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 II ₁ factor ${\mathcal{M}}$. We use this local form of doubly stochastic maps to develop a notion of joint majorization between n-tuples 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 ${\mathcal{M}}$ can be embedded into a separable abelian C*-subalgebra of ${\mathcal{M}}$ with diffuse spectral measure.