2-positive contractive projections on noncommutative \(\mathrm{L}^p\)-spaces

We prove the first theorem on projections on general noncommutative \(\mathrm{L}^p\)-spaces associated with non-type I von Neumann algebras where \(1 \leqslant p < \infty\). This is the first progress on this topic since the seminal work of Arazy and Friedman [Memoirs AMS 1992] where the problem of the description of contractively complemented subspaces of noncommutative \(\mathrm{L}^p\)-spaces is explicitly raised. We show that the range of a 2-positive contractive projection on an arbitrary noncommutative \(\mathrm{L}^p\)-space is completely order isometrically isomorphic to some noncommutative \(\mathrm{L}^p\)-space. This result is sharp and is even new for Schatten spaces \(S^p\). Our approach relies on non-tracial Haagerup's noncommutative \(\mathrm{L}^p\)-spaces in an essential way, even in the case of a projection acting on a Schatten space and is unrelated to the methods of Arazy and Friedman.