Schur-Horn theorems in II∞-factors

Abstract

We describe majorization between selfadjoint operators in a ρ-finite II∞ factor (M, τ) in terms of simple spectral relations. For a diffuse abelian von Neumann subalgebra A ⊂ M that admits a (necessarily unique) tracepreserving conditional expectation, denoted by EA, we characterize the closure in the measure topology of the image through EA of the unitary orbit of a selfadjoint operator in M in terms of majorization (i.e., a Schur-Horn theorem). We also obtain similar results for the contractive orbit of positive operators in M and for the unitary and contractive orbits of τ-integrable operators in M.