$ℓ_{∞}$-sums and the Banach space $ℓ_{∞}/c₀$

• Autorzy: Christina Brech , Piotr Koszmider
• Języki publikacji: EN

Abstrakty

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 $ℓ_{∞}/c₀$.

Kategorie tematyczne

• 46B03: Isomorphic theory (including renorming) of Banach spaces
• 03E75: Applications of set theory

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Fundamenta Mathematicae
• Rocznik: 2014
• Tom: 224
• Numer: 2
• Strony: 175-185

Daty

wydano

2014

Twórcy

autor

Christina Brech

• Departamento de Matemática, Instituto de Matemática e Estatística, Universidade de São Paulo, Caixa Postal 66281, 05314-970, São Paulo, Brazil

autor

Piotr Koszmider

• Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, 00-956 Warszawa, Poland

Identyfikatory

DOI

10.4064/fm224-2-3