On biorthogonal systems whose functionals are finitely supported

Christina Brech¹ ; Piotr Koszmider²

¹ Instituto de Matemática, Estatística e Computação Científica Universidade Estadual de Campinas Rua Sérgio Buarque de Holanda 651 13083-859, Campinas, Brazil and Departamento de Matemática Instituto de Matemática e Estatística Universidade de São Paulo Rua do Matão 1010 05508-090, São Paulo, Brazil
² Instytut Matematyki Politechniki /L/odzkiej W/olcza/nska 215 90-924 /L/od/x, Poland and Institute of Mathematics Polish Academy of Sciences Śniadeckich 8 P.O. Box 21 00-956 Warszawa, Poland

Fundamenta Mathematicae, Tome 213 (2011) no. 1, pp. 43-66.

Résumé

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 $k>n$.

Détail

DOI : 10.4064/fm213-1-3

Keywords: each natural number consistent there compact hausdorff totally disconnected space has uncountable semi biorthogonal sequence omega where xis atomic measures supports consisting n points has biorthogonal systems omega where xis atomic measures supports consisting points complements result todorcevic which implies consistent spaces exist proves its consistent nonmetrizable compact hausdorff totally disconnected space banach space has uncountable biorthogonal system where functionals measures form delta delta omega follows results consistent irredundance boolean algebra clop totally disconnected banach algebra strictly smaller sizes biorthogonal systems compact spaces exhibit interesting behaviour respect known cardinal functions hereditary density powers countable uncountable even spread uncountable