# Wydawnictwa / Czasopisma IMPAN / Fundamenta Mathematicae / Wszystkie zeszyty

## On biorthogonal systems whose functionals are finitely supported

### Tom 213 / 2011

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

#### Streszczenie

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 $k>n$.

#### Autorzy

• Christina BrechInstituto de Matemática, Estatística
e Computação Científica
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
e-mail
• Piotr KoszmiderInstytut Matematyki Politechniki /L/odzkiej
W/olcza/nska 215
90-924 /L/od/x, Poland
and
Institute of Mathematics
Polish Academy of Sciences