Abstract

We consider a tracial state \(\varphi\) on a von Neumann algebra \(\mathcal{A}\) and assume that projections \(P,Q\) of \(\mathcal{A}\) are independent if \(\varphi(PQ)=\varphi(P)\varphi(Q)\). First we present the general criterion of a projection pair independence. We then give a geometric criterion for independence of different pairs of projections. If atoms \(P\) and \(Q\) are independent then \(\varphi(P)=\varphi(Q)\). Also here we deal with an analog of a ‘‘symmetric difference’’ for a pair of projections \(P\) and \(Q\), namely, the projection \(R\equiv P\vee Q-P\wedge Q\). If \(R\neq 0,I\), the pairs \(\{P,R\}\) and \(\{Q,R\}\) are independent then \(\varphi(P)=\varphi(Q)=1/2\) and \(\varphi(P\wedge Q+P\vee Q)=1\). If, moreover, \(P\) and \(Q\) are independent, then \(\varphi(P\wedge Q)\leq 1/4\) and \(\varphi(P\vee Q)\geq 3/4\). For an atomless von Neumann algebra \(\mathcal{A}\) a tracial normal state is determined by its specification of independent events. We clarify our considerations with examples of projection pairs with the differemt mutual independency relations. For the full matrix algebra we give several equivalent conditions for the independence of pairs of projections.

中文翻译：

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非交换概率论中事件的独立性

摘要

我们认为一个tracial状态\（\ varphi \）在冯·诺依曼代数\（\ mathcal {A} \） ，并假定预测\（P，Q \）的\（\ mathcal {A} \）是独立的，如果\ (\varphi(PQ)=\varphi(P)\varphi(Q)\)。首先，我们提出了投影对独立性的一般标准。然后我们给出了不同投影对独立性的几何标准。如果原子\(P\)和\(Q\)是独立的，那么\(\varphi(P)=\varphi(Q)\)。同样在这里，我们处理一对投影\(P\)和\(Q\)的“对称差异”的模拟，即投影\(R\equiv P\vee QP\wedge Q\)。如果\(R\neq 0,I\)，对\(\{P,R\}\)和\(\{Q,R\}\)是独立的，那么\(\varphi(P)=\varphi (Q)=1/2\)和\(\varphi(P\wedge Q+P\vee Q)=1\)。此外，如果\(P\)和\(Q\)是独立的，那么\(\varphi(P\wedge Q)\leq 1/4\)和\(\varphi(P\vee Q)\geq 3 /4\)。对于无原子冯诺依曼代数\(\mathcal{A}\)轨迹正常状态由其对独立事件的规范决定。我们通过具有不同相互独立关系的投影对的例子来阐明我们的考虑。对于全矩阵代数，我们给出了投影对独立性的几个等价条件。