On extensions of $c_0$-valued operators

We study pairs of Banach spaces $(X,Y)$, with $Y\subset X$, for which the thesis of Sobczyk's theorem holds, namely, such that every bounded $c_0$-valued operator defined in $Y$ extends to $X$. We are mainly concerned with the case when $X$ is a $C(K)$ space and $Y\equiv C(L)$ is a Banach subalgebra of $C(K)$. The main result of the article states that, if $K$ is a compact line and $L$ is countable, then every bounded $c_0$-valued operator defined in $C(L)$ extends to $C(K)$.

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