The local form of doubly stochastic maps and joint majorization in II$_1$ factors

Details The local form of doubly stochastic maps and joint majorization in II$_1$ factors The local form of doubly stochastic maps and joint majorization in II$_1$ factors

2006-06-02

arxiv.org/abs/math/0606060v1

Integral Equations Operator Theory 61 (2008), no. 1, 1-19

Mathematics

Operator Algebras

v1: Jan 20, 2006. 13 pages

Scientific paper

We find a description of the restriction of doubly stochastic maps to separable abelian $C^*$-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 $C^*$-subalgebra of $\cM$ can be embedded into a separable abelian $C^*$-subalgebra of $\cM$ with diffuse spectral measure.

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