Schur-Horn theorems in II${}_{\mathrm{\infty }}$-factors

We describe majorization between selfadjoint operators in a $\sigma$-finite II${}_{\mathrm{\infty }}$ 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 $\mathcal{M}$.