On Independence of Events in Noncommutative Probability Theory

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.