We describe majorization between selfadjoint operators in a $\sigma$-finite II$_\infty$ factor $(\mathcal{ M},\tau)$ in terms of simple spectral relations. For a diffuse abelian von Neumann subalgebra $\mathcal{ A}\subset \mathcal{ M}$ with trace-preserving conditional expectation $E_{ \mathcal{ A}}$, we characterize the closure in the measure topology of the image through $E_{ \mathcal{ A}}$ of the unitary orbit of a selfadjoint operator in $\mathcal{ M}$ in terms of majorization (i.e., a Schur-Horn theorem). We also obtain similar results for the contractive orbit of positive operators in $\mathcal{ M}$ and for the unitary and contractive orbits of $\tau$-integrable operators in $\mathcal{ M}$.
Categories: math.OA