Schur-Horn Theorems In Ii∞-Factors

Abstract

We describe majorization between selfadjoint operators in a semi-finite II_\infty factor (M;\tau) in terms of simple spectral relations. For a difuse abelian von Neumann subalgebra A of M that admits a (necessarily unique) trace-preserving conditional expectation, denoted by E_A, we characterize the closure in the measure topology of the image through E_A 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 \tau-integrable selfadjoint operators in M.