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1
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A space C(K) where all nontrivial complemented subspaces have big densities

100%
Koszmider P.
Studia Mathematica
|
2005
|
tom 168
|
nr 2
109-127
EN
Using the method of forcing we prove that consistently there is a Banach space (of continuous functions on a totally disconnected compact Hausdorff space) of density κ bigger than the continuum where all operators are multiplications by a continuous function plus a weakly compact operator and which has no infinite-dimensional complemented subspaces of density continuum or smaller. In particular no separable infinite-dimensional subspace has a complemented superspace of density continuum or smaller, consistently answering a question of Johnson and Lindenstrauss of 1974.
2
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A C(K) Banach space which does not have the Schroeder-Bernstein property

100%
Koszmider P.
Studia Mathematica
|
2012
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tom 212
|
nr 2
95-117
EN
We construct a totally disconnected compact Hausdorff space K₊ which has clopen subsets K₊'' ⊆ K₊' ⊆ K₊ such that K₊'' is homeomorphic to K₊ and hence C(K₊'') is isometric as a Banach space to C(K₊) but C(K₊') is not isomorphic to C(K₊). This gives two nonisomorphic Banach spaces (necessarily nonseparable) of the form C(K) which are isomorphic to complemented subspaces of each other (even in the above strong isometric sense), providing a solution to the Schroeder-Bernstein problem for Banach spaces of the form C(K). The subset K₊ is obtained as a particular compactification of the pairwise disjoint union of an appropriately chosen sequence $(K_{1,n} ∪ K_{2,n})_{n ∈ ℕ}$ of Ks for which C(K)s have few operators. We have $K₊' = K₊∖K_{1,0}$ and $K₊'' = K₊∖(K_{1,0} ∪ K_{2,0})$.
3
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On families of Lindelöf and related subspaces of $2^{ω₁}$

64%
Junqueira L., Koszmider P.
Fundamenta Mathematicae
|
2001
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tom 169
|
nr 3
205-231
EN
We consider the families of all subspaces of size ω₁ of $2^{ω₁}$ (or of a compact zero-dimensional space X of weight ω₁ in general) which are normal, have the Lindelöf property or are closed under limits of convergent ω₁-sequences. Various relations among these families modulo the club filter in $[X]^{ω₁}$ are shown to be consistently possible. One of the main tools is dealing with a subspace of the form X ∩ M for an elementary submodel M of size ω₁. Various results with this flavor are obtained. Another tool used is forcing and in this case various preservation or nonpreservation results of topological and combinatorial properties are proved. In particular we prove that there may be no c.c.c. forcing which destroys the Lindelöf property of compact spaces, answering a question of Juhász. Many related questions are formulated.
4
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On automorphisms of the Banach space $ℓ_{∞}/c₀$

64%
Koszmider P., Rodríguez-Porras C.
Fundamenta Mathematicae
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2016
|
tom 235
|
nr 1
49-99
EN
We investigate Banach space automorphisms $T: ℓ_{∞}/c₀ → ℓ_{∞}/c₀$ focusing on the possibility of representing their fragments of the form $T_{B,A}: ℓ_{∞}(A)/c₀(A) → ℓ_{∞}(B)/c₀(B)$ for A,B ⊆ ℕ infinite by means of linear operators from $ℓ_{∞}(A)$ into $ℓ_{∞}(B)$, infinite A×B-matrices, continuous maps from B* = βB∖B into A*, or bijections from B to A. This leads to the analysis of general bounded linear operators on $ℓ_{∞}/c₀$. We present many examples, introduce and investigate several classes of operators, for some of them we obtain satisfactory representations and for others give examples showing that this is impossible. In particular, we show that there are automorphisms of $ℓ_{∞}/c₀$ which cannot be lifted to operators on $ℓ_{∞}$, and assuming OCA+MA we show that every automorphism T of $ℓ_{∞}/c₀$ with no fountains or with no funnels is locally induced by a bijection, i.e., $T_{B,A}$ is induced by a bijection from some infinite B ⊆ ℕ to some infinite A ⊆ ℕ. This additional set-theoretic assumption is necessary as we show that the Continuum Hypothesis implies the existence of counterexamples of diverse flavours. However, many basic problems, some of which are listed in the last section, remain open.
5
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On biorthogonal systems whose functionals are finitely supported

64%
Brech C., Koszmider P.
Fundamenta Mathematicae
|
2011
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tom 213
|
nr 1
43-66
EN
We show that for each natural number n > 1, it is consistent that there is a compact Hausdorff totally disconnected space $K_{2n}$ such that $C(K_{2n})$ has no uncountable (semi)biorthogonal sequence $(f_ξ,μ_ξ)_{ξ∈ω₁}$ where $μ_ξ$'s are atomic measures with supports consisting of at most 2n-1 points of $K_{2n}$, but has biorthogonal systems $(f_ξ,μ_ξ)_{ξ∈ω₁}$ where $μ_ξ$'s are atomic measures with supports consisting of 2n points. This complements a result of Todorcevic which implies that it is consistent that such spaces do not exist: he proves that its is consistent that for any nonmetrizable compact Hausdorff totally disconnected space K, the 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 < ω₁ and $x_ξ,y_ξ ∈ K$. It also follows from our results that it is consistent that the irredundance of the Boolean algebra Clop(K) for a totally disconnected K or of the Banach algebra C(K) 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.
6
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The Arkhangel’skiĭ–Tall problem: a consistent counterexample

64%
Gruenhage G., Koszmider P.
Fundamenta Mathematicae
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1996
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tom 149
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nr 2
143-166
EN
We construct a consistent example of a normal locally compact metacompact space which is not paracompact, answering a question of A. V. Arkhangel'skiĭ and F. Tall. An interplay between a tower in P(ω)/Fin, an almost disjoint family in $[ω]^ω$, and a version of an (ω,1)-morass forms the core of the proof. A part of the poset which forces the counterexample can be considered a modification of a poset due to Judah and Shelah for obtaining a Q-set by a countable support iteration.
7
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$ℓ_{∞}$-sums and the Banach space $ℓ_{∞}/c₀$

64%
Brech C., Koszmider P.
Fundamenta Mathematicae
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2014
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tom 224
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nr 2
175-185
EN
This paper is concerned with the isomorphic structure of the Banach space $ℓ_{∞}/c₀$ and how it depends on combinatorial tools whose existence is consistent with but not provable from the usual axioms of ZFC. Our main global result is that it is consistent that $ℓ_{∞}/c₀$ does not have an orthogonal $ℓ_{∞}$-decomposition, that is, it is not of the form $ℓ_{∞}(X)$ for any Banach space X. The main local result is that it is consistent that $ℓ_{∞}(c₀(𝔠))$ does not embed isomorphically into $ℓ_{∞}/c₀$, where 𝔠 is the cardinality of the continuum, while $ℓ_{∞}$ and c₀(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 $ℓ_{∞}/c₀$ is isomorphic to its $ℓ_{∞}$-sum and in particular it contains an isomorphic copy of all Banach spaces of the form $ℓ_{∞}(X)$ for any subspace X of $ℓ_{∞}/c₀$.
8
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On complemented copies of c₀(ω₁) in C(Kⁿ) spaces

64%
Candido L., Koszmider P.
Studia Mathematica
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2016
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tom 233
|
nr 3
209-226
EN
Given a compact Hausdorff space K we consider the Banach space of real continuous functions C(Kⁿ) or equivalently the n-fold injective tensor product $⊗̂^{n}_{ε}C(K)$ or the Banach space of vector valued continuous functions C(K,C(K,C(K...,C(K)...). We address the question of the existence of complemented copies of c₀(ω₁) in $⊗̂^{n}_{ε} C(K)$ under the hypothesis that C(K) contains such a copy. This is related to the results of E. Saab and P. Saab that $X ⊗̂_{ε} Y$ contains a complemented copy of c₀ if one of the infinite-dimensional Banach spaces X or Y contains a copy of c₀, and of E. M. Galego and J. Hagler that it follows from Martin's Maximum that if C(K) has density ω₁ and contains a copy of c₀(ω₁), then C(K×K) contains a complemented copy of c₀(ω₁). Our main result is that under the assumption of ♣ for every n ∈ ℕ there is a compact Hausdorff space Kₙ of weight ω₁ such that C(K) is Lindelöf in the weak topology, C(Kₙ) contains a copy of c₀(ω₁), C(Kₙⁿ) does not contain a complemented copy of c₀(ω₁), while $C(Kₙ^{n+1})$ does contain a complemented copy of c₀(ω₁). This shows that additional set-theoretic assumptions in Galego and Hagler's nonseparable version of Cembrano and Freniche's theorem are necessary, as well as clarifies in the negative direction the matter unsettled in a paper of Dow, Junnila and Pelant whether half-pcc Banach spaces must be weakly pcc.
9
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The Arkhangel'skiĭ–Tall problem under Martin’s Axiom

64%
Gruenhage G., Koszmider P.
Fundamenta Mathematicae
|
1996
|
tom 149
|
nr 3
275-285
EN
We show that MA$_{σ-centered}(ω_1)$ implies that normal locally compact metacompact spaces are paracompact, and that MA($ω_1$) implies normal locally compact metalindelöf spaces are paracompact. The latter result answers a question of S. Watson. The first result implies that there is a model of set theory in which all normal locally compact metacompact spaces are paracompact, yet there is a normal locally compact metalindelöf space which is not paracompact.
10
Content available remote

On Marczewski-Burstin representable algebras

51%
Balcerzak M., Bartoszewicz A., Koszmider P.
Colloquium Mathematicum
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2004
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tom 99
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nr 1
55-60
EN
We construct algebras of sets which are not MB-representable. The existence of such algebras was previously known under additional set-theoretic assumptions. On the other hand, we prove that every Boolean algebra is isomorphic to an MB-representable algebra of sets.
11
Content available remote

Maximal left ideals of the Banach algebra of bounded operators on a Banach space

45%
Dales H., Kania T., Kochanek T., Koszmider P., Laustsen N.
Studia Mathematica
|
2013
|
tom 218
|
nr 3
245-286
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