The Schur-Horn theorem in von Neumann algebras

Abstract

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 $I{I}_1$ 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 $I{I}_1$ 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 $I{I}_{\infty }$ and type $III$ factors as well as finite von Neumann algebras.

29 pages, one figure, errors corrected and presentation improved

Cited by in corpus (7)

• Majorization and a Schur-Horn Theorem for positive compact operators, the nonzero kernel case
• Multivariable Schur-Horn theorems
• Diagonality and idempotents with applications to problems in operator theory and frame theory
• Thompson's theorem for compact operators and diagonals of unitary operators
• Closed Convex Hulls of Unitary Orbits in C${}^{\ast }$-Algebras of Real Rank Zero
• The Schur-Horn theorem for unbounded operators with discrete spectrum
• Majorisation and Kadison's Carpenter's Theorem