II $_1$因子中双随机映射的局部形式与联合优控制
The local form of doubly stochastic maps and joint majorization in II$_1$ factors
摘要：
我们找到了双随机映射对可分交换$C^*$ -子代数的限制的一个描述，其中$C^*$ -子代数是一个II $_1$因子$\cM$。我们使用这种局部形式的双随机映射发展的概念，联合优化之间的$n$ -元组的相互交换的自伴算子，扩展那些Kamei（单自伴算子）和Hiai（单正规算子）在II $_1$因子的情况下。这种联合优化的几个特征。作为一个副产品，我们证明了任何可分离的交换$C^*$ -子代数的$\cM$可以嵌入到一个可分离的交换$C^*$ -子代数的$\cM$与扩散谱测度。
英文摘要：
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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