Second derivatives of norms and contractive complementation in vector valued spaces

We consider 1-complemented subspaces (ranges of contractive projections) of vector-valued spaces l(p)(X), where X is a Banach space with a 1-unconditional basis and p is an element of (1,2) boolean OR (2, infinity). If the norm of X is twice continuously differentiable and satisfies certain conditions connecting the norm and the notion of disjointness with respect to the basis, then we prove that every 1-complemented subspace of l(p)(X) admits a basis of mutually disjoint elements. Moreover, we show that every contractive projection is then an averaging operator. We apply our results to the space l(p)(l(q)) with p,q is an element of (1,2) boolean OR (2, infinity) and obtain a complete characterization of its 1-complemented subspaces.