The local form of doubly stochastic maps and joint majorization in II₁ factors

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We find a description of the restriction of doubly stochastic maps to separable abelian C*-subalgebras of a II₁ factor. 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 can be embedded into a separable abelian C*-subalgebra of with diffuse spectral measure.