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On biorthogonal systems whose functionals are finitely supported

100%

Brech C., Koszmider P.

Fundamenta Mathematicae

2011

tom 213

nr 1

43-66

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.

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

100%

Brech C., Koszmider P.

Fundamenta Mathematicae

2014

tom 224

nr 2

175-185

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 $ℓ_{∞}/c₀$.

Ograniczanie wyników

2 Fundamenta Mathematicae

2 Brech C.

2 Koszmider P.

1 2014

1 2011

Wyniki wyszukiwania

On biorthogonal systems whose functionals are finitely supported

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