${\ell }_{\infty }$-sums and the Banach space ${\ell }_{\infty }/c₀$

Christina Brech, and Piotr Koszmider. "$ℓ_{∞}$-sums and the Banach space $ℓ_{∞}/c₀$." Fundamenta Mathematicae 224.2 (2014): 175-185. <http://eudml.org/doc/283298>.

@article{ChristinaBrech2014,
abstract = {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 $ℓ_\{∞\}(c₀())$ does not embed isomorphically into $ℓ_\{∞\}/c₀$, where is the cardinality of the continuum, while $ℓ_\{∞\}$ and c₀() 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 $ℓ_\{∞\}/c₀$.},
author = {Christina Brech, Piotr Koszmider},
journal = {Fundamenta Mathematicae},
keywords = {Banach space },
language = {eng},
number = {2},
pages = {175-185},
title = {$ℓ_\{∞\}$-sums and the Banach space $ℓ_\{∞\}/c₀$},
url = {http://eudml.org/doc/283298},
volume = {224},
year = {2014},
}

TY - JOUR
AU - Christina Brech
AU - Piotr Koszmider
TI - $ℓ_{∞}$-sums and the Banach space $ℓ_{∞}/c₀$
JO - Fundamenta Mathematicae
PY - 2014
VL - 224
IS - 2
SP - 175
EP - 185
AB - 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 $ℓ_{∞}(c₀())$ does not embed isomorphically into $ℓ_{∞}/c₀$, where is the cardinality of the continuum, while $ℓ_{∞}$ and c₀() 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 $ℓ_{∞}/c₀$.
LA - eng
KW - Banach space
UR - http://eudml.org/doc/283298
ER -