We continue our investigation of contractive projections on noncommutative $\mathrm{ L}^p$-spaces where $1 < p < \infty$ started in \cite{ ArR19}. We improve the results of \cite{ ArR19} and we characterize precisely the positive contractive projections on a noncommutative $\mathrm{ L}^p$-space associated with a $\sigma$-finite von Neumann algebra. We connect this topic to the theory of $\mathrm{ JW}^*$-algebras. More precisely, in large cases, we are able to show that the range of a positive contractive projection is isometric to a nonassociative $\mathrm{ L}^p$-space associated to a $\mathrm{ JW}^*$-algebra.
Categories: math.OA math.FA