On the c0c0-extension property for compact lines

We present a characterization of the continuous increasing surjections ϕ:K→Lϕ:K→L between compact lines K and L   for which the corresponding subalgebra ϕ⁎C(L)ϕ⁎C(L) has the c0c0-extension property in C(K)C(K). A natural question arising in connection with this characterization is shown to be independent of the axioms of ZFC.