On extensions of c0-valued operators

We study pairs of Banach spaces (X,Y), with Y⊂X, for which the thesis of Sobczyk's theorem holds, namely, such that every bounded c0-valued operator defined in Y extends to X. We are mainly concerned with the case when X is a C(K) space and Y≡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 c0-valued operator defined in C(L) extends to C(K).