Projections in the J-sums of Banach spaces

We study some families of projections in the J-sums of Banach spaces \(J(\Phi )\) $J(\mathrm{\Phi })$ and  \({\hat{J}}(\Phi )\) $\widehat{J}(\mathrm{\Phi })$ introduced by Bellenot. As an application, we show that, under some conditions, \(J(\Phi )\) $J(\mathrm{\Phi })$ and \({\hat{J}}(\Phi )\) $\widehat{J}(\mathrm{\Phi })$ are subprojective, i.e., every closed infinite-dimensional subspace of either of them contains a complemented infinite-dimensional subspace.