自由Fisher信息量与合并自由

Free Fisher Information and Amalgamated Freeness

LMAM, School of Mathematical Science, Peking University, Beijing 100871, P. R. China

摘要: 在算子值非交换概率空间中引入算子值自由Fisher信息量的概念，这一定义是对D.Voiculescu在有迹的von Neumann代数上定义的自由Fisher信息量的推广．证明了算子值自由Fisher信息量与合并自由性是密切相关的，即证明了若干个算子值随机变量的自由Fisher信息量的可加性等价于这些随机变量的合并自由性．并且也类似地得到了Cramer-Rao不等式．
- Hilbert C*-模 /
- 算子值随机变量 /
- 自由Fisher信息量
Abstract: The notion of operator-valued free Fisher information was introduced.It is a generalization of free Fisher information which was defined by D.Voiculescu on tracial von Neumann algebras.It is proved that the operator-valued free Fisher information is closely related to amalgamated freeness,i.e.the operator-valued free Fisher information of some random variables is additive if and only if these random variables are a free family with amalgamation over a subalgebra.Cramer-Raoinequality in operator-valued settings is also obtained.
- Hilbert C^*-module /
- operator-valued random variable /
- free Fisher information

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