Chain conditions in free Banach lattices
We show that for an arbitrary Banach space $E$, the free Banach lattice $FBL[E]$ generated by $E$ satisfies the $σ$-bounded chain condition.
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arXiv subjects
Grzegorz Plebanek
Publications and source records attributed to Grzegorz Plebanek.
Mardešić conjecture and free products of Boolean algebras
We show that for every $d\ge 1$, if $L_1,\ldots, L_d$ are linearly ordered compact spaces and there is a continuous surjection \[ L_1\times L_2\times \dots\times L_d\to K_1\times K_2\times\ldots\times K_{d}\times K_{d+1},\] where all the spaces $K_i$ are infinite, then $K_i, K_j$ are metrizable for some $1\le i<j\le d+1$. This answers a problem posed by S. Mardešić. We present some related results on Boolean algebras not containing free products with too many uncountable factors. In particular, we answer a problem on initial chain algebras that was posed by L. Baur and L. Heindorf.
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Extension operators and twisted sums of $c_0$ and $C(K)$ spaces
We investigate the following problem posed by Cabello Sanchéz, Castillo, Kalton, and Yost: Let $K$ be a nonmetrizable compact space. Does there exist a nontrivial twisted sum of $c_0$ and $C(K)$, i.e., does there exist a Banach space $X$ containing a non-complemented copy $Z$ of $c_0$ such that the quotient space $X/Z$ is isomorphic to $C(K)$? Using additional set-theoretic assumptions we give the first examples of compact spaces $K$ providing a negative answer to this question. We show that under Martin's axiom and the negation of the continuum hypothesis, if either $K$ is the Cantor cube $2^{ω_1}$ or $K$ is a separable scattered compact space of height $3$ and weight $ω_1$, then every twisted sum of $c_0$ and $C(K)$ is trivial. We also construct nontrivial twisted sums of $c_0$ and $C(K)$ for $K$ belonging to several classes of compacta. Our main tool is an investigation of pairs of compact spaces $K\subseteq L$ which do not admit an extension operator $C(K)\to C(L)$.
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Weakly Radon-Nikodým Boolean algebras and independent sequences
A compact space is said to be weakly Radon-Nikodým (WRN) if it can be weak*-embedded into the dual of a Banach space not containing $\ell_1$. We investigate WRN Boolean algebras, i.e. algebras whose Stone space is WRN compact. We show that the class of WRN algebras and the class of minimally generated algebras are incomparable. In particular, we construct a minimally generated nonWRN Boolean algebra whose Stone space is a separable Rosenthal compactum, answering in this way a question of W. Marciszewski. We also study questions of J. Rodríguez and R. Haydon concerning measures and the existence of nontrivial convergent sequences on WRN compacta, obtaining partial results on some natural subclasses.
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On properties of compacta that do not reflect in small continuous images
Assuming that there is a stationary set in $ω_{2}$ of ordinals of countable cofinality that does not reflect, we prove that there exists a compact space which is not Corson compact and whose all continuous images of weight at most $ω_1$ are Eberlein compacta. This yields an example of a Banach space of density $ω_{2}$ which is not weakly compactly generated but all its subspaces of density $ω_{1}$ are weakly compactly generated. We also prove that under Martin's axiom countable functional tightness does not reflect in small continuous images of compacta.
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Tukey classification of some ideals in $ω$ and the lattices of weakly compact sets in Banach spaces
We study the lattice structure of the family of weakly compact subsets of the unit ball $B_X$ of a separable Banach space $X$, equipped with the inclusion relation (this structure is denoted by $\mathcal{K}(B_X)$) and also with the parametrized family of almost inclusion relations $K \subseteq L+εB_X$, where $ε>0$ (this structure is denoted by $\mathcal{AK}(B_X)$). Tukey equivalence between partially ordered sets and a suitable extension to deal with $\mathcal{AK}(B_X)$ are used. Assuming the axiom of analytic determinacy, we prove that separable Banach spaces fall into four categories, namely: $\mathcal{K}(B_X)$ is equivalent either to a singleton, or to $ω^ω$, or to the family $\mathcal{K}(\mathbb{Q})$ of compact subsets of the rational numbers, or to the family $[\mathfrak{c}]^{<ω}$ of all finite subsets of the continuum. Also under the axiom of analytic determinacy, a similar classification of $\mathcal{AK}(B_X)$ is obtained. For separable Banach spaces not containing $\ell^1$, we prove in ZFC that $\mathcal{K}(B_X) \sim \mathcal{AK}(B_X)$ are equivalent to either $\{0\}$, $ω^ω$, $\mathcal{K}(\mathbb{Q})$ or $[\mathfrak{c}]^{<ω}$. The lattice structure of the family of all weakly null subsequences of an unconditional basis is also studied.
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Compactifications of $ω$ and the Banach space $c_0$
We investigate for which compactifications $γω$ of the discrete space of natural numbers $ω$, the natural copy of the Banach space $c_0$ is complemented in $C(γω)$. We show, in particular, that the separability of the remainder of $γω$ is neither sufficient nor necessary for $c_0$ being complemented in $C(γω)$ (for the latter our result is proved under the continuum hypothesis). We analyse, in this context, compactifications of $ω$ related to embeddings of the measure algebra into $P(ω)/fin$. We also prove that a Banach space $C(K)$ contains a rich family of complemented copies of $c_0$ whenever the compact space $K$ admits only measures of countable Maharam type.
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Measures on Suslinean spaces
We study the existence of non-separable compact spaces that support a measure and are small from the topological point of view. In particular, we show that under Martin's axiom there is a non-separable compact space supporting a measure which has countable $π$-character and which cannot be mapped continuously onto $[0,1]^{ω_1}$. On the other hand, we prove that in the random model there is no non-separable compact space having countable $π$-character and supporting a measure.
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A normal measure on a compact connected space
We present a construction of a compact connected space which supports a normal probability measure.
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Nonseparable growth of the integers supporting a measure
Assuming $\mathfrak b = \mathfrak c$ (or some weaker statement), we construct a compactification $γω$ of $ω$ such that its remainder $γω\setminusω$ is nonseparable and carries a strictly positive measure.
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Countable tightness in the spaces of regular probability measures
We prove that if $K$ is a compact space and the space $P(K\times K)$ of regular probability measures on $K\times K$ has countable tightness in its $weak^*$ topology, then $L_1(μ)$ is separable for every $μ\in P(K)$. It has been known that such a result is a consequence of Martin's axiom MA$(ω_1)$. Our theorem has several consequences; in particular, it generalizes a theorem due to Bourgain and Todorčević on measures on Rosenthal compacta.
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Representations of ideals in Polish groups and in Banach spaces
We investigate ideals of the form $\{A \subseteq ω\colon \sum_{n\in A} x_n$ is unconditionally convergent $\}$, where $(x_n)_{n\inω}$ is a sequence in a Polish group or in a Banach space. If an ideal on $ω$ can be seen in this form for some sequence in $X$, then we say that it is representable in $X$. After numerous examples we show the following theorems: (1) An ideal is representable in a Polish Abelian group iff it is an analytic P-ideal. (2) An ideal is representable in a Banach space iff it is a non-pathological analytic P-ideal. We focus on the family of ideals representable in $c_0$. We prove that the trace of the null ideal, Farah's ideal, and Tsirelson ideals are not representable in $c_0$, and that a tall $F_σ$ P-ideal is representable in $c_0$ iff it is a summable ideal. Also, we provide an example of a peculiar ideal which is representable in $\ell_1$ but not in $\mathbb{R}$. Finally, we summarize some open problems of this topic.
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On isomorphisms of Banach spaces of continuous functions
We prove that if $K$ and $L$ are compact spaces and $C(K)$ and $C(L)$ are isomorphic as Banach spaces then $K$ has a $π$-base consisting of open sets $U$ such that $\bar{U}$ is a continuous image of some compact subspace of $L$. This gives some information on isomorphic classes of the spaces of the form $C([0,1]^κ)$ and $C(K)$ where $K$ is Corson compact.
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On Borel structures in the Banach space C(βω)
M. Talagrand showed that, for the Cech-Stone compactification βω of the space of natural numbers, the norm and the weak topology generate different Borel structures in the Banach space C(βω). We prove that the Borel structures in C(βω) generated by the weak and the pointwise topology are also different. We also show that in C(ω*), where ω*=βω- ω, there is no countable family of pointwise Borel sets separating functions from C(ω*).
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On positive embeddings of C(K) spaces
We investigate isomorphic embeddings $T: C(K)\to C(L)$ between Banach spaces of continuous functions. We show that if such an embedding $T$ is a positive operator then $K$ is an image of $L$ under a upper semicontinuous set-function having finite values. Moreover we show that $K$ has a $π$-base of sets which closures a continuous images of compact subspaces of $L$. Our results imply in particular that if $C(K)$ can be positively embedded into $C(L)$ then some topological properties of $L$, such as countable tightness of Frechetness, pass to the space $K$. We show that some arbitrary isomorphic embeddings $C(K)\to C(L)$ can be, in a sense, reduced to positive embeddings.
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On Baire measurability in spaces of continuous functions
Let C(K) be the Banach space of all continuous functions on a given compact space K. We investigate the w*-sequential closure in C(K)* of the set of all finitely supported probabilities on K. We discuss the coincidence of the Baire sigma-algebras on C(K) associated to the weak and pointwise convergence topologies.
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A weak* separable C(K)* space whose unit ball is not weak* separable
We provide a ZFC example of a compact space K such that C(K)* is w*-separable but its closed unit ball is not w*-separable. All previous examples of such kind had been constructed under CH. We also discuss the measurability of the supremum norm on that C(K) equipped with its weak Baire sigma-algebra.
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Measurability in C(2^k) and Kunen cardinals
A cardinal k is called a Kunen cardinal if the sigma-algebra on k x k generated by all products AxB, coincides with the power set of k x k. For any cardinal k, let C(2^k) be the Banach space of all continuous real-valued functions on the Cantor cube 2^k. We prove that k is a Kunen cardinal if and only if the Baire sigma-algebra on C(2^k) for the pointwise convergence topology coincides with the Borel sigma-algebra on C(2^k) for the norm topology. Some other links between Kunen cardinals and measurability in Banach spaces are also given.
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