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

Guzmán, Osvaldo; Hrušák, Michael; Koszmider, Piotr

doi:10.4064/fm780-8-2020

Cited by 3 publications

(4 citation statements)

References 23 publications

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“…In fact, an almost disjoint family A of infinite subsets of N is R-embeddable if and only if there is an injective φ : N → Q such that the sets φ[A] for A ∈ A are ranges of sequences converging to distinct irrational reals. This follows from Lemma 1 and Lemma 2 of [17]. Lemma 14.…”

Section: Concentration In Spheres Ofmentioning

confidence: 87%

“…On the other hand, we show that certain almost disjoint families which are not -embeddable, known as Luzin families (Definition 18), induce the Banach space such that the sphere of admits an uncountable -equilateral subset (Proposition 20) and the sphere of admits an uncountable -separated subset for any (Proposition 21). For more on -embeddability, see [17]. Note that A. Dow showed in [8] that assuming (the proper forcing axiom) every maximal almost disjoint family contains a Luzin subfamily.…”

Section: Introductionmentioning

confidence: 99%

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Banach Spaces in Which Large Subsets of Spheres Concentrate

Koszmider

2022

J. Inst. Math. Jussieu

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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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Section: Concentration In Spheres Ofmentioning

confidence: 87%

Section: Introductionmentioning

confidence: 99%

Banach Spaces in Which Large Subsets of Spheres Concentrate

Koszmider

2022

J. Inst. Math. Jussieu

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“…By Lemma 1 and Lemma 2 of [14] an almost disjoint family A of infinite subsets of N is R-embeddable if and only if there is an injective φ :…”

Section: 3mentioning

confidence: 99%

“…2 )-separated subset (Proposition 20). For more on R-embeddability see [14]. Note that A. Dow showed in [6] that assuming PFA every maximal almost disjoint family contains a Luzin subfamily.…”

Section: Introductionmentioning

confidence: 99%

Banach spaces in which large subsets of spheres concentrate

Koszmider¹

2021

Preprint

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We construct a nonseparable Banach space X (actually, of density continuum) such that any uncountable subset Y of the unit sphere of X contains uncountably many points distant by less than 1 (in fact, by less then 1 − ε for some ε > 0). This solves in the negative the central problem of the search for a nonseparable version of Kottman's theorem which so far produced many deep positive results for special classes of Banach spaces and related the global properties of the spaces to the distances between pairs of points of uncountable subsets of the unit sphere. The property of our space is strong enough to imply that it contains no uncountable Auerbach system nor an uncountable equilateral set. The space is a strictly convex renorming of the Johnson-Lindenstrauss space induced by an R-embeddable almost disjoint family of subsets of N. We also show that this special property of the almost disjoint family is essential to obtain the above properties.

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