On Contractively Complemented Subspaces

Abstract. It is shown that for an L1-predual space X and a countable linearly independent subset of ext(BX∗) whose norm-closed linear span Y in X∗ is w∗-closed, there exists a w∗-continuous contractive projection from X∗ onto Y . This result combined with those of Pelczynski and Bourgain yields a simple proof of the Lazar-Lindenstrauss theorem that every separable L1-predual with non-separable dual contains a contractively complemented subspace isometric to C(∆), the Banach space of functions continuous on the Cantor discontinuum ∆. It is further shown that if X∗ is isometric to l1 and (e ∗ n) is a basis for X∗ isometrically equivalent to the usual l1-basis, then there exists a w∗-convergent subsequence (e∗mn) of (e ∗ n) such that the closed linear subspace of X∗ generated by the sequence (e∗m2n − e ∗ m2n−1 ) is the range of a w∗-continuous contractive projection in X∗. This yields a new proof of Zippin’s result that c0 is isometric to a contractively complemented subspace of X.