(ω_1)$ in $\hat\bigotimes_{\varepsilon}C(K)$ under the hypothesis that $C(K)$ contains an isomorphic copy of $c_0(ω_1)$. This is related to the results of E. Saab and P. Saab that $X\hat\otimes_\varepsilon Y$ contains a complemented copy of $c_0$, if one of the infinite dimensional Banach spaces $X$ or $Y$ contains a copy of $c_0$ and of E. M. Galego and J. Hagler that it follows from Martin's Maximum that if $C(K)$ has density $ω_1$ and contains a copy of $c_0(ω_1)$, then $C(K\times K)$ contains a complemented copy $c_0(ω_1)$. The main result is that under the assumption of $\clubsuit$ for every $n\in N$ there is a compact Hausdorff space $K_n$ of weight $ω_1$ such that $C(K)$ is Lindelöf in the weak topology, $C(K_n)$ contains a copy of $c_0(ω_1)$, $C(K_n^n)$ does not contain a complemented copy of $c_0(ω_1)$ while $C(K_n^{n+1})$ does contain a complemented copy of $c_0(ω_1)$. This shows that additional set-theoretic assumptions in Galego and Hagler's nonseparable version of Cembrano and Freniche's theorem are necessary as well as clarifies in the negative direction the matter unsettled in a paper of Dow, Junnila and Pelant whether half-pcc Banach spaces must be weakly pcc.

math.FA↗
arXiv2013

On embeddings of $C_0(K)$ spaces into $C_0(L,X)$ spaces

Let $C_0(K, X)$ denote the space of all continuous $X$-valued functions defined on the locally compact Hausdorff space $K$ which vanish at infinity, provided with the supremum norm. If $X$ is the scalar field, we denote $C_0(K, X)$ by simply $C_0(K)$. In this paper we prove that for locally compact Hausdorff spaces $K$ and $L$ and for Banach space $X$ containing no copy of $c_0$, if there is a isomorphic embedding of $C_0(K)$ into $C_0(L,X)$ where either $X$ is separable or $X^*$ has the Radon-Nikodým property, then either $K$ is finite or $|K|\leq |L|$. As a consequence of this result, if there is a isomorphic embedding of $C_0(K)$ into $C_0(L,X)$ where $X$ contains no copy of $c_0$ and $L$ is scattered, then $K$ must be scattered.

math.FA↗