The Schur-Horn Theorem in von Neumann algebras

Dedicated to the memory of William Arveson(1934-2011) Abstract. A few years ago, Richard Kadison thoroughly analysed the diagonals of projection operators on Hilbert spaces and asked the following question: Let A be a masa in a type II1 factor M and let A ∈ A be a positive contraction. Letting E be the canonical normal conditional expectation from M to A, can one find a projection P ∈M so that E(P) = A? In a later paper, Kadison and Arveson, as an extension, conjectured a Schur-Horn theorem in type II1 factors. In this paper, I give a proof of this conjecture of Arveson and Kadison. I also prove versions of the Schur-Horn theorem for type II ∞ and type III factors as well as finite von Neumann algebras. 1