arXiv subjects

Wiesław Śliwa

Publications and source records attributed to Wiesław Śliwa.

5 recordsLinked to original sources

arXiv2026

Reflexivity of Grothendieck Spaces and Strongly Normal Sequences

A normalized sequence $(y_n)$ in the dual $E^{*}$ of a Banach space $E$ is called normal if it is weak-star null, and strongly normal if [ G(y_n)=\left{x\in E:\sum_{n=1}^{\infty}|y_n(x)|<\infty\right} ] is dense in $E$. We prove that, within the class of Grothendieck Banach spaces, reflexivity is characterized by the subsequence behavior of normal sequences: a Grothendieck space $E$ is reflexive if and only if every normal sequence in $E^{*}$ contains a strongly normal subsequence. If $E$ is nonreflexive and Grothendieck, then $E^{*}$ contains a normalized weakly null sequence none of whose subsequences is strongly normal; more strongly, it contains a closed hereditarily $\ell_{1}$ subspace $Y$ with a normalized weakly null Schauder basis such that no sequence in $S(Y)$ is strongly normal. We also obtain an exact quotient criterion valid for arbitrary Banach spaces: a normal sequence has a strongly normal subsequence precisely when, for some subsequence $z=(z_n)$, the quotient $E/\bigcap_n\ker z_n$ is separable. The proofs combine a weak-star basic-sequence argument with results of Haydon, Johnson--Maurey--Schechtman and Popov. Concrete realizations are given in $(L_{\infty}(μ))^{*}$ for nonatomic probability spaces (and, more generally, for $σ$-finite spaces with a nonatomic part) and in $\ell_{\infty}^{*}$.

math.FA↗

arXiv2026

Continuous Linear Surjections from $C_p(X)$ onto Symmetric Sequence Ideals in $c_0$

Rosenthal's classical theorem says that, for every infinite compact space $X$, the Banach space $C(X)$ admits a quotient isomorphic to either $c_0$ or $\ell_2$. The corresponding question for $C_p(X)$, the space $C(X)$ endowed with the topology of pointwise convergence, is much subtler and still open; the only compact spaces for which the existence of an infinite-dimensional metrizable quotient is not presently settled in ZFC are Efimov compacta. We prove that the Banach space case cannot be reproduced with the usual sequence spaces carrying their pointwise topologies: Let $X$ be a Tychonoff space and let $E\subseteq c_0$ be a non-trivial symmetric sequence ideal endowed with the subspace topology inherited from $\mathbb{R}^{\mathbb{N}}$. Then the existence of a continuous linear surjection $T:C_p(X)\rightarrow E_p$ implies $E=c_0$, where $E_p$ means $E$ with the topology inherited from $\mathbb{R}^{\mathbb{N}}$. Hence, no proper non-zero symmetric sequence ideal of $c_0$ can be realized as a continuous linear image of a $C_p$-space. Combining this result with the characterization of the Josefson--Nissenzweig property for $C_p(X)$ obtained by Banakh, Kąkol, and Śliwa, we derive a complete characterization of all pairs $(X,E)$ for which such a surjection exists. In particular, for every $0<q<\infty$, there is no continuous linear surjection $C_p(X)\rightarrow(\ell_q)_p$.

math.FA↗

arXiv2026

The two-disjoint-copies property for compact spaces, homogeneity and connection with $C_p$-theory

A Tychonoff space $X$ has the two-disjoint-copies property (2DCP) if there exists a sequence $(K_n)_{n\inω}$ of non-empty compact subsets of $X$ such that each $K_n$ contains two disjoint subsets homeomorphic to $K_{n+1}$. Banakh, Kąkol and Śliwa showed that 2DCP yields an infinite-dimensional metrizable quotient of $C_p(X)$, while it is still a long-standing open question whether $C_p(X)$ has such a quotient for any infinite compact space $X$. The above concept as well as the last problem are closely related to Efimov's problem that has remained open for 40 years. We will discuss a number of conditions that imply 2DCP. For example, every locally homogeneous compact space, every space containing a copy of $βω$ or $2^ω$ has 2DCP although compact $h$-homogeneous spaces with 2DCP without such copies exist in ZFC. We prove that no scattered compact space has 2DCP and there exist in ZFC compact perfect spaces without 2DCP. This implies that for compact metric spaces $X$ the 2DCP is equivalent to uncountability of $X$. There exist explicit uncountable separable compact spaces failing 2DCP, for example the Isbell-Mrówka compacta. We give positive classes among zero-dimensional compact spaces; for example, the Brech, as well as the Sobota-Zdomskyy compact spaces of Efimov type have 2DCP. Open questions are included.

math.GN↗

arXiv2025

An Elementary Proof Of The Josefson-Nissenzweig Theorem For Banach Spaces C(KxL)

In [8] probabilistic methods, in particular a variant of the Weak Law of Large Numbers related to the Bernoulli distribution, have been used to show that for every infinite compact spaces K and L there exists a sequence $(μ_n)$ of normalized signed measures on $K\times L$ with finite supports which converges to $0$ with respect to the weak topology of the dual Banach space $C(K\times L).$ In this paper, we return to this construction, limiting ourselves only to elementary combinatorial calculus. The main efects of this construction are additional information about the measures $μ_n$, this is particularly clearly seen (among the others) in the resulting inequalities $$\frac{1}{2\sqrtπ}\frac{1}{\sqrt{n}} <\sup_{A\times B\subset X\times Y} |μ_n(A\times B)|<\frac{2}{\sqrtπ}\frac{1}{\sqrt{n}},$$ $n\in\mathbb{N}$, with $μ_n(f) \to_n 0$ for every $f\in C(X \times Y);$ where X and Y are arbitrary Tychonoff spaces containing infinite compact subsets, respectively. As an application we explicitly describe for Banach spaces $C(X\times Y)$ some complemented subspaces isomorphic to $c_0$. This result generalizes the classical theorem of Cembranos and Freniche, which states that for every infinite compact spaces K and L, the Banach space $C(K\times L)$ contains a complemented copy of the Banach space $c_0.$

math.FA↗

arXiv2025

Lotz-Peck-Porta and Rosenthal's theorems for spaces $C_p(X)$

For a Tychonoff space $X$ by $C_p(X)$ we denote the space $C(X)$ of continuous real valued functions on $X$ endowed with the pointwise topology. We prove that an infinite compact space $X$ is scattered if and only if every closed infinite-dimensional subspace in $C_p(X)$ contains a copy of $c_0$ (with the pointwise topology) which is complemented in the whole space $C_p(X)$. This provides a $C_p$-version of the theorem of Lotz, Peck and Porta for Banach spaces $C(X)$ and $c_0$. Applications will be provided. We prove also a $C_p$-version of Rosenthal's theorem by showing that for an infinite compact $X$ the space $C_p(X)$ contains a closed copy of $c_{0}(Γ)$ (with the pointwise topology) for some uncountable set $Γ$ if and only if $X$ admits an uncountable family of pairwise disjoint open subsets of $X$. Illustrating examples, additional supplementing $C_p$-theorems and comments are included.

math.FA↗