On extensions of -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 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 ≡ 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 ) .