article

Title: A Schur-Horn theorem in II$_1$ factors

Authors: Martin Argerami and Pedro Gustavo Massey

Issue: Volume 56 (2007), Issue 5, 2051-2060

Abstract:

$Given a II$_1$ factor $\mathcal{M}$ and a diffuse abelian von Neumann subalgebra $\mathcal{A} \subset \mathcal{M}$, we prove a version of the Schur-Horn theorem, namely \[ \overline{E_{\mathcal{A}}(\mathcal{U}_{\mathcal{M}}(b))}^{\sigma\mbox{-}\mathrm{sot}} = \{a \in \mathcal{A}^{\mathrm{sa}}: a \prec b \}, \quad b \in \mathcal{M}^{\mathrm{sa}}, \] where $\prec$ denotes spectral majorization, $E_{\mathcal{A}}$ the unique trace-preserving conditional expectation onto $\mathcal{A}$, and $\mathcal{U}_{\mathcal{M}}(b)$ the unitary orbit of $b$ in $\mathcal{M}$. This result is inspired by a recent problem posed by Arveson and Kadison.$