On surjections between Banach spaces of continuous functions on separable nonmetrizable compact lines
Volume 246 / 2019
Fundamenta Mathematicae 246 (2019), 301-320
MSC: Primary 46E15, 46B03, 54F05.
DOI: 10.4064/fm600-9-2018
Published online: 15 April 2019
Abstract
For a compact subset $K$ of $[0,1]$ and a subset $A$ of $K$, we denote by $K_A$ the modification of the two-arrows space with base $K$ and duplicated set $A$. We study necessary conditions for the existence of continuous linear surjections between the Banach spaces $C(K_A)$ of all real continuous functions on $K_A$ spaces. We show that if there exists a continuous linear surjection from $C(K_A)$ onto $C(L_B)$ and $A$ is a member of the additive Borel class $\Sigma _\alpha $ for some ordinal number $1\leq \alpha \leq \omega _1$, then $B\in \Sigma _{\max\{3,1+\alpha \}}$.
