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

Abstract

We prove the first theorem on projections on general noncommutative ${\mathrm{L}}^p$-spaces associated with non-type I von Neumann algebras where $1⩽p<\infty$. 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.

28 pages, final version, minor typos corrected. arXiv admin note: text overlap with arXiv:1909.00391

References in corpus (5)

• Banach embedding properties of non-commutative L^p-spaces
• A reduction method for noncommutative $L_p$-spaces and applications
• Composition Operators on Haagerup $L^p$-spaces
• Positive contractive projections on noncommutative ${\mathrm{L}}^p$-spaces and nonassociative ${\mathrm{L}}^p$-spaces
• Nonassociative ${\mathrm{L}}^p$-spaces and embeddings in noncommutative ${\mathrm{L}}^p$-spaces