On extensions of $c_0$-valued operators

TL;DR

This paper investigates conditions under which $c_0$-valued operators defined on subspaces of certain Banach spaces, especially $C(K)$ spaces with compact line $K$, can be extended to the whole space, revealing specific extension properties.

Contribution

It characterizes when $c_0$-valued operators on subalgebras of $C(K)$ spaces extend to the entire space, focusing on compact line $K$ and countable $L$.

Findings

If $K$ is a compact line and $L$ is countable, then every bounded $c_0$-valued operator on $C(L)$ extends to $C(K)$.

The extension property holds for certain pairs of Banach spaces related to $C(K)$ spaces.

The results provide new insights into the structure of $c_0$-valued operators and their extension properties in Banach space theory.

Abstract

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 $C(K)$.