arXiv subjects

Jakub Rondoš

Publications and source records attributed to Jakub Rondoš.

12 recordsLinked to original sources

arXiv2026

A positive version of Miljutin's Theorem

In this paper we provide a positive version of Miljutin's theorem. Namely, we show that for any uncountable compact metric spaces $K$ and $L$ there exists a positive isomorphism $T:C(K)\longrightarrow C(L)$. Moreover, the inverse of $T$ is a regular operator, and $T$ may be chosen so that $\|T\|_\mathrm{r} \cdot \|T^{-1}\|_\mathrm{r} \leq 9+6\sqrt3$. The method also gives positive versions of classical nonmetrizable results for Miljutin and Dugundji compacta. In particular, if $G$ and $H$ are infinite compact groups of the same weight, then $C(G)$ and $C(H)$ are positively and regularly isomorphic with the same uniform quantitative estimate.

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arXiv2026

The classification of $C(K)$ spaces for countable compacta by positive isomorphisms

We study the classification of spaces of continuous functions $C(K)$ under positive linear maps. For infinite countable compacta, we show that whenever $C(K)$ and $C(L)$ are isomorphic, there exists an isomorphism $T:C(K)\to C(L)$ satisfying either $T\geq 0$ or $T^{-1}\geq 0$. We also prove that for any compact spaces $K$ and $L$, the existence of a positive embedding $T: C(K) \to C(L)$ implies that the Cantor--Bendixson height of $K$ does not exceed the height of $L$. Further, we introduce a one-sided positive Banach-Mazur distance and compute it in several families of the corresponding spaces of continuous functions. Our estimates also yield new exact values for the classical Banach-Mazur distance between such spaces.

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arXiv2026

Borel complexity of isometry classes of $\mathcal{C}(K)$ spaces with countable compacta

For every countable compact space $K$, we determine the exact Borel complexity of the isometry class of the Banach space $\mathcal{C}(K)$. As a byproduct, we also determine the precise Borel complexity of the homeomorphism class of a fixed countable compact space $K$, improving earlier results of Cenzer and Mauldin. The above results provide concrete and natural examples of sets with arbitrarily high, still exactly determined, Borel complexity. Moreover, we find a new characterization of those real $L_1$-preduals that are isometric to $\mathcal{C}(K)$ for some zero-dimensional compact space $K$ and we determine the precise Borel complexity of $\mathcal{C}(2^{\mathbb{N}})$.

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arXiv2026

Complementability of separable spaces $\mathcal{C}(K)$ in Banach spaces

For a metric compact space $L$ and a Banach space $E$, we provide a characterization of the complementability of the Banach space $\mathcal{C}(L)$ of continuous functions on $L$ inside $E$ in terms of the existence of a certain tree in the product $E \times E^*$, based on new descriptions of the Banach spaces $\mathcal{C}([1, ω^α])$ for countable ordinal numbers $α$ and $\mathcal{C}(2^ω)$. Applying this general result in the case where $E=\mathcal{C}(K)$ for some compact space $K$, we further obtain a characterization of the existence of a positively $1$-complemented positively isometric copy of $\mathcal{C}(L)$ inside $\mathcal{C}(K)$ in terms of the topology of $K$ and the space of probability Radon measures on $K$. In the process, we also prove a variant of the classical Holsztyński theorem for isometric embeddings onto complemented subspaces.

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arXiv2024

On embedding separable spaces $\mathcal{C}(L)$ in arbitrary spaces $\mathcal{C}(K)$

Supplementing and expanding classical results, for compact spaces $K$ and $L$, $L$ metric, and their Banach spaces $\mathcal{C}(L)$ and $\mathcal{C}(K)$ of continuous real-valued functions, we provide several characterizations of the existence of isometric, resp. isomorphic, embeddings of $\mathcal{C}(L)$ into $\mathcal{C}(K)$. In particular, we show that if the embedded space $\mathcal{C}(L)$ is separable, then the classical theorems of Holsztyński and Gordon become equivalences. We also obtain new results describing the relative cellularities of the perfect kernel of a given compact space $K$ and of the Cantor--Bendixson derived sets of $K$ of countable order in terms of the presence of isometric copies of specific spaces $\mathcal{C}(L)$ inside $\mathcal{C}(K)$.

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arXiv2024

Complemented subspaces of Banach spaces $C(K\times L)$

We prove that, for every compact spaces $K_1,K_2$ and compact group $G$, if both $K_1$ and $K_2$ map continuously onto $G$, then the Banach space $C(K_1 \times K_2)$ contains a complemented subspace isometric to the Banach space $C(G)$. Consequently, $C(K_1\times K_2)$ contains a complemented copy of $C([0,1])$ for every non-scattered $K_1,K_2$. Also, answering a question of Alspach and Galego, we get that $C(βω\timesβω)$ contains a complemented copy of $C([0,1]^κ)$ for every cardinal number $1\leκ\le{\mathfrak c}$ and hence a complemented copy of $C(K)$ for every metric compact space $K$. On the other hand, for the pointwise topology, we show that $C_p(βω\timesβω)$ contains no complemented copy of $C_p(2^ω)$.

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arXiv2023

Boundary integral representation of multipliers of fragmented affine functions and other intermediate function spaces

We develop a theory of abstract intermediate function spaces on a compact convex set $X$ and study the behaviour of multipliers and centers of these spaces. In particular, we provide some criteria for coincidence of the center with the space of multipliers and a general theorem on boundary integral representation of multipliers. We apply the general theory in several concrete cases, among others to strongly affine Baire functions, to the space $A_f(X)$ of fragmented affine functions, to the space $(A_f(X))^μ$, the monotone sequential closure of $A_f(X)$, to their natural subspaces formed by Borel functions, or, in some special cases, to the space of all strongly affine functions. In addition, we prove that the space $(A_f(X))^μ$ is determined by extreme points and provide a large number of illustrating examples and counterexamples.

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arXiv2023

On isomorphisms of $\mathcal{C}(K)$ spaces and cardinal invariants of derivatives of $K$

We present a necessary condition for a pair of $\mathcal{C}(K)$ spaces to be isomorphic in terms of topological properties of Cantor-Bendixon derivatives of $K$. This in particular gives a completely new information about the perfect kernels of such $K$. In the process, we extend known lower estimates of the Banach-Mazur distance between a pair of spaces of continuous functions from the case of scattered compact spaces to a more general setting. Next, we apply this general result to deduce some new information about isomorphisms of spaces of continuous functions over Eberlein compacta of height $ω+1$. Further, we show that isomorphisms of $\mathcal{C}(K)$ spaces preserve the spread of $K$, and we also prove some new results for pairs of spaces of continuous functions whose Banach-Mazur distance is less than $3$.

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arXiv2022

Isomorphisms of $\mathcal{C}(K, E)$ spaces and height of $K$

Let $K_1$, $K_2$ be compact Hausdorff spaces and $E_1, E_2$ be Banach spaces not containing a copy of $c_0$. We establish lower estimates of the Banach-Mazur distance between the spaces of continuous functions $\mathcal{C}(K_1, E_1)$ and $\mathcal{C}(K_2, E_2)$ based on the ordinals $ht(K_1)$, $ht(K_2)$, which are new even for the case of spaces of real valued functions on ordinal intervals. As a corollary we deduce that $\mathcal{C}(K_1, E_1)$ and $\mathcal{C}(K_2, E_2)$ are not isomorphic if $ht(K_1)$ is substantially different from $ht(K_2)$.

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arXiv2020

On the Banach-Mazur distance between continuous function spaces with scattered boundaries

We study the dependence of the Banach-Mazur distance between two subspaces of vector-valued continuous functions on the scattered structure of their boundaries. In the spirit of a result of Gordon, we show that the constant $2$ appearing in the Amir-Cambern theorem may be replaced by $3$ for some class of subspaces. This we achieve by showing that the Banach-Mazur distance of two function spaces is at least 3, if the height of the set of weak peak points of one of the spaces is larger than the height of a closed boundary of the second space. Next we show that this estimate can be improved, if the considered heights are finite and significantly different. As a corollary, we obtain new results even for the case of $\mathcal{C}(K, E)$ spaces.

math.FA↗

arXiv2020

An Amir-Cambern theorem for subspaces of Banach lattice-valued continuous functions

For $i=1,2$, let $E_i$ be a reflexive Banach lattice over $\mathbb{R}$ with a certain parameter $λ^+(E_i)>1$, let $K_i$ be a locally compact (Hausdorff) topological space and let $\mathcal{H}_i$ be a closed subspace of $\mathcal{C}_0(K_i, E_i)$ such that each point of the Choquet boundary $\mathcal{Ch}_{\mathcal{H}_i} K_i$ of $\mathcal{H}_i$ is a weak peak point. We show that if there exists an isomorphism $T\colon \mathcal{H}_1 \to \mathcal{H}_2$ with $\Vert T \Vert \cdot \Vert T^{-1} \Vert<\min \lbrace λ^+(E_1), λ^+(E_2) \rbrace$ such that $T$ and $T^{-1}$ preserve positivity, then $\mathcal{Ch}_{\mathcal{H}_1} K_1$ is homeomorphic to $\mathcal{Ch}_{\mathcal{H}_2} K_2$.

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arXiv2019

Isomorphisms of subspaces of vector-valued continuous functions

We deal with isomorphic Banach-Stone type theorems for closed subspaces of vector-valued continuous functions. Let $\mathbb{F}=\mathbb{R}$ or $\mathbb{C}$. For $i=1,2$, let $E_i$ be a reflexive Banach space over $\mathbb{F}$ with a certain parameter $λ(E_i)>1$, which in the real case coincides with the Schaffer constant of $E_i$, let $K_i$ be a locally compact (Hausdorff) topological space and let $\mathcal{H}_i$ be a closed subspace of $\mathcal{C}_0(K_i, E_i)$ such that each point of the Choquet boundary $\mathcal{Ch}_{\mathcal{H}_i} K_i$ of $\mathcal{H}_i$ is a weak peak point. We show that if there exists an isomorphism $T:\mathcal{H}_1 \rightarrow \mathcal{H}_2$ with $\Vert T \Vert \cdot \Vert T^{-1} \Vert<\min \lbrace λ(E_1), λ(E_2) \rbrace$, then $\mathcal{Ch}_{\mathcal{H}_1} K_1$ is homeomorphic to $\mathcal{Ch}_{\mathcal{H}_2} K_2$. Next we provide an analogous version of the weak vector-valued Banach-Stone theorem for subspaces, where the target spaces do not contain an isomorphic copy of $c_0$.

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