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

Avilés, Antonio; Brech, Christina

doi:10.1016/j.topol.2011.05.022

Cited by 13 publications

(26 citation statements)

References 11 publications

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“…The hypothesis is consistent by a result of Brech and Koszmider [6]. The paper [1] contains further results on the existence of spaces of universal disposition for S under different cardinality assumptions.…”

Section: Lemma 45 a C 0 -Valued Operator Defined On A Finite-dimenssupporting

confidence: 80%

Banach spaces of universal disposition

Avilés

Sánchez

Castillo

et al. 2011

Journal of Functional Analysis

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In this paper we present a method to obtain Banach spaces of universal and almost-universal disposition with respect to a given class $\mathfrak M$ of normed spaces. The method produces, among other, the Gurari\u{\i} space $\mathcal G$ (the only separable Banach space of almost-universal disposition with respect to the class $\mathfrak F$ of finite dimensional spaces), or the Kubis space $\mathcal K$ (under {\sf CH}, the only Banach space with the density character the continuum which is of universal disposition with respect to the class $\mathfrak S$ of separable spaces). We moreover show that $\mathcal K$ is not isomorphic to a subspace of any $C(K)$-space -- which provides a partial answer to the injective space problem-- and that --under {\sf CH}-- it is isomorphic to an ultrapower of the Gurari\u{\i} space. We study further properties of spaces of universal disposition: separable injectivity, partially automorphic character and uniqueness properties

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Section: Lemma 45 a C 0 -Valued Operator Defined On A Finite-dimenssupporting

confidence: 80%

Banach spaces of universal disposition

Avilés

Sánchez

Castillo

et al. 2011

Journal of Functional Analysis

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“…We therefore need only to find some strictly saturated graph ← − G with no infinite path, such that G is of the first type. We apply a bookkeeping argument, similar to the one used in the proof of existence of Avilés-Brech Boolean algebra in [1]. Let { α |α < c} be a partition of c into sets of cardinality c, and such that min α ≥ α.…”

Section: From Proposition 9 Followsmentioning

confidence: 99%

“…However, it turns out that some applications of back-and-forth argument can be carried out only assuming that c is regular, or even in ZFC alone. For example, Avilés and Brech generalized Parovičenko Theorem, by introducing a property stronger than the countable separation property, which guarantees the uniqueness of certain Boolean algebra with this property, when c is regular [1]. A different generalization was obtained by Dow and Hart [2].…”

Section: Introductionmentioning

confidence: 99%

On countably saturated linear orders and certain class of countably saturated graphs

Kostana

2020

Arch. Math. Logic

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The idea of this paper is to explore the existence of canonical countably saturated models for different classes of structures. It is well-known that, under CH, there exists a unique countably saturated linear order of cardinality $$\mathfrak {c}$$ c . We provide some examples of pairwise non-isomorphic countably saturated linear orders of cardinality $$\mathfrak {c}$$ c , under different set-theoretic assumptions. We give a new proof of the old theorem of Harzheim, that the class of countably saturated linear orders has a uniquely determined one-element basis. From our proof it follows that this minimal linear order is a Fraïssé limit of certain Fraïssé class. In particular, it is homogeneous with respect to countable subsets. Next we prove the existence and uniqueness of the uncountable version of the random graph. This graph is isomorphic to $$(H(\omega _1),\in \cup \ni )$$ ( H ( ω 1 ) , ∈ ∪ ∋ ) , where $$H(\omega _1)$$ H ( ω 1 ) is the set of hereditarily countable sets, and two sets are connected if one of them is an element of the other. In the last section, an example of a prime countably saturated Boolean algebra is presented.

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“…The key notion of this paper is that of a tightly

σ

‐filtered Boolean algebra, studied in and under different names in . We generalize the fact that tightly

σ

‐filtered algebras do not contain

ω_{2}

‐chains [, Proposition 2.5] to a more general principle (Theorem ) asserting that if an

ω_{2}

‐sequence of elements often satisfies some Boolean equation, then this equation must also be often satisfied in a permuted order.…”

Section: Introductionmentioning

confidence: 99%

A 1‐separably injective Banach space that does not contain ℓ∞

Avilés

Koszmider

2018

Bulletin of London Math Soc

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We show that the problem whether every 1-separably injective Banach space contains an isomorphic copy of ∞ is undecidable. Namely, unlike under the continuum hypothesis, assuming Martin's axiom and the negation of the continuum hypothesis, there is an 1-separably injective Banach space of the form C(K) (which means that K is an F -space) without an isomorphic copy of ∞.This result is a consequence of our study of ω2-subsets of tightly σ-filtered Boolean algebras introduced by Koppelberg for which we obtain some general principles useful when transferring properties of Boolean algebras to the level of Banach spaces.

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

Cited by 13 publications

References 11 publications

Banach spaces of universal disposition

Banach spaces of universal disposition

On countably saturated linear orders and certain class of countably saturated graphs

A 1‐separably injective Banach space that does not contain ℓ∞

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