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Jiří Spurný

Publications and source records attributed to Jiří Spurný.

17 recordsLinked to original sources
arXiv2026

Operators on injective tensor products of L1-preduals

Let X be an L1-predual and E,F be Banach spaces. We use the fact that an unconditionally converging operator T from the injective tensor product of X and E to F is strongly bounded and extend T to an operator S on continuous F-valued functions on the dual unit ball of X with the preservation of properties of T. This procedure provides a unified approach for proving properties of the tensor product of X and E based on the properties of E.

math.FA↗
arXiv2025

Simpliciality of vector-valued function spaces

We investigate integral representation of vector-valued function spaces, i.e., of subspaces $H\subset C(K,E)$, where $K$ is a compact space and $E$ is a (real or complex) Banach space. We point out that there are two possible ways of generalizing representation theorems known from the scalar case -- either one may represent (all) functionals from $H^*$ using $E^*$-valued vector measures on $K$ (as it is done in the literature) or one may represent (some) operators from $L(H,E)$ by scalar measures on $K$ using the Bochner integral. These two ways lead to two different notions of simpliciality which we call `vector simpliciality' and `weak simpliciality'. It turns out that these two notions are in general incomparable. Moreover, the weak simpliciality is not affected by renorming the target space $E$, while vector simpliciality may be affected. Further, if $H$ contains constants, vector simpliciality is strictly stronger and admits several characterizations (partially analogous to the characterizations known in the scalar case). We also study orderings of measures inspired by C.J.K.~Batty which may be (in special cases) used to characterize $H$-boundary measures. Finally, we give a finer version of representation theorem using positive measures on $K\times B_{E^*}$ and characterize uniqueness in this case.

math.FA↗
arXiv2024

Transference of measures via disintegration

Given a compact space $K$ and a Banach space $E$ we study the structure of positive measures on the product space $K\times B_{E^*}$ representing functionals on $C(K,E)$, the space of $E$-valued continuous functions on $K$. Using the technique of disintegration we provide an alternative approach to the procedure of transference of measures introduced by Batty (1990). This enables us to substantially strengthen some of his results, to discover a rich order structure on these measures, to identify maximal and minimal elements and to relate them to the classical Choquet order.

math.FA↗
arXiv2024

On simpliciality of function spaces not containing constants

We investigate simpliciality of function spaces without constants. We prove, in particular, that several properties characterizing simpliciality in the classical case differ in this new setting. We also show that it may happen that a given point is not represented by any measure pseudosupported by the Choquet boundary, illustrating so limits of possible generalizations of the representation theorem. Moreover, we address the abstract Dirichlet problem in the new setting and establish some common points and nontrivial differences with the classical case.

math.FA↗
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.

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 $\mathcal{Ch}_{\mathcal{H}_2} K_2$.

math.FA↗
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 $c_0$.

math.FA↗
arXiv2018

The minimum principle for affine functions with the point of continuity property and isomorphisms of spaces of continuous affine function

Let X be a compact convex set and let ext X stand for the set of extreme points of X. We show that an affine function with the point of continuity property on X satisfies the minimum principle. As a corollary we obtain a generalization of a theorem by H.B. Cohen and C.H. Chu by proving the following result. Let X,Y be compact convex sets such that every extreme point of X and Y is a weak peak point and let the Banach-Mazur distance between spaces of affine continuous functions on X and Y is smaller then 2. Then ext X is homeomorphic to ext Y.

math.FA↗
arXiv2016

Baire classes of affine vector-valued functions

We investigate Baire classes of strongly affine mappings with values in Fréchet spaces. We show, in particular, that the validity of the vector-valued Mokobodzki's result on affine functions of the first Baire class is related to the approximation property of the range space. We further extend several results known for scalar functions on Choquet simplices or on dual balls of $L_1$-preduals to the vector-valued case. This concerns, in particular, affine classes of strongly affine Baire mappings, the abstract Dirichlet problem and the weak Dirichlet problem for Baire mappings. Some of these results have weaker conclusions than their scalar versions. We also establish an affine version of the Jayne-Rogers selection theorem.

math.FA↗
arXiv2015

Preserving affine Baire classes by perfect affine maps

Let $φ\colon X\to Y$ be an affine continuous surjection between compact convex sets. Suppose that the canonical copy of the space of real-valued affine continuous functions on $Y$ in the space of real-valued affine continuous functions on $X$ is complemented. We show that if $F$ is a topological vector space, then $f\colon Y\to F$ is of affine Baire class $α$ whenever the composition $f\circφ$ is of affine Baire class $α$. This abstract result is applied to extend known results on affine Baire classes of strongly affine Baire mappings.

math.FA↗
arXiv2014

Quantification of the Banach-Saks property

We investigate possible quantifications of the Banach-Saks property and the weak Banach-Saks property. We prove quantitative versions of relationships of the Banach-Saks property of a set with norm compactness and weak compactness. We further establish a quantitative version of the characterization of the weak Banach-Saks property of a set using uniform weak convergence and $\ell_1$-spreading models. We also study the case of the unit ball and in this case we prove a dichotomy which is an analogue of the James distortion theorem for $\ell_1$-spreading models.

math.FA↗
arXiv2014

Typical martingale diverges at a typical point

We investigate convergence of martingales adapted to a given filtration of finite $σ$-algebras. To any such filtration we associate a canonical metrizable compact space $K$ such that martingales adapted to the filtration can be canonically represented on $K$. We further show that (except for trivial cases) typical martingale diverges at a comeager subset of $K$. `Typical martingale' means a martingale from a comeager set in any of the standard spaces of martingales. In particular we show that a typical $L^1$-bounded martingale of norm at most one converges almost surely to zero and has maximal possible oscillation on a comeager set.

math.PR↗
arXiv2013

On quantitative Schur and Dunford-Pettis properties

We show that the dual to any subspace of $c_0(Γ)$ has the strongest possible quantitative version of the Schur property. Further, we establish relationship between the quantitative Schur property and quantitative versions of the Dunford-Pettis property. Finally, we apply these results to show, in particular, that any subspace of the space of compact operators on $\ell_p$ ($1<p<\infty$) with Dunford-Pettis property satisfies automatically both its quantitative versions.

math.FA↗
arXiv2012

Quantitative Dunford-Pettis property

We investigate possible quantifications of the Dunford-Pettis property. We show, in particular, that the Dunford-Pettis property is automatically quantitative in a sense. Further, there are two incomparable mutually dual stronger versions of a quantitative Dunford-Pettis property. We prove that $L^1$ spaces and $C(K)$ spaces posses both of them. We also show that several natural measures of weak non-compactness are equal in $L^1$ spaces.

math.FA↗
arXiv2011

On a difference between quantitative weak sequential completeness and the quantitative Schur property

We study quantitative versions of the Schur property and weak sequential completeness, proceeding thus with investigations started by G. Godefroy, N. Kalton and D. Li and continued by H. Pfitzner and the authors. We show that the Schur property of $\ell_1$ holds quantitatively in the strongest possible way and construct an example of a Banach space which is quantitatively weakly sequentially complete, has the Schur property but fails the quantitative form of the Schur property.

math.FA↗
arXiv2010

A quantitative version of James' compactness theorem

We introduce two measures of weak non-compactness $Ja_E$ and $Ja$ that quantify, via distances, the idea of boundary behind James' compactness theorem. These measures tell us, for a bounded subset $C$ of a Banach space $E$ and for given $x^*\in E^*$, how far from $E$ or $C$ one needs to go to find $x^{**}\in \bar{C}^{w^*}\subset E^{**}$ with $x^{**}(x^*)=\sup x^* (C)$. A quantitative version of James' compactness theorem is proved using $Ja_E$ and $Ja$, and in particular it yields the following result: {\it Let $C$ be a closed convex bounded subset of a Banach space $E$ and $r>0$. If there is an element $x_0^{**}$ in $\bar{C}^{w^*}$ whose distance to $C$ is greater than $r$, then there is $x^*\in E^*$ such that each $x^{**}\in\bar{C}^{w^*}$ at which $\sup x^*(C)$ is attained has distance to $E$ greater than $r/2$.} We indeed establish that $Ja_E$ and $Ja$ are equivalent to other measures of weak non-compactness studied in the literature. We also collect particular cases and examples showing when the inequalities between the different measures of weak non-compactness can be equalities and when the inequalities are sharp.

math.FA↗