The PHC criterion and the realignment criterion for pure states in infinite-dimensional bipartite quantum systems are given. Furthermore, several equivalent conditions for pure states to be separable are generalized to infinite-dimensional systems.
Let A be a factor von Neumann algebra and Φ be a nonlinear surjective map from A onto itself.We prove that,if Φ satisfies that Φ(A)Φ(B)-Φ(B)Φ(A)= AB-BA* for all A,B ∈ A,then there exist a linear bijective map Ψ:A → A satisfying Ψ(A)Ψ(B)-Ψ(B)Ψ(A)= AB-BA* for A,B ∈A and a real functional h on A with h(0) = 0 such that Φ(A) = Ψ(A) + h(A)I for every A ∈A.In particular,if A is a type I factor,then,Φ(A) = cA + h(A)I for every A ∈ A,where c = ±1.
让 M\mathcal { M } 并且 N\mathcal { N } von Neumann 代数学被组 SL 2 的合理行动导致() 并且它上面的一半上的亚群 P 飞机 \mathbbH\mathbb { H } 。我们显示出那 N\mathcal { N } 是空间的对组 von Neumann 代数学 LP\mathcal 同形 { L }_P 和描绘的 M\mathcal { M } 并且它的 commutant
Let N and M be nests on Banach spaces X and Y over the real or complex field F,respectively,with the property that if M∈M such that M-=M,then M is complemented in Y.Let AlgN and AlgM be the associated nest algebras.Assume that Φ:AlgN→AlgM is a bijective map.It is proved that,if dim X=∞ and if there is a nontrivial element in N which is complemented in X,then Φ is Lie multiplicative (i.e.Φ([A,B])=[Φ(A),Φ(B)] for all A,B∈AlgN) if and only if Φ has the form Φ(A)=-TA*T-1+τ(A) for all A∈AlgN or Φ(A)=TAT-1+τ(A) for all A∈AlgN,where T is an invertible linear or conjugate linear operator and τ:AlgN→FI is a map with τ([A,B])=0 for all A,B∈AlgN.The Lie multiplicative maps are also characterized for the case dim X<∞.
