arXiv subjects

Natalia Maślany

Publications and source records attributed to Natalia Maślany.

4 recordsLinked to original sources

arXiv2026

Raja's covering index of $L_p$ spaces

We study Raja's covering index $Θ_X(n)$ for classical $L_p$-spaces and their non-commutative counterparts. For infinite-dimensional Hilbert spaces we compute the covering index exactly, proving \[ Θ_H(n)=n^{-1/2}\qquad(n\in\mathbb N); \] in particular $Θ_H(2)=1/\sqrt2$, thus answering a question of Raja about the precise two-piece covering index of $\elltwo$. For scalar-valued Lebesgue spaces $L_p(μ)$, $1\le p<\infty$, we construct an explicit block decomposition of the unit ball yielding the upper bound $Θ_{L_p(μ)}(n)\le n^{-1/p}$ for all $n\in\mathbb{N}$; in particular $Θ_{\ell_p}(n)\le n^{-1/p}$. For $1<p<\infty$, under the corresponding $p$-AUS renormability hypothesis, this combines with Raja's general lower bound to give the sharp asymptotic estimate $Θ_{L_p(μ)}(n)\asymp n^{-1/p}$. We also obtain uniform upper bounds $Θ_{L_p(μ;E)}(n)\le n^{-1/p}$ for Bochner spaces $L_p(μ;E)$ over non-atomic $σ$-finite measure spaces, with constants independent of the Banach space $E$; this shows that, at the level of power-type upper estimates, the covering index decays at the same rate regardless of the asymptotic geometry of~$E$ and provides a partial negative answer to a problem of Raja. Finally, using non-commutative Clarkson inequalities, we derive power-type lower bounds $Θ_{L_p(M,τ)}(n)\gtrsim n^{-1/r}$ for non-commutative $L_p(M,τ)$ spaces associated with semifinite von Neumann algebras, where $r=\min\{p,2\}$. We do not attempt to optimise the exponent or constants in the non-commutative setting.

math.FA↗

arXiv2024

Differential embeddings into algebras of topological stable rank 1

We identify a class of smooth Banach *-algebras that are differential subalgebras of commutative C*-algebras whose openness of multiplication is completely determined by the topological stable rank of the target C*-algebra. We then show that group algebras of Abelian groups of unbounded exponent fail to have uniformly open convolution. Finally, we completely characterise in the complex case (uniform) openness of multiplication in algebras of continuous functions in terms of the covering dimension.

math.OA↗

arXiv2023

On isometries and Tingley's problem for the spaces $T[θ, S_α], 1 \leq α<ω_{1}$

We extend the existing results on surjective isometries of unit spheres in the Tsirelson space $T\left[\frac{1}{2}, S_1\right]$ to the class $T[θ,S_α]$ for any integer $θ^{-1} \geq 2$ and $1 \leqslant α< ω_1$, where $S_α$ denotes the Schreier family of order $α$. This positively answers Tingley's problem for these spaces, which asks whether every surjective isometry between unit spheres can be extended to a surjective linear isometry of the entire space. Furthermore, we improve the result stating that every linear isometry on $T[θ, S_1]$ ($θ\in \left(0, \frac{1}{2}\right]$) is determined by a permutation of the first $\lceil θ^{-1} \rceil$ elements of the canonical unit basis, followed by a possible sign change of the corresponding coordinates and a sign change of the remaining coordinates. Specifically, we prove that only the first $\lfloor θ^{-1} \rfloor$ elements can be permuted. This finding enables us to establish a sufficient condition for being a linear isometry in these spaces.

math.FA↗

arXiv2023

Isometries of combinatorial Tsirelson spaces

We extend existing results that characterize isometries on the Tsirelson-type spaces $T\big[\frac{1}{n}, \mathcal{S}_1\big]$ ($n\in \mathbb{N}, n\geq 2$) to the class $T[θ, \mathcal{S}_α]$ ($θ\in \big(0, \frac{1}{2}\big]$, $1\leqslant α< ω_1$), where $\mathcal{S}_α$ denote the Schreier families of order $α$. We prove that every isometry on $T[θ, \mathcal{S}_1]$ ($θ\in \big(0, \frac{1}{2}\big]$) is determined by a permutation of the first $\lceil θ^{-1} \rceil$ elements of the canonical unit basis followed by a possible sign-change of the corresponding coordinates together with a sign-change of the remaining coordinates. Moreover, we show that for the spaces $T[θ, \mathcal{S}_α]$ ($θ\in \big(0, \frac{1}{2}\big]$, $2\leqslant α< ω_1$) the isometries exhibit a more rigid character, namely, they are all implemented by a sign-change operation of the vector coordinates.

math.FA↗