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1
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The Dirichlet problem for Baire-one functions

100%
Spurný J.
Open Mathematics
|
2004
|
tom 2
|
nr 2
260-271
EN
Let X be a compact convex set and let ext X stand for the set of all extreme points of X. We characterize those bounded function defined on ext X which can be extended to an affine Baire-one function on the whole set X.
2
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Distances to spaces of affine Baire-one functions

100%
Spurný J.
Studia Mathematica
|
2010
|
tom 199
|
nr 1
23-41
EN
Let E be a Banach space and let $ℬ₁(B_{E*})$ and $𝔄₁(B_{E*})$ denote the space of all Baire-one and affine Baire-one functions on the dual unit ball $B_{E*}$, respectively. We show that there exists a separable L₁-predual E such that there is no quantitative relation between $dist(f,ℬ₁(B_{E*}))$ and $dist(f,𝔄₁(B_{E*}))$, where f is an affine function on $B_{E*}$. If the Banach space E satisfies some additional assumption, we prove the existence of some such dependence.
3
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$F_{σ}$-additive covers of Čech complete and scattered-K-analytic spaces

100%
Spurný J.
Fundamenta Mathematicae
|
2008
|
tom 199
|
nr 2
131-138
EN
We prove that an $F_{σ}$-additive cover of a Čech complete, or more generally scattered-K-analytic space, has a σ-scattered refinement. This generalizes results of G. Koumoullis and R. W. Hansell.
4
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Descriptive properties of elements of biduals of Banach spaces

64%
Ludvík P., Spurný J.
Studia Mathematica
|
2012
|
tom 209
|
nr 1
71-99
EN
If E is a Banach space, any element x** in its bidual E** is an affine function on the dual unit ball $B_{E*}$ that might possess a variety of descriptive properties with respect to the weak* topology. We prove several results showing that descriptive properties of x** are quite often determined by the behaviour of x** on the set of extreme points of $B_{E*}$, generalizing thus results of J. Saint Raymond and F. Jellett. We also prove a result on the relation between Baire classes and intrinsic Baire classes of L₁-preduals which were introduced by S. A. Argyros, G. Godefroy and H. P. Rosenthal (2003). Also, several examples witnessing natural limits of our positive results are presented.
5
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$F_{σ}$-mappings and the invariance of absolute Borel classes

64%
Holický P., Spurný J.
Fundamenta Mathematicae
|
2004
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tom 182
|
nr 3
193-204
EN
It is proved that $F_{σ}$-mappings preserve absolute Borel classes, which improves results of R. W. Hansell, J. E. Jayne and C. A. Rogers. The proof is based on the fact that any $F_{σ}$-mapping f: X → Y of an absolute Suslin metric space X onto an absolute Suslin metric space Y becomes a piecewise perfect mapping when restricted to a suitable $F_{σ}$-set $X_{∞} ⊂ X$ satisfying $f(X_{∞}) = Y$.
6
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Quantification of the reciprocal Dunford-Pettis property

64%
Kalenda O., Spurný J.
Studia Mathematica
|
2012
|
tom 210
|
nr 3
261-278
EN
We prove in particular that Banach spaces of the form C₀(Ω), where Ω is a locally compact space, enjoy a quantitative version of the reciprocal Dunford-Pettis property.
7
Content available remote

Baire classes of affine vector-valued functions

64%
Kalenda O., Spurný J.
Studia Mathematica
|
2016
|
tom 233
|
nr 3
227-277
EN
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 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 L₁-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.
8
Content available remote

Affine Baire functions on Choquet simplices

64%
Kačena M., Spurný J.
Open Mathematics
|
2011
|
tom 9
|
nr 1
127-138
EN
We construct a metrizable simplex X such that for each n ɛ ℕ there exists a bounded function f on ext X of Baire class n that cannot be extended to a strongly affine function of Baire class n. We show that such an example cannot be constructed via the space of harmonic functions.
9
Content available remote

Uniqueness of Cartesian Products of Compact Convex Sets

51%
Lipecki Z., Losert V., Spurný J.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2011
|
tom 59
|
nr 2
175-183
EN
Let $X_i$, i∈ I, and $Y_j$, j∈ J, be compact convex sets whose sets of extreme points are affinely independent and let φ be an affine homeomorphism of $∏_{i∈ I} X_i$ onto $∏_{j∈ J} Y_j$. We show that there exists a bijection b: I → J such that φ is the product of affine homeomorphisms of $X_i$ onto $Y_{b(i)}$, i∈ I.
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