arXiv subjects

Christina Brech

Publications and source records attributed to Christina Brech.

12 recordsLinked to original sources

arXiv2025

Isometries of James-Schreier and Lorentz spaces

We study the group of surjective linear isometries on certain real Banach sequence spaces using the preservation of extreme points in the closed unit ball. Our main result provides a characterization of the extreme points of the dual unit ball of the James-Schreier space $V_1$. As a consequence, we show that the only isometries on $V_1$ are $\pm Id$. We also obtain a Banach-Stone-type result for Lorentz sequence spaces, analogous to one proved in \cite{CarothersIsometrisLorentz} for Lorentz function spaces.

math.FA↗

arXiv2024

Density cardinals

How many permutations are needed so that every infinite-coinfinite set of natural numbers with asymptotic density can be rearranged to no longer have the same density? We prove that the density number $\mathfrak{dd}$, which answers this question, is equal to the least size of a non-meager set of reals, $\mathsf{non} (\mathcal{M})$. The same argument shows that a slight modification of the rearrangement number $\mathfrak{rr}$ of~\cite{BBBHHL20} is equal to $\mathsf{non} (\mathcal{M})$, and similarly for a cardinal invariant related to large-scale topology introduced by Banakh~\cite{Ba23}, thus answering a question of the latter. We then consider variants of $\mathfrak{dd}$ given by restricting the possible densities of the original set and / or of the permuted set, providing lower and upper bounds for these cardinals and proving consistency of strict inequalities. We finally look at cardinals defined in terms of relative density and of asymptotic mean, and relate them to the rearrangement numbers of~\cite{BBBHHL20}.

math.LO↗

arXiv2020

Banach-Stone-like results for combinatorial Banach spaces

We show that under a certain topological assumption on two compact hereditary families $\F$ and $\G$ on some infinite cardinal $κ$, the corresponding combinatorial spaces $X_\F$ and $X_\G$ are isometric if and only if there is a permutation of $κ$ inducing a homeomorphism between $\F$ and $\G$. We also prove that two different regular families $\F$ and $\G$ on $ω$ cannot be permuted one to the other. Both these results strengthen the main result of \cite{BrechFerencziTcaciuc}.

math.FA↗

arXiv2017

Homogeneous families on trees and subsymmetric basic sequences

We study density requirements on a given Banach space that guarantee the existence of subsymmetric basic sequences by extending Tsirelson's well-known space to larger index sets. We prove that for every cardinal $κ$ smaller than the first Mahlo cardinal there is a reflexive Banach space of density $κ$ without subsymmetric basic sequences. As for Tsirelson's space, our construction is based on the existence of a rich collection of homogeneous families on large index sets for which one can estimate the complexity on any given infinite set. This is used to describe detailedly the asymptotic structure of the spaces. The collections of families are of independent interest and their existence is proved inductively. The fundamental stepping up argument is the analysis of such collections of families on trees.

math.FA↗

arXiv2015

An isometrically universal Banach space induced by a non-universal Boolean algebra

Given a Boolean algebra $A$, we construct another Boolean algebra $B$ with no uncountable well-ordered chains such that the Banach space of real valued continuous functions $C(K_A)$ embeds isometrically into $C(K_B)$, where $K_A$ and $K_B$ are the Stone spaces of $A$ and $B$ respectively. As a consequence we obtain the following: If there exists an isometrically universal Banach space for the class of Banach spaces of a given uncountable density $κ$, then there is another such space which is induced by a Boolean algebra which is not universal for Boolean algebras of cardinality $κ$. Such a phenomenon cannot happen on the level of separable Banach spaces and countable Boolean algebras. This is related to the open question if the existence of an isometrically universal Banach space and of a universal Boolean algebra are equivalent on the nonseparable level (both are true on the separable level).

math.FA↗

arXiv2012

$\ell_\infty$-sums and the Banach space $\ell_\infty/c_0$

This paper is concerned with the isomorphic structure of the Banach space $\ell_\infty/c_0$ and how it depends on combinatorial tools whose existence is consistent but not provable from the usual axioms of ZFC. Our main global result is that it is consistent that $\ell_\infty/c_0$ does not have an orthogonal \break $\ell_\infty$-decomposition that is, it is not of the form $\ell_\infty(X)$ for any Banach space $X$. The main local result is that it is consistent that $\ell_\infty(c_0(\mathfrak{c}))$ does not embed isomorphically into $\ell_\infty/c_0$, where $\mathfrak{c}$ is the cardinality of the continuum, while $\ell_\infty$ and $c_0(\mathfrak{c})$ always do embed quite canonically. This should be compared with the results of Drewnowski and Roberts that under the assumption of the continuum hypothesis $\ell_\infty/c_0$ is isomorphic to its $\ell_\infty$-sum and in particular it contains an isomorphic copy of all Banach spaces of the form $\ell_\infty(X)$ for any subspace $X$ of $\ell_\infty/c_0$.

math.FA↗

arXiv2011

On universal spaces for the class of Banach spaces whose dual balls are uniform Eberlein compacts

For k being the first uncountable cardinal w_1 or k being the cardinality of the continuum c, we prove that it is consistent that there is no Banach space of density k in which it is possible to isomorphically embed every Banach space of the same density which has a uniformly Gâteaux differentiable renorming or, equivalently, whose dual unit ball with the weak* topology is a subspace of a Hilbert space (a uniform Eberlein compact space). This complements a consequence of results of M. Bell and of M. Fabian, G. Godefroy, V. Zizler that assuming the continuum hypothesis, there is a universal space for all Banach spaces of density k=c=w_1 which have a uniformly Gâteaux differentiable renorming. Our result implies, in particular, that βN-N may not map continuously onto a compact subset of a Hilbert space with the weak topology of density k=w_1 or k=c and that a C(K) space for some uniform Eberlein compact space K may not embed isomorphically into l_\infty/c_0.

math.FA↗

arXiv2010

A Boolean algebra and a Banach space obtained by push-out iteration

Under the assumption that the continuum c is a regular cardinal, we prove the existence and uniqueness of a Boolean algebra B of size c defined by sharing the main structural properties that P(N)/fin has under CH and in the aleph2-Cohen model. We prove a similar result in the category of Banach spaces.

math.LO↗

arXiv2010

On universal Banach spaces of density continuum

We consider the question whether there exists a Banach space $X$ of density continuum such that every Banach space of density not bigger than continuum isomorphically embeds into $X$ (called a universal Banach space of density $\cc$). It is well known that $\ell_\infty/c_0$ is such a space if we assume the continuum hypothesis. However, some additional set-theoretic assumption is needed, as we prove in the main result of this paper that it is consistent with the usual axioms of set-theory that there is no universal Banach space of density $\cc$. Thus, the problem of the existence of a universal Banach space of density $\cc$ is undecidable using the usual axioms of set-theory. We also prove that it is consistent that there are universal Banach spaces of density $\cc$, but $\ell_\infty/c_0$ is not among them. This relies on the proof of the consistency of the nonexistence of an isomorphic embedding of $C([0,\cc])$ into $\ell_\infty/c_0$.

math.FA↗

arXiv2010

On biorthogonal systems whose functionals are finitely supported

We show that for each natural $n>1$ it is consistent that there is a compact Hausdorff space $K_{2n}$ such that in $C(K_{2n})$ there is no uncountable (semi)biorthogonal sequence $(f_ξ,μ_ξ)_{ξ\in ω_1}$ where $μ_ξ$'s are atomic measures with supports consisting of at most $2n-1$ points of $K_{2n}$, but there are biorthogonal systems $(f_ξ,μ_ξ)_{ξ\in ω_1}$ where $μ_ξ$'s are atomic measures with supports consisting of $2n$ points. This complements a result of Todorcevic that it is consistent that each nonseparable Banach space $C(K)$ has an uncountable biorthogonal system where the functionals are measures of the form $δ_{x_ξ}-δ_{y_ξ}$ for $ξ<ω_1$ and $x_ξ,y_ξ\in K$. It also follows that it is consistent that the irredundance of the Boolean algebra $Clop(K)$ or the Banach algebra $C(K)$ for $K$ totally disconnected can be strictly smaller than the sizes of biorthogonal systems in $C(K)$. The compact spaces exhibit an interesting behaviour with respect to known cardinal functions: the hereditary density of the powers $K_{2n}^k$ is countable up to $k=n$ and it is uncountable (even the spread is uncountable) for $k>n$.

math.FA↗

arXiv2010

On the density of Banach {$C(K)$} spaces with the Grothendieck property

Using the method of forcing we prove that consistently there is a Banach space of continuous functions on a compact Hausdorff space with the Grothendieck property and with density less than the continuum. It follows that the classical result stating that ``no nontrivial complemented subspace of a Grothendieck $C(K)$ space is separable'' cannot be strengthened by replacing ``is separable'' by ``has density less than that of $l_\infty$'', without using an additional set-theoretic assumption. Such a strengthening was proved by Haydon, Levy and Odell, assuming Martin's axiom and the negation of the continuum hypothesis. Moreover, our example shows that certain separation properties of Boolean algebras are quite far from the Grothendieck property.

math.FA↗

arXiv2010

Thin-very tall compact scattered spaces which are hereditarily separable

We strengthen the property $Δ$ of a function $f:[ω_2]^2\rightarrow [ω_2]^{\leq ω}$ considered by Baumgartner and Shelah. This allows us to consider new types of amalgamations in the forcing used by Rabus, Juhász and Soukup to construct thin-very tall compact scattered spaces. We consistently obtain spaces $K$ as above where $K^n$ is hereditarily separable for each $n\in\N$. This serves as a counterexample concerning cardinal functions on compact spaces as well as having some applications in Banach spaces: the Banach space $C(K)$ is an Asplund space of density $\aleph_2$ which has no Fréchet smooth renorming, nor an uncountable biorthogonal system.

math.FA↗