On biorthogonal systems whose functionals are finitely supported

Abstract

We show that for each natural $n>1$ it is consistent that there is a compact Hausdorff space $K_{2n}$ such that in $C(K_{2n})$ there is no uncountable (semi)biorthogonal sequence $(f_ξ,μ_ξ)_{ξ\in ω_1}$ where $μ_ξ$'s are atomic measures with supports consisting of at most $2n-1$ points of $K_{2n}$, but there are biorthogonal systems $(f_ξ,μ_ξ)_{ξ\in ω_1}$ where $μ_ξ$'s are atomic measures with supports consisting of $2n$ points. This complements a result of Todorcevic that it is consistent that each nonseparable Banach space $C(K)$ has an uncountable biorthogonal system where the functionals are measures of the form $δ_{x_ξ}-δ_{y_ξ}$ for $ξ<ω_1$ and $x_ξ,y_ξ\in K$. It also follows that it is consistent that the irredundance of the Boolean algebra $Clop(K)$ or the Banach algebra $C(K)$ for $K$ totally disconnected 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 $k>n$.