arXiv subjects

Andre Ostrak

Publications and source records attributed to Andre Ostrak.

8 recordsLinked to original sources

arXiv2026

A counterexample for the Daugavet index of thickness in $\ell_1$-sums

We give a negative answer to a question of Haller-Langemets-Lima-Nadel-Rueda Zoca asking whether, for all Banach spaces $X$ and $Y$, the Daugavet index of thickness satisfies \[ T(X\oplus_1 Y)=\min\{T(X),T(Y)\}. \] We show that this equality does hold whenever one of the two summands has the Daugavet property. On the other hand, if $D$ is a Banach space with the Daugavet property and $N$ is a suitable absolute norm, then for $X=D\oplus_N D$, one has $T(X\oplus_1 X)<T(X)$.

math.FA↗

arXiv2025

Duality of Lipschitz-free spaces over ultrametric spaces

We give a metric characterisation of when the Lipschitz-free space over a separable ultrametric space is a dual Banach space. In the case where the Lipschitz-free space has a predual, we show that this predual is M-embedded if and only if the metric space is proper. We show that for ultrametric spaces, the little Lipschitz space is always an M-ideal in the corresponding space of Lipschitz functions, and we show that this is not the case for metric spaces in general, thus answering a question posed by Werner in the negative. Finally, we show that the space of Lipschitz functions of an ultrametric space contains a strongly extreme point.

math.FA↗

arXiv2025

The local diameter two property and the diameter two property in spaces of Lipschitz functions

We separate the local diameter two property from the diameter two property and their weak-star counterparts from each other in spaces of Lipschitz functions. We characterise the $w^*$-LD$2$P, the $w^*$-D$2$P, the LD$2$P, and the SD$2$P in these spaces. We introduce a generalised version of cyclical monotonicity to study functionals on spaces of Lipschitz functions.

math.FA↗

arXiv2024

Separating diameter two properties from their weak-star counterparts in spaces of Lipschitz functions

We address some open problems concerning Banach spaces of real-valued Lipschitz functions. Specifically, we prove that the diameter two properties differ from their weak-star counterparts in these spaces. In particular, we establish the existence of dual Banach spaces lacking the symmetric strong diameter two property but possessing its weak-star counterpart. We show that there exists an octahedral Lipschitz-free space whose bidual is not octahedral. Furthermore, we prove that the Banach space of real-valued Lipschitz functions from any infinite subset of $\ell_1$ possesses the symmetric strong diameter two property. These results are achieved by introducing new sufficient conditions, providing new examples and clarifying the status of known ones.

math.FA↗

arXiv2022

Diameter two properties for spaces of Lipschitz functions

We solve some open problems regarding diameter two properties within the class of Banach spaces of real-valued Lipschitz functions by using the de Leeuw transform. Namely, we show that: the diameter two property, the strong diameter two property, and the symmetric strong diameter two property are all different for these spaces of Lipschitz functions; the space $\operatorname{Lip}_0(K_n)$ has the symmetric strong diameter two property for every $n\in \mathbb{N}$, including the case of $n=2$; every local norm-one Lipschitz function is a Daugavet point.

math.FA↗

arXiv2022

The Lipschitz-free space over length space is locally almost square but never almost square

We prove that the Lipschitz-free space over a metric space M is locally almost square whenever M is a length space. Consequently, the Lipschitz-free space is locally almost square if and only if it has the Daugavet property. We also show that a Lipschitz-free space is never almost square.

math.FA↗

arXiv2020

On the duality of the symmetric strong diameter $2$ property in Lipschitz spaces

We characterise the weak$^*$ symmetric strong diameter $2$ property in Lipschitz function spaces by a property of its predual, the Lipschitz-free space. We call this new property decomposable octahedrality and study its duality with the symmetric strong diameter $2$ property in general. For a Banach space to be decomposably octahedral it is sufficient that its dual space has the weak$^*$ symmetric strong diameter $2$ property. Whether it is also a necessary condition remains open.

math.FA↗

arXiv2019

Characterisation of the weak-star symmetric strong diameter 2 property in Lipschitz spaces

We give a characterisation of the weak* symmetric strong diameter 2 property for Lipschitz function spaces in terms of a property of the corresponding metric space. Using this characterisation we show that the weak* symmetric strong diameter 2 property is different from the weak* strong diameter 2 property in Lipschitz spaces, thereby answering a question posed in a recent paper by Haller, Langemets, Lima, and Nadel.

math.FA↗