2010/05/19 by Christina Brech, Brech, Christina, Piotr Koszmider +1
Mathematics · #FOS: Mathematics #Functional Analysis (math.FA) #math.FA
paper · pdf · doi:10.48550/arxiv.1005.3532
arxiv created 2010/05/19 · arxiv updated 2010/05/20
We show that for each natural n>1 it is consistent that there is a compact Hausdorff space K2n such that in C(K2n) there is no uncountable (semi)biorthogonal sequence (fξ,μξ)ξ∈ ω1 where μξ's are atomic measures with supports consisting of at most 2n-1 points of K2n, but there are biorthogonal systems (fξ,μξ)ξ∈ ω1 where μξ'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 δxξ-δyξ for ξ<ω1 and xξ,yξ∈ 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 K2nk is countable up to k=n and it is uncountable (even the spread is uncountable) for k>n.