On biorthogonal systems whose functionals are finitely supported - Mathematics > Functional AnalysisReport as inadecuate




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Abstract: We show that for each natural $n>1$ it is consistent that there is a compactHausdorff space $K {2n}$ such that in $CK {2n}$ there is no uncountablesemibiorthogonal 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 there are 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 that it is consistent thateach nonseparable Banach space $CK$ has an uncountable biorthogonal systemwhere 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 that it isconsistent that the irredundance of the Boolean algebra $ClopK$ or the Banachalgebra $CK$ for $K$ totally disconnected can be strictly smaller than thesizes of biorthogonal systems in $CK$. The compact spaces exhibit aninteresting behaviour with respect to known cardinal functions: the hereditarydensity of the powers $K {2n}^k$ is countable up to $k=n$ and it is uncountableeven the spread is uncountable for $k>n$.



Author: Christina Brech, Piotr Koszmider

Source: https://arxiv.org/



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