On biorthogonal systems whose functionals are finitely supported

Christina Brech, and Piotr Koszmider. "On biorthogonal systems whose functionals are finitely supported." Fundamenta Mathematicae 213.1 (2011): 43-66. <http://eudml.org/doc/283120>.

@article{ChristinaBrech2011,
abstract = {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.},
author = {Christina Brech, Piotr Koszmider},
journal = {Fundamenta Mathematicae},
keywords = {biorthogonal system; space; irredundant set; unordered -split Cantor set},
language = {eng},
number = {1},
pages = {43-66},
title = {On biorthogonal systems whose functionals are finitely supported},
url = {http://eudml.org/doc/283120},
volume = {213},
year = {2011},
}

TY - JOUR
AU - Christina Brech
AU - Piotr Koszmider
TI - On biorthogonal systems whose functionals are finitely supported
JO - Fundamenta Mathematicae
PY - 2011
VL - 213
IS - 1
SP - 43
EP - 66
AB - 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.
LA - eng
KW - biorthogonal system; space; irredundant set; unordered -split Cantor set
UR - http://eudml.org/doc/283120
ER -