arXiv subjects

Filip Talimdjioski

Publications and source records attributed to Filip Talimdjioski.

4 recordsLinked to original sources

arXiv2024

Lipschitz-free spaces over strongly countable-dimensional spaces and approximation properties

Let $T$ be a compact, metrisable and strongly countable-dimensional topological space. Let $\mathcal{M}^T$ be the set of all metrics $d$ on $T$ compatible with its topology, and equip $\mathcal{M}^T$ with the topology of uniform convergence, where the metrics are regarded as functions on $T^2$. We prove that the set $\mathcal{A}^{T,1}$ of metrics $d\in\mathcal{M}^T$ for which the Lipschitz-free space $\mathcal{F}(T,d)$ has the metric approximation property is residual in $\mathcal{M}^T$.

math.FA↗

arXiv2023

Lipschitz-free spaces over properly metrisable spaces and approximation properties

Let $T$ be a topological space admitting a compatible proper metric, that is, a locally compact, separable and metrisable space. Let $\mathcal{M}^T$ be the non-empty set of all proper metrics $d$ on $T$ compatible with its topology, and equip $\mathcal{M}^T$ with the topology of uniform convergence, where the metrics are regarded as functions on $T^2$. We prove that the set $\mathcal{A}^{T,1}$ of metrics $d\in\mathcal{M}^T$ for which the Lipschitz-free space $\mathcal{F}(T,d)$ has the metric approximation property is a dense set in $\mathcal{M}^T$, and is furthermore residual in $\mathcal{M}^T$ when $T$ is zero-dimensional. We also prove that if $T$ is uncountable then the set $\mathcal{A}^T_f$ of metrics $d\in\mathcal{M}^T$ for which $\mathcal{F}(T,d)$ fails the approximation property is dense in $\mathcal{M}^T$. Combining the last statement with a result of Dalet, we conclude that for any `properly metrisable' space $T$, $\mathcal{A}^T_f$ is either empty or dense in $\mathcal{M}^T$.

math.FA↗

arXiv2023

Lipschitz-free spaces over Cantor sets and approximation properties

Let $K=2^\mathbb{N}$ be the Cantor set, let $\mathcal{M}$ be the set of all metrics $d$ on $K$ that give its usual (product) topology, and equip $\mathcal{M}$ with the topology of uniform convergence, where the metrics are regarded as functions on $K^2$. We prove that the set of metrics $d\in\mathcal{M}$ for which the Lipschitz-free space $\mathcal{F}(K,d)$ has the metric approximation property is a residual $F_{σδ}$ set in $\mathcal{M}$, and that the set of metrics $d\in\mathcal{M}$ for which $\mathcal{F}(K,d)$ fails the approximation property is a dense meager set in $\mathcal{M}$. This answers a question posed by G. Godefroy.

math.FA↗

arXiv2022

Lipschitz-Free Spaces over Manifolds and the Metric Approximation Property

Let $\|\cdot\|$ be a norm on $\mathbb{R}^N$ and let $M$ be a closed $C^1$-submanifold of $\mathbb{R}^N$. Consider the pointed metric space $(M,d)$, where $d$ is the metric given by $d(x,y)=\|x-y\|$, $x,y\in M$. Then the Lipschitz-free space $\mathcal{F}(M)$ has the Metric Approximation Property.

math.FA↗