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Eva Pernecká

Publications and source records attributed to Eva Pernecká.

14 recordsLinked to original sources
arXiv2026

A solution to the extreme point problem and other applications of Choquet theory to Lipschitz-free spaces

We prove that every element of a Lipschitz-free space admits an expression as a convex series of elements with compact support. As a consequence, we conclude that all extreme points of the unit ball of Lipschitz-free spaces are elementary molecules, solving a long-standing problem. We also deduce that all elements of a Lipschitz-free space with the Radon-Nikodým property can be expressed as convex integrals of molecules. Our results are based on a recent theory of integral representation for functionals on Lipschitz spaces which draws on classical Choquet theory, due to the third named author.

math.FA↗
arXiv2025

On the uniform continuity of homeomorphisms between the spheres of $\ell_\infty^k$ and $\ell_1^k$

We consider the problem of whether there is a sequence of homeomorphisms $(F_k)_k$ between the unit spheres of the $k$-dimensional Banach spaces $\ell_\infty^k$ and $\ell_1^k$ which is also equi-uniformly continuous. We prove that this cannot be the case if the sequence $(F_k)_k$ either (1) does not increase support sizes (which is a property strictly weaker than support preservation) or (2) is step preserving (which is a property strictly weaker than being equivariant with respect to permutations of the canonical basis). We also provide quantitative estimates relating the moduli of uniform continuity of the maps to the dimension of the spaces. This gives partial answers to a question of W. B. Johnson and it is related to the problem of whether $c_0$ has Kasparov and Yu's Property (H). Our results also apply to more general spaces other than $\ell_1$ such as spaces with unconditional bases which are not equivalent to the standard $c_0$ basis. Finally, we derive an asymptotic concentration inequality that must be satisfied by step preserving equi-uniformly continuous maps defined on the positive parts of these unit spheres.

math.FA↗
arXiv2025

Pełczyński's property (V$^*$) in Lipschitz-free spaces

We prove that Pelczyński's property (V$^*$) is locally determined for Lipschitz-free spaces, and obtain several sufficient conditions for it to hold. We deduce that $\mathcal{F}(M)$ has property (V$^*$) when the complete metric space $M$ is locally compact and purely 1-unrectifiable, a Hilbert space, or belongs to a class of Carnot-Carathéodory spaces satisfying a bi-Hölder condition, including Carnot groups.

math.FA↗
arXiv2025

Equivariant liftings in Lipschitz-free spaces

We consider Banach spaces $X$ that can be linearly lifted into their Lipschitz-free spaces $\mathcal{F}(X)$ and, for a group $G$ acting on $X$ by linear isometries, we study the possible existence of $G$-equivariant linear liftings. In particular, we prove that such lifting exists when $G$ is compact in the strong operator topology, or an increasing union of such groups and $\mathcal{F}(X)$ is complemented in its bidual by an equivariant projection. As an example of application, we define and study a complex version of the Lipschitz-free space $\mathcal{F}(X)$ when $X$ is a subset of a complex Banach space stable under the action of the circle group.

math.FA↗
arXiv2024

Convex integrals of molecules in Lipschitz-free spaces

We introduce convex integrals of molecules in Lipschitz-free spaces $\mathcal{F}(M)$ as a continuous counterpart of convex series considered elsewhere, based on the de Leeuw representation. Using optimal transport theory, we show that these elements are determined by cyclical monotonicity of their supports, and that under certain finiteness conditions they agree with elements of $\mathcal{F}(M)$ that are induced by Radon measures on $M$, or that can be decomposed into positive and negative parts. We also show that convex integrals differ in general from convex series of molecules. Finally, we present some standalone results regarding extensions of Lipschitz functions which, combined with the above, yield applications to the extremal structure of $\mathcal{F}(M)$. In particular, we show that all elements of $\mathcal{F}(M)$ are convex series of molecules when $M$ is uniformly discrete and identify all extreme points of the unit ball of $\mathcal{F}(M)$ in that case.

math.FA↗
arXiv2021

Integral representation and supports of functionals on Lipschitz spaces

We analyze the relationship between Borel measures and continuous linear functionals on the space $\mathrm{Lip}_0(M)$ of Lipschitz functions on a complete metric space $M$. In particular, we describe continuous functionals arising from measures and vice versa. In the case of weak$^\ast$ continuous functionals, i.e. members of the Lipschitz-free space $\mathcal{F}(M)$, measures on $M$ are considered. For the general case, we show that the appropriate setting is rather the uniform (or Samuel) compactification of $M$ and that it is consistent with the treatment of $\mathcal{F}(M)$. This setting also allows us to give a definition of support for all elements of $\mathrm{Lip}_0(M)^\ast$ with similar properties to those in $\mathcal{F}(M)$, and we show that it coincides with the support of the representing measure when such a measure exists. We deduce that the members of $\mathrm{Lip}_0(M)^\ast$ that can be expressed as the difference of two positive functionals admit a Jordan-like decomposition into a positive and a negative part.

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arXiv2020

Topological groups with invariant linear spans

Given a topological group $G$ that can be embedded as a topological subgroup into some topological vector space (over the field of reals) we say that $G$ has invariant linear span if all linear spans of $G$ under arbitrary embeddings into topological vector spaces are isomorphic as topological vector spaces. For an arbitrary set $A$ let $\mathbb{Z}^{(A)}$ be the direct sum of $|A|$-many copies of the discrete group of integers endowed with the Tychonoff product topology. We show that the topological group $\mathbb{Z}^{(A)}$ has invariant linear span. This answers a question of D. Dikranjan et al. in positive. We prove that given a non-discrete sequential space $X$, the free abelian topological group $A(X)$ over $X$ is an example of a topological group that embeds into a topological vector space but does not have invariant linear span.

math.GN↗
arXiv2020

Supports in Lipschitz-free spaces and applications to extremal structure

We show that the class of Lipschitz-free spaces over closed subsets of any complete metric space $M$ is closed under arbitrary intersections, improving upon the previously known finite-diameter case. This allows us to formulate a general and natural definition of supports for elements in a Lipschitz-free space $\mathcal F(M)$. We then use this concept to study the extremal structure of $\mathcal F(M)$. We prove in particular that $(δ(x) - δ(y))/d(x,y)$ is an exposed point of the unit ball of $\mathcal F(M)$ whenever the metric segment $[x,y]$ is trivial, and that any extreme point which can be expressed as a finitely supported perturbation of a positive element must be finitely supported itself. We also characterise the extreme points of the positive unit ball: they are precisely the normalized evaluation functionals on points of $M$.

math.FA↗
arXiv2020

Supports and extreme points in Lipschitz-free spaces

For a complete metric space $M$, we prove that the finitely supported extreme points of the unit ball of the Lipschitz-free space $\mathcal{F}(M)$ are precisely the elementary molecules $(δ(p)-δ(q))/d(p,q)$ defined by pairs of points $p,q$ in $M$ such that the triangle inequality $d(p,q)<d(p,r)+d(q,r)$ is strict for any $r\in M$ different from $p$ and $q$. To this end, we show that the class of Lipschitz-free spaces over closed subsets of $M$ is closed under arbitrary intersections when $M$ has finite diameter, and that this allows a natural definition of the support of elements of $\mathcal{F}(M)$.

math.FA↗
arXiv2015

On uniformly differentiable mappings from $\ell_\infty(Γ)$

In 1970 Haskell Rosenthal proved that if $X$ is a Banach space, $Γ$ is an infinite index set, and $T:\ell_\infty(Γ)\to X$ is a bounded linear operator such that $\inf_{γ\inΓ}\|T(e_γ)\|>0$ then $T$ acts as an isomorphism on $\ell_\infty(Γ')$, for some $Γ'\subsetΓ$ of the same cardinality as $Γ$. Our main result is a nonlinear strengthening of this theorem. More precisely, under the assumption of GCH and the regularity of $Γ$, we show that if ${F}:B_{\ell_\infty(Γ)}\to X$ is uniformly differentiable and such that $\inf_{γ\inΓ}\|{F}(e_γ){-F(0)}\|>0$ then there exists $x\in B_{\ell_\infty(Γ)}$ such that $d{F}(x)[\cdot]$ is a bounded linear operator which acts as an isomorphism on $\ell_\infty(Γ')$, for some $Γ'\subsetΓ$ of the same cardinality as $Γ$.

math.FA↗
arXiv2015

Approximation and Schur properties for Lipschitz free spaces over compact metric spaces

We prove that for any separable Banach space $X$, there exists a compact metric space which is homeomorphic to the Cantor space and whose Lipschitz-free space contains a complemented subspace isomorphic to $X$. As a consequence we give an example of a compact metric space which is homeomorphic to the Cantor space and whose Lipschitz-free space fails the approximation property and we prove that there exists an uncountable family of topologically equivalent distances on the Cantor space whose free spaces are pairwise non isomorphic. We also prove that the free space over a countable compact metric space has the Schur property. These results answer questions by G. Godefroy.

math.FA↗
arXiv2015

The Metric Approximation Property and Lipschitz-Free Spaces over Subsets of $\mathbb{R}^N$

We prove that for certain subsets $M \subseteq \mathbb{R}^N$, $N \geqslant 1$, the Lipschitz-free space $\mathcal{F}(M)$ has the metric approximation property (MAP), with respect to any norm on $\mathbb{R}^N$. In particular, $\mathcal{F}(M)$ has the MAP whenever $M$ is a finite-dimensional compact convex set. This should be compared with a recent result of Godefroy and Ozawa, who showed that there exists a compact convex subset $M$ of a separable Banach space, for which $\mathcal{F}(M)$ fails the approximation property.

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