Second derivatives of norms and contractive complementation in vector-valued spaces

We consider 1-complemented subspaces (ranges of contractive projections) of vector-valued spaces ${\mathrm{\ell }}_p(X)$, where $X$ is a Banach space with a 1-unconditional basis and $p\in (1,2)\cup (2,\mathrm{\infty })$. 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 ${\mathrm{\ell }}_p(X)$ 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 ${\mathrm{\ell }}_p({\mathrm{\ell }}_q)$ with $p,q\in (1,2)\cup (2,\mathrm{\infty })$ and obtain a complete characterization of its 1-complemented subspaces.