2-positive contractive projections on noncommutative Lp-spaces

Abstract

We prove the first theorem on projections on general noncommutative Lp-spaces associated with non-type I von Neumann algebras where 1⩽p<∞. This is the first progress on this topic since the seminal work of Arazy and Friedman (1992) [6] where the problem of the description of contractively complemented subspaces of noncommutative Lp-spaces is explicitly raised. We show that the range of a 2-positive contractive projection on an arbitrary noncommutative Lp-space is completely order isometrically isomorphic to some noncommutative Lp-space. This result is sharp and is even new for Schatten spaces Sp. Our approach relies on non-tracial Haagerup's noncommutative Lp-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.