arXiv subjects
Publications and source records attributed to Grzegorz Plebanek.
Every almost disjoint family $A$ of infinite subsets of $ω$ gives rise to a scattered compact space $K_A$ and a family of Johnson-Lindenstrauss spaces JL$_p(A)$. We consider almost disjoint families arising from finitely branching trees and investigate whether they define homeomorphic compacta and isomorphic Banach spaces.
In this working note we study the one-sided positive Banach-Mazur distance between some pairs of $C(K)$ Banach spaces. Building on methods developed in [4], we solve, in particular, one of the problems posed in [2].
The classical Banach spaces $L_\infty[0,1]$ and $\ell_\infty$ are isomorphic. We present here some lower and upper bounds for their Banach-Mazur distance.
Assuming $\mathfrak p=\mathfrak c$, we show that for every Eberlein compact space $L$ of weight $\mathfrak c$ there exists a short exact sequence $0\to c_0\to X\to C(L)\to 0$, where the Banach space $X$ is not isomorphic to a $C(K)$-space.
We present several results providing lower bounds for the Banach-Mazur distance \[d_{BM}(C(K), C(L))\] between Banach spaces of continuous functions on compact spaces. The main focus is on the case where $C(L)$ represents the classical Banach space $c$ of convergent sequences. In particular, we obtain generalizations and refinements of recent results from \cite{GP24} and \cite{MP25}. Currently, it seems that one of the most interesting questions is when $K = [0, ω]$ is a convergent sequence with a limit and $L = [0,ω]\times 3$ consists of three convergent sequences. In this case, we obtain \[3.53125 \leq d_{BM}(C([0,ω]\times 3),C[0,ω]) \leq 3.87513\]
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 $C_p(2^ω)$.
Given a compact space $K$, we denote by $P(K)$ the space of all Radon probability measures on $K$, equipped with the $weak^\ast$ topology inherited from $C(K)^\ast$. For nonmetrizable compacta $K$ even basic properties of $P(K)$ spaces depend of additional axioms of set theory. We discuss here older and quite recent results on the subject.
For any cardinal number $κ$ and an index set $Γ$, $Σ_κ$-product of real lines consists of elements of ${\mathbb R}^Γ$ having $<κ$ nonzero coordinates. A compact space $K$ is $κ$-Corson compact if it can be embedded into such a space for some $Γ$. The class of ($ω_1$-)Corson compact spaces has been intensively studied over last decades. We discuss properties of $κ$-Corson compacta for various cardinal numbers $κ$ as well as properties of related Boolean algebras and spaces of continuous functions. We present here a detailed discussion of the class of $ω$-Corson compacta extending the results of Nakhmanson and Yakovlev. For $κ>ω$, our results on $κ$-Corson compact spaces are related to the line of research originated by Kalenda and Bell and Marciszewski, and continued by Bonnet, Kubis and Todorcevic in their recent paper.
We consider a separable compact line $K$ and its extension $L$ consisting of $K$ and a countable number of isolated points. The main object of study is the existence of a bounded extension operator $E: C(K)\to C(L)$. We show that if such an operator exists then there is one for which $\|E\|$ is an odd natural number. We prove that if the topological weight of $K$ is bigger than or equal to the least cardinality of a set $X \subseteq [0,1]$ that cannot be covered by a sequence of closed sets of measure zero then there is an extension $L$ of $K$ admitting no bounded extension operator.
We prove that there is a compact space $L$ and a 1-complemented subspace of the Banach space $C(L)$ which is not isomorphic to a space of continuous functions.
It follows from a theorem of Rosenthal that a compact space is $ccc$ if and only if every Eberlein continuous image is metrizable. Motivated by this result, for a class of compact spaces $\mathcal{C}$ we define its orthogonal $\mathcal{C}^\perp$ as the class of all compact spaces for which every continuous image in $\mathcal{C}$ is metrizable. We study how this operation relates classes where centeredness is scarce with classes where it is abundant (like Eberlein and $ccc$ compacta), and also classes where independence is scarce (most notably weakly Radon-Nikodým compacta) with classes where it is abundant. We study these problems for zero-dimensional compact spaces with the aid of Boolean algebras and show the main difficulties arising when passing to the general setting. Our main results are the constructions of several relevant examples.
We present a connected version of the compact $L$-space constructed by Kenneth Kunen under CH. We show that this provides a Corson compact space $K$ such that the Banach space $C(K)$ is isomorphic to no space of continuous function on a zero-dimensional compactum.
For a compact space $K$ we consider the space $P(K)$, of probability regular Borel measures on $K$, equipped with the $weak^\ast$ topology inherited from $C(K)^\ast$. We discuss possible characterizations of those compact spaces $K$ for which $P(K)$ is $\aleph_0$-monolithic. The main result states that under $\diamondsuit$ there exists a nonseparable Corson compact space $K$ such that $P(K)$ is $\aleph_0$-monolithic but $K$ supports a measure of uncountable type.
We consider the class of Banach space $Y$ for which $c_0$ admits a nontrivial twisted sum with $Y$. We present a characterization of such space $Y$ in terms of properties of the $weak^\ast$ topology on $Y^\ast$. We prove that under the continuum hypothesis $c_0$ has a nontrivial twisted sum with every space of the form $Y=C(K)$, where $K$ is compact and not metrizable. This gives a consistent positive solution to a problem posed by Cabello, Castillo, Kalton and Yost.
We discuss three problems of Koszmider on the structure of the spaces of continuous functions on the Stone compact $K_{\mathcal A}$ generated by an almost disjoint family $\mathcal A$ of infinite subsets of $ω$ -- we present a solution to two problems and develop a previous results of Marciszewski and Pol answering the third one. We will show, in particular, that assuming Martin's axiom the space $C(K_{\mathcal A})$ is uniquely determined up to isomorphism by the cardinality of $\mathcal A$ whenever $|{\mathcal A}|<{\mathfrak c}$, while there are $2^{\mathfrak c}$ nonisomorphic spaces $C(K_{\mathcal A})$ with $|{\mathcal A}|= {\mathfrak c}$. We also investigate Koszmider's problems in the context of the class of separable Rosenthal compacta and indicate the meaning of our results in the language of twisted sums of $c_0$ and some $C(K)$ spaces.
We introduce the notion that a zero-dimensional compact space $L$ is a \emph{Boolean image} of an arbitrary compact space $K$. When $K$ is also zero-dimensional, this just means that $L$ is a continuous image of $K$. However, a number of interesting questions arise when we consider connected compacta $K$.
We investigate reflection-type problems on the class SPM, of Boolean algebras carrying strictly positive finitely additive measures. We show, in particular, that in the constructible universe there is a Boolean algebra $\mathfrak A$ which is not in SPM but every subalgebra of $\mathfrak A$ of cardinality $\mathfrak c$ admits a strictly positive measure. This result is essentially due to Farah and Velickovic.
We prove that if a compact line is fragmentable, then it is a Radon-Nikodým compact space.