On biorthogonal systems whose functionals are finitely supported

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

Abstrakty

EN

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.

Kategorie tematyczne

• 46B26: Nonseparable Banach spaces
• 03E35: Consistency and independence results
• 54D80: Special constructions of spaces (spaces of ultrafilters, etc.)

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Fundamenta Mathematicae
• Rocznik: 2011
• Tom: 213
• Numer: 1
• Strony: 43-66

Daty

wydano

2011

Twórcy

autor

Christina Brech

• Instituto de Matemática, Estatística, e Computação Científica, Universidade Estadual de Campinas, Rua Sérgio Buarque de Holanda 651, 13083-859, Campinas, Brazil
• Departamento de Matemática, Instituto de Matemática e Estatística, Universidade de São Paulo, Rua do Matão 1010, 05508-090, São Paulo, Brazil

autor

Piotr Koszmider

• Instytut Matematyki Politechniki Łódzkiej, Wólczańska 215, 90-924 Łódź, Poland
• Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, P.O. Box 21, 00-956 Warszawa, Poland

Identyfikatory

DOI

10.4064/fm213-1-3