C~*-代数中元素的τ-期望与τ-方差(英文)

τ-Expectations and τ-Variances of Elements of a Unital C~*-Algebra

引入了有单位元的C*-代数A的元素A的τ-期望Expτ(A)与τ-方差Varτ(A),它们分别定义为Expτ(A)=τ(A)与Varτ(A)=Expτ((A Expτ(A)1AI)2),建立了关于它们的若干等式与不等式,给出了Expτ(AB)Expτ(A)Expτ(B)与Varτ(A+B)Varτ(A)+Varτ(A)的一些估计.引入并研究了元素A,B的τ-协方差Covτ(A,B),研究了元素的τ-独立性.

The expectation Exp τ （A） and the variance Var τ （A） of an element A of a unital C＊-algebra A with respect to a given state τ of A are defined as Exp τ （A）=τ（A） and Var τ （A）=Exp τ （（A Exp τ （A）1 A ） 2 ）. Some equalities and inequalities about the expectations and the variances are established. Especially, some estimates of the difference of Exp τ （AB） and Exp τ （A）Exp τ （B）, Var τ （A ＋ B） and Var τ （A） ＋ Var τ （A） are obtained, respectively. Moreover, the τ-covariance Cov τ （A, B） of A, B is introduced and the τ-independence of two elements is characterized.