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2011 | 213 | 1 | 43-66
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On biorthogonal systems whose functionals are finitely supported

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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.
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  • 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
  • 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
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bwmeta1.element.bwnjournal-article-doi-10_4064-fm213-1-3
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