The Local Form of Doubly Stochastic Maps and Joint Majorization in II 1 Factors

Abstract

We find a description of the restriction of doubly stochastic maps to separable abelian C ∗ -subalgebras of a II1 factor M. We use this local form of doubly stochastic maps to develop a notion of joint majorization between ntuples of mutually commuting self-adjoint operators that extends those of Kamei (for single self-adjoint operators) and Hiai (for single normal operators) in the II1 factor case. Several characterizations of this joint majorization are obtained. As a byproduct we prove that any separable abelian C ∗ -subalgebra of M can be embedded into a separable abelian C ∗ -subalgebra of M with diffuse spectral measure.