Positive contractive projections in Schatten Spaces

We characterize the positively 1-complemented subspaces of $S^p$, for $1\le p<\mathrm{\infty }$, where $S^p$ denotes the Schatten spaces. Building on the work of Arazy and Friedman, who described the 1-complemented subspaces of $S^p$, for $1\le p\mathrm{\ne }2<\mathrm{\infty }$, we establish that there are five mutually distinct types of indecomposable positively 1-complemented subspaces in $S^p$. Moreover, every positively 1-complemented subspace of $S^p$ can be expressed as a direct sum of some of these indecomposable subspaces.