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The Schur-Horn theorem in von Neumann algebras

A few years ago, Richard Kadison thoroughly analysed the diagonals of projection operators on Hilbert spaces and asked the following question: Let $\mathcal{A}$ be a masa in a type $II_1$ factor $\mathcal{M}$ and let $A \in \mathcal{A}$ be a positive contraction. Letting $E$ be the canonical normal conditional expectation from $\mathcal{M}$ to $\mathcal{A}$, can one find a projection $P \in \mathcal{M}$ so that [E(P) = A?] In a later paper, Kadison and Arveson, as an extension, conjectured a Schur-Horn theorem in type $II_1$ 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 $II_{\infty}$ and type $III$ factors as well as finite von Neumann algebras.