On biorthogonal systems whose functionals are finitely supported

Tom 213 / 2011

Christina Brech, Piotr Koszmider Fundamenta Mathematicae 213 (2011), 43-66 MSC: Primary 46B26; Secondary 03E35, 54D80. DOI: 10.4064/fm213-1-3


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 $(f_\xi,\mu_\xi)_{\xi\in \omega_1}$ where $\mu_\xi$'s are atomic measures with supports consisting of at most $2n-1$ points of $K_{2n}$, but has biorthogonal systems $(f_\xi,\mu_\xi)_{\xi\in \omega_1}$ where $\mu_\xi$'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 $\delta_{x_\xi}-\delta_{y_\xi}$ for $\xi<\omega_1$ and $x_\xi,y_\xi\in K$. It also follows from our results that it is consistent that the irredundance of the Boolean algebra ${\rm Clop}(K)$ 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 $k>n$.


  • Christina BrechInstituto 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
  • Piotr KoszmiderInstytut Matematyki Politechniki /L/odzkiej
    W/olcza/nska 215
    90-924 /L/od/x, Poland
    Institute of Mathematics
    Polish Academy of Sciences
    Śniadeckich 8
    P.O. Box 21
    00-956 Warszawa, Poland

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