Unconditionality in spaces of smooth functions

Our main result shows that subspaces of L(1)([0, 1]) on which the blow-up operators act compactly are isometric to dual spaces, and their natural predual belongs to the Banach-Mazur closure of quotient spaces of c(0)(N). Related general results are shown for subspaces X of C(0)(Omega) or of reflexive Kothe function spaces, which imply that when X consists of smooth functions it embeds into a Banach space with an unconditional basis.