arXiv subjects

Piotr Koszmider

Publications and source records attributed to Piotr Koszmider.

At least 19 recordsLinked to original sources

arXiv2026

Small masas of the Calkin algebra in the Cohen model

We show that maximal abelian C*-subalgebras (masas) of the Calkin algebra (the algebra of all bounded operators on the separable Hilbert space modulo compact operators) may consistently have their densities strictly less than continuum and we describe many isomorphism types of such masas. Specifically, we prove that after adding any number of Cohen reals to a model of CH the algebra $C(K_{\mathcal A})$ of all complex-valued continuous functions on the Stone space $K_{\mathcal A}$ of a Boolean algebra $\mathcal A$ of cardinality $ω_1$ is $*$-isomorphic to a masa of the Calkin algebra if and only if $\mathcal A$ does not admit a countably generated ultrafilter. Moreover, for every such Boolean algebra we obtain $ω_2$ pairwise unitarily non-equivalent such masas, none of which has a commutative lift. We also show in ZFC that if a C*-algebra of the form $C(K)$ for any compact Hausdorff $K$ is $*$-isomorphic to a masa of the Calkin algebra, then no point of $K$ may have character smaller than $\mathfrak p$. Therefore, consistently, there may not be any masa of the Calkin algebra of density less than continuum.

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arXiv2026

On masas of the Calkin algebra generated by projections

Assuming the continuum hypothesis CH, we obtain complete $*$-isomorphic classification of maximal abelian self-adjoint subalgebras (masas) of the Calkin algebra $\mathcal Q(\ell_2)$ (bounded operators on a separable Hilbert space modulo compact operators) generated by projections. In particular, for any compact totally disconnected Hausdorff space $K$ of weight not exceeding the continuum and not admitting $G_δ$ points we construct under CH a masa of $\mathcal Q(\ell_2)$ which is $*$-isomorphic to the algebra $C(K)$ of complex-valued continuous functions on $K$. This, among others, shows that masas of the Calkin algebra could have rather unexpected properties compared to the previously known three $*$-isomorphic types of them generated by projections: $\ell_\infty/c_0$, $L_\infty$ and $\ell_\infty/c_0\oplus L_\infty$. It can be shown that some additional set-theoretic hypothesis, like CH, is necessary for such results. However, without making any additional set-theoretic assumptions we still construct a family of maximal possible cardinality (of the power set of $\mathbb R$) of pairwise non-$*$-isomorphic masas of $\mathcal Q(\ell_2)$ generated by projections and with properties unlike the three above examples.

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arXiv2026

Counting spaces of functions on separable compact lines

We investigate the following general problem, closely related to the problem of isomorphic classification of Banach spaces $C(K)$ of continuous real-valued functions on a compact space $K$, equipped with the supremum norm: Let $\mathcal{K}$ be a class of compact spaces. How many isomorphism types of Banach spaces $C(K)$ are there, for $K\in \mathcal{K}$? We prove that for any uncountable regular cardinal number $κ$, there exist exactly $2^κ$ isomorphism types of spaces $C(K)$ for compact spaces of weight $κ$. We show that, for the class $\mathcal{L}_{ω_1}$ of separable compact linearly ordered spaces of weight $ω_1$, the answer to the above question depends on additional set-theoretic axioms. In particular, assuming the continuum hypothesis, there are $2^{ω_1}$ isomorphism types of $C(L)$, for $L\in \mathcal{L_{ω_1}}$, and assuming a certain axiom proposed by Baumgartner, there is only one type.

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arXiv2025

Almost disjoint families and some automorphic and injective properties of $\ell_\infty/c_0$

Answering questions of A. Avilés, F. Cabello Sánchez, J. Castillo, M. González and Y. Moreno we show that the following statements are independent of the usual axioms ZFC with arbitrarily large continuum: for every (some) $ω<κ<2^ω$ (1) any linear bounded operator $T: c_0(κ)\rightarrow\ell_\infty/c_0$ extends to any superspace of $c_0(κ)$. (2) any isomorphism between any two copies of $c_0(κ)$ inside $\ell_\infty/c_0$ extends to an automorphism of $\ell_\infty/c_0$. This contrasts with Boolean, Banach algebraic or isometric levels, where the objects known as Hausdorff gap and Luzin gap witness the failure in ZFC of the corresponding properties for the corresponding structures already at the first uncountable cardinal $κ=ω_1$. In particular, consistently, any two pairwise disjoint families in $\wp(\mathbb N)/Fin$ of the same cardinality $ω<κ<2^ω$ can be mapped onto each other by a linear automorphism of $\ell_\infty/c_0$ regardless of their different combinatorial, algebraic or topological positions in $\wp(\mathbb N)/Fin$. Our positive consistency results use a restricted version of Martin's axiom for a partial order that adds an infinite block diagonal matrix of an operator on $\ell_\infty$ which induces an operator on $\ell_\infty/c_0$. The construction of its finite blocks relies on a lemma of Bourgain and Tzafriri on finite dimensional Banach spaces. Our negative consistency results rely on an analysis of almost disjoint families of $\mathbb N$, the embeddings of $c_0(κ)$ into $\ell_\infty/c_0$ they induce and their extensions to $\ell_\infty^c(κ)$.

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arXiv2025

On Subspaces of Indecomposable Banach Spaces

We address the following question: what is the class of Banach spaces isomorphic to subspaces of indecomposable Banach spaces? We show that this class includes all Banach spaces of density not bigger than the continuum which do not admit $\ell_\infty$ as a quotient (equivalently do not admit a subspace isomorphic to $\ell_1(\cc)$). This includes all Asplund spaces and all weakly Lindelöf determined Banach spaces of density not bigger than the continuum. However, we also show that this class includes some Banach spaces admitting $\ell_\infty$ as a quotient. This sheds some light on the question asked in [S. Argyros, R. Haydon, \emph{Bourgain-Delbaen $L^\infty$-spaces, the scalar-plus-compact property and related problems}, Proceedings of the International Congress of Mathematicians (ICM 2018), Vol. III, 1477--1510. Page 1502] whether all Banach spaces not containing $\ell_\infty$ embed in some indecomposable Banach spaces. Our method of constructing indecomposable Banach spaces above a given Banach space is a considerable modification of the method of constructing Banach spaces of continuous functions with few$^*$ operators developed before by the first-named author.

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arXiv2024

Products of C*-algebras that do not embed into the Calkin algebra

We consider the Calkin algebra $\mathcal{Q}(\ell_2)$, i.e., the quotient of the algebra $\mathcal B(\ell_2)$ of all bounded linear operators on the separable Hilbert space $\ell_2$ divided by the ideal $\mathcal K(\ell_2)$ of all compact operators on $\ell_2$. We show that in the Cohen model of set theory ZFC there is no embedding of the product $(c_0(2^ω))^{\mathbb{N}}$ of infinitely many copies of the abelian C*-algebra $c_0(2^ω)$ into $\mathcal{Q}(\ell_2)$ (while $c_0(2^ω)$ always embeds into $\mathcal{Q}(\ell_2)$). This enlarges the collection of the known examples due to Vaccaro and to McKenney and Vignati of abelian algebras, asymptotic sequence algebras, reduced products and coronas of stabilizations which consistently do not embed into the Calkin algebra. As in the Cohen model the rigidity of quotient structures fails in general, our methods do not rely on these rigidity phenomena as is the case of most examples mentioned above. The results should be considered in the context of the result of Farah, Hirshberg and Vignati which says that consistently all C*-algebras of density up to $2^ω$ do embed into $\mathcal{Q}(\ell_2)$. In particular, the algebra $(c_0(2^ω))^{\mathbb{N}}$ consistently embeds into the Calkin algebra as well.

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arXiv2024

On Ramsey-type properties of the distance in nonseparable spheres

Given an uncountable subset $\mathcal Y$ of a nonseparable Banach space, is there an uncountable $\mathcal Z\subseteq \mathcal Y$ such that the distances between any two distinct points of $\mathcal Z$ are more or less the same? If an uncountable subset $\mathcal Y$ of a nonseparable Banach space does not admit an uncountable $\mathcal Z\subseteq \mathcal Y$, where any two points are distant by more than $r>0$, is it because $\mathcal Y$ is the countable union of sets of diameters not bigger than $r$? We investigate connections between the set-theoretic phenomena involved and the geometric properties of uncountable subsets of nonseparable Banach spaces of densities up to $2^ω$ related to uncountable $(1+)$-separated sets, equilateral sets or Auerbach systems. The results include geometric dichotomies for a wide range of classes of Banach spaces, some in ZFC, some under the assumption of OCA+MA and some under a hypothesis on the descriptive complexity of the space as well as constructions (in ZFC or under CH) of Banach spaces where the geometry of the unit sphere displays anti-Ramsey properties. This complements classical theorems for separable spheres and the recent results of Hájek, Kania, Russo for densities above $2^ω$ as well as offers a synthesis of possible phenomena and categorization of examples for uncountable densities up to $2^ω$ obtained previously by the author and Guzmán, Hrušák, Ryduchowski and Wark. It remains open if the dichotomies may consistently hold for all Banach spaces of the first uncountable density or if the strong anti-Ramsey properties of the distance on the unit sphere of a Banach space can be obtained in ZFC.

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arXiv2023

Countably tight dual ball with a nonseparable measure

We construct a compact Hausdorff space $K$ such that the space $P(K)$ of Radon probabiblity measures on $K$ considered with the weak$^*$ topology (induced from the space of continuous functions $C(K)$) is countably tight which is a generalization of sequentiality (i.e., if a measure $μ$ is in the closure of a set $M$, there is a countable $M'\subseteq M$ such that $μ$ is in the closure of $M'$) but $K$ carries a Radon probability measure which has uncountable Maharam type (i.e., $L_1(μ)$ is nonseparable). The construction uses (necessarily) an additional set-theoretic assumption (the $\diamondsuit$ principle) as it was already known, by a result of Fremlin, that it is consistent that such spaces do not exist. This should be compared with the result of Plebanek and Sobota who showed that countable tightness of $P(K\times K)$ implies that all Radon measures on $K$ have countable type. So, our example shows that the tightness of $P(K\times K)$ and of $P(K)\times P(K)$ can be different as well as $P(K)$ may have Corson property (C) while $P(K\times K)$ fails to have it answering a question of Pol. Our construction is also a relevant example in the general context of injective tensor products of Banach spaces complementing recent results of Avilés, Martínez-Cervantes, Rodríguez and Rueda Zoca.

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arXiv2023

Equilateral and separated sets in some Hilbert generated Banach spaces

We study Hilbert generated versions of nonseparable Banach spaces $\mathcal X$ considered by Shelah, Steprāns and Wark where the behavior of the norm on nonseparable subsets is so irregular that it does not allow any linear bounded operator on $\mathcal X$ other than a diagonal operator (or a scalar multiple of the identity) plus a separable range operator. We address the questions if these spaces admit uncountable equilateral sets and if their unit spheres admit uncountable $(1+)$-separated or $(1+\varepsilon)$-separated sets. We resolve some of the above questions for two types of these spaces by showing both absolute and undecidability results. The corollaries are that the continuum hypothesis (in fact: the existence of a nonmeager set of reals of the first uncountable cardinality) implies the existence of an equivalent renorming of the nonsepareble Hilbert space $\ell_2(ω_1)$ which does not admit any uncountable equilateral set and it implies the existence of a nonseparable Hilbert generated Banach space containing an isomorphic copy of $\ell_2$ in each nonseparable subspace, whose unit sphere does not admit an uncountable equilateral set and does not admit an uncountable $(1+\varepsilon)$-separated set for any $\varepsilon>0$. This could be compared with a recent result by Hájek, Kania and Russo saying that all nonseparable reflexive Banach spaces admit uncountable $(1+\varepsilon)$-separated sets in their unit spheres.

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arXiv2022

Almost disjoint families and the geometry of nonseparable spheres

We consider uncountable almost disjoint families of subsets of $\mathbb N$, the Johnson-Lindenstrauss Banach spaces $(\mathcal X_{\mathcal A}, \|\ \|_\infty)$ induced by them, and their natural equivalent renormings $(\mathcal X_{\mathcal A}, \|\ \|_{\infty, 2})$. We introduce a partial order $\mathbb P_{\mathcal A}$ and characterize some geometric properties of the spheres of $(\mathcal X_{\mathcal A}, \|\ \|_{\infty})$ and of $(\mathcal X_{\mathcal A}, \|\ \|_{\infty, 2})$ in terms of combinatorial properties of $\mathbb P_{\mathcal A}$. Exploiting the extreme behavior of some known and some new almost disjoint families among others we show the existence of Banach spaces where the unit spheres display surprising geometry: 1) There is a Banach space of density continuum whose unit sphere is the union of countably many sets of diameters strictly less than $1$. 2) It is consistent that for every $ρ>0$ there is a nonseparable Banach space, where for every $δ>0$ there is $\varepsilon>0$ such that every uncountable $(1-\varepsilon)$-separated set of elements of the unit sphere contains two elements distant by less than $1$ and two elements distant at least by $2-ρ-δ$. It should be noted that for every $\varepsilon>0$ every nonseparable Banach space has a plenty of uncountable $(1-\varepsilon)$-separated sets by the Riesz Lemma. We also obtain a consistent dichotomy for the spaces of the form $(\mathcal X_{\mathcal A}, \|\ \|_{\infty, 2})$: The Open Coloring Axiom implies that the unit sphere of every Banach space of the form $(\mathcal X_{\mathcal A}, \|\ \|_{\infty, 2})$ either is the union of countably many sets of diameter strictly less than $1$ or it contains an uncountable $(2-\varepsilon)$-separated set for every $\varepsilon>0$.

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arXiv2021

On the existence of overcomplete sets in some classical nonseparable Banach spaces

For a Banach space $X$ its subset $Y\subseteq X$ is called overcomplete if $|Y|=dens(X)$ and $Z$ is linearly dense in $X$ for every $Z\subseteq Y$ with $|Z|=|Y|$. In the context of nonseparable Banach spaces this notion was introduced recently by T. Russo and J. Somaglia but overcomplete sets have been considered in separable Banach spaces since the 1950ties. We prove some absolute and consistency results concerning the existence and the nonexistence of overcomplete sets in some classical nonseparable Banach spaces. For example: $c_0(ω_1)$, $C([0,ω_1])$, $L_1(\{0,1\}^{ω_1})$, $\ell_p(ω_1)$, $L_p(\{0,1\}^{ω_1})$ for $p\in (1, \infty)$ or in general WLD Banach spaces of density $ω_1$ admit overcomplete sets (in ZFC). The spaces $\ell_\infty$, $\ell_\infty/c_0$, spaces of the form $C(K)$ for $K$ extremally disconnected, superspaces of $\ell_1(ω_1)$ of density $ω_1$ do not admit overcomplete sets (in ZFC). Whether the Johnson-Lindenstrauss space generatedin $\ell_\infty$ by $c_0$ and the characteristic functions of elements of an almost disjoint family of subsets of $\mathbb N$ of cardinality $ω_1$ admits an overcomplete set is undecidable. The same refers to all nonseparable Banach spaces with the dual balls of density $ω_1$ which are separable in the weak$^*$ topology. The results proved refer to wider classes of Banach spaces but several natural open questions remain open.

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arXiv2021

Banach spaces in which large subsets of spheres concentrate

We construct a nonseparable Banach space $\mathcal X$ (actually, of density continuum) such that any uncountable subset $\mathcal Y$ of the unit sphere of $\mathcal X$ contains uncountably many points distant by less than $1$ (in fact, by less then $1-\varepsilon$ for some $\varepsilon>0$). This solves in the negative the central problem of the search for a nonseparable version of Kottman's theorem which so far has produced many deep positive results for special classes of Banach spaces and has related the global properties of the spaces to the distances between points of uncountable subsets of the unit sphere. The property of our space is strong enough to imply that it contains neither an uncountable Auerbach system nor an uncountable equilateral set. The space is a strictly convex renorming of the Johnson-Lindenstrauss space induced by an $\mathbb R$-embeddable almost disjoint family of subsets of $\mathbb N$. We also show that this special feature of the almost disjoint family is essential to obtain the above properties.

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arXiv2021

On coverings of Banach spaces and their subsets by hyperplanes

Given a Banach space we consider the $σ$-ideal of all of its subsets which are covered by countably many hyperplanes and investigate its standard cardinal characteristics as the additivity, the covering number, the uniformity, the cofinality. We determine their values for separable Banach spaces, and approximate them for nonseparable Banach spaces. The remaining questions reduce to deciding if the following can be proved in ZFC for every nonseparable Banach space $X$: (1) $X$ can be covered by $ω_1$-many of its hyperplanes; (2) All subsets of $X$ of cardinalities less than ${\rm cf}([{\rm dens}(X)]^ω)$ can be covered by countably many hyperplanes. We prove (1) and (2) for all Banach spaces in many well-investigated classes and that they are consistent with any possible size of the continuum. (1) is related to the problem whether every compact Hausdorff space which has small diagonal is metrizable and (2) to large cardinals.

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arXiv2021

Large Banach spaces with no infinite equilateral sets

A subset of a Banach space is called equilateral if the distances between any two of its distinct elements are the same. It is proved that there exist non-separable Banach spaces (in fact of density continuum) with no infinite equilateral subset. These examples are strictly convex renormings of $\ell_1([0,1])$. A wider class of renormings of $\ell_1([0,1])$ which admit no uncountable equilateral sets is also considered.

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arXiv2021

A Banach space induced by an almost disjoint family, admitting only few operators and decompositions

We consider the closed subspace of $\ell_\infty$ generated by $c_0$ and the characteristic functions of elements of an uncountable, almost disjoint family $\mathcal A$ of infinite subsets of $\mathbb N$. This Banach space has the form $C_0(K_{\mathcal A})$ for a locally compact Hausdorff space $K_{\mathcal A}$ that is known under many names, such as $Ψ$-space and Isbell--Mrówka space. We construct an uncountable, almost disjoint family ${\mathcal A}$ such that the Banach algebra of all bounded linear operators on $C_0(K_{\mathcal A})$ is as small as possible in the sense that every bounded linear operator on $C_0(K_{\mathcal A})$ is the sum of a scalar multiple of the identity and an operator that factors through $c_0$ (which in this case is equivalent to having separable range). This implies that $C_0(K_{\mathcal A})$ has the fewest possible decompositions: whenever $C_0(K_{\mathcal A})=X\oplus Y$ with $dim({X})=\infty$, $dim({Y})=\infty$, either ${X}$ is isomorphic to $C_0(K_{\mathcal A})$ and ${Y}$ to $c_0$, or vice versa. These results improve previous work of the first named author in which an extra set-theoretic hypothesis was required. We also discuss the consequences of these results for the algebra of all bounded linear operators on our Banach space $C_0(K_{\mathcal A})$ concerning the lattice of closed ideals, characters and automatic continuity of homomorphisms. To exploit the perfect set property for Borel sets as in the classical construction of an almost disjoint family of Mrówka we need to deal with $\mathbb N \times \mathbb N$-matrices rather than with the usual partitioners. This noncommutative setting requires new ideas inspired by the theory of compact and weakly compact operators and the use of an extraction principle due to F. van Engelen, K. Kunen and A. Miller concerning Borel subsets of the square.

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arXiv2020

A non-diagonalizable pure state

We construct a pure state on the C*-algebra $\mathcal B(\ell_2)$ of all bounded linear operators on $\ell_2$ which is not diagonalizable, i.e., it is not of the form $\lim_u\langle T(e_k), e_k\rangle$ for any orthonormal basis $(e_k)_{k\in \mathbb N}$ of $\ell_2$ and an ultrafilter $u$ on $\mathbb N$. This constitutes a counterexample to Anderson's conjecture without additional hypothesis and improves results of C. Akemann, N. Weaver, I. Farah and I. Smythe who constructed such states making additional set-theoretic assumptions. It follows from results of J. Anderson and the positive solution to the Kadison-Singer problem due to A. Marcus, D. Spielman, N. Srivastava that the restriction of our pure state to any atomic masa $D((e_k)_{k\in \mathbb N})$ of diagonal operators with respect to an orthonormal basis $(e_k)_{k\in \mathbb N}$ is not multiplicative on $D((e_k)_{k\in \mathbb N})$.

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arXiv2019

Rosenthal families, pavings and generic cardinal invariants

Following D. Sobota we call a family $\mathcal F$ of infinite subsets of $\mathbb N$ a Rosenthal family if it can replace the family of all infinite subsets of $\mathbb N$ in classical Rosenthal's Lemma concerning sequences of measures on pairwise disjoint sets. We resolve two problems on Rosenthal families: every ultrafilter is a Rosenthal family and the minimal size of a Rosenthal family is exactly equal to the reaping cardinal $\mathfrak r$. This is achieved through analyzing nowhere reaping families of subsets of $\mathbb N$ and through applying a paving lemma which is a consequence of a paving lemma concerning linear operators on $\ell_1^n$ due to Bourgain. We use connections of the above results with free set results for functions on $\mathbb N$ and with linear operators on $c_0$ to determine the values of several other derived cardinal invariants.

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arXiv2019

On $\mathbb R$-embeddability of almost disjoint families and Akemann-Doner C*-algebras

An almost disjoint family $\mathcal A$ of subsets of $\mathbb N$ is said to be $\mathbb R$-embeddable if there is a function $f:\mathbb N\rightarrow \mathbb R$ such that the sets $f[A]$ are ranges of real sequences converging to distinct reals for distinct $A\in \mathcal A$. It is well known that almost disjoint families which have few separations, such as Luzin families, are not $\mathbb R$-embeddable. We study extraction principles related to $\mathbb R$-embeddability and separation properties of almost disjoint families of $\mathbb N$ as well as their limitations. An extraction principle whose consistency is our main result is: every almost disjoint family of size continuum contains an $\mathbb R$-embeddable subfamily of size continuum. It is true in the Sacks model. The Cohen model serves to show that the above principle does not follow from the fact that every almost disjoint family of size continuum has two separated subfamilies of size continuum. We also construct in ZFC an almost disjoint family, where no two uncountable subfamilies can be separated but always a countable subfamily can be separated from any disjoint subfamily. Using a refinement of the $\mathbb R$-embeddability property called a controlled $\mathbb R$-embedding property we obtain the following results concerning Akemann-Doner C*-algebras which are induced by uncountable almost disjoint families: a) In ZFC there are Akemann-Doner C*-algebras of density $\mathfrak c$ with no commutative subalgebras of density $\mathfrak c$, b) It is independent from ZFC whether there is an Akemann-Doner algebra of density $\mathfrak c$ with no nonseparable commutative subalgebra. This completes an earlier result that there is in ZFC an Akemann-Doner algebra of density $ω_1$ with no nonseparable commutative subalgebra.

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