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 $C^{\ast }$-subalgebras of a II${}_1$ factor $\cM$. 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 II${}_1$ factor case. Several characterizations of this joint majorization are obtained. As a byproduct we prove that any separable abelian $C^{\ast }$-subalgebra of $\cM$ can be embedded into a separable abelian $C^{\ast }$-subalgebra of $\cM$ with diffuse spectral measure.

arXiv commentv1: Jan 20, 2006. 13 pages