Let A and B be unital rings, and M be an （A, B）-bimodule, which is faithful as a left A-module and also as a right B-module. Let U = Tri（A,M, B） be the triangular algebra. In this paper, we give some different characterizations of Lie higher derivations on U.
The positive partial transposition(PPT)criterion and the realignment criterion constitute two of the most important criteria for detecting entanglement.The generalized partial transposition(GPT)criterion which contains the PPT criterion and the realignment criterion as special cases,provides a necessary condition for a multipartite state to be separable.Here we extend the GPT criterion to the infinitedimensional multipartite(bipartite)case and show that it includes multipartite(bipartite)PPT criterion and multipartite(bipartite)realignment criterion as special cases as well.
Let X1 and X2 be complex Banach spaces with dimension at least three, A1 and A2 be standard operator algebras on X1 and X2, respectively. For k ≥ 2, let （i1, i2, . . . , im） be a finite sequence such that {i1, i2, . . . , im} = {1, 2, . . . , k} and assume that at least one of the terms in （i1, . . . , im） appears exactly once. Define the generalized Jordan productT1 o T2 o··· o Tk = Ti1Ti2··· Tim ＋ Tim··· Ti2Ti1 on elements in Ai. This includes the usual Jordan product A1A2 ＋ A2A1, and the Jordan triple A1A2A3 ＋ A3A2A1. Let Φ ： A1 → A2 be a map with range containing all operators of rank at most three. It is shown that Φ satisfies that σπ（Φ（A1） o··· o Φ（Ak）） = σπ（A1 o··· o Ak） for all A1, . . . , Ak, where σπ（A） stands for the peripheral spectrum of A, if and only if Φ is a Jordan isomorphism multiplied by an m-th root of unity.
We discuss the fidelity of states in the infinite-dimensional systems and give an elementary proof of the infinite-dimensional version of Uhlmann's theorem.This theorem is used to generalize several properties of the fidelity of the finite-dimensional case to the infinite-dimensional case.These are somewhat different from those for the finite-dimensional case.
Let U = Tri（fit, M, B） be a triangular ring, where A and B are unital rings, and M is a faithful （A, B）-bimodule. It is shown that an additive map φ on U is centralized at zero point （i.e., ,φ（A）B = A,φ（B） = 0 whenever AB = 0） if and only if it is a centralizer. Let 5 ： U →U be an additive map. It is also shown that the following four conditions are equivalent： （1） 5 is specially generalized derivable at zero point, i.e., 5（AB） = δ（A）B ＋ AS（B） - Aδ（I）B whenever AB = 0; （2） 5 is generalized derivable at zero point, i.e., there exist additive maps τ1 and τ2 on U derivable at zero point such that δ（AB） = δ（A）B ＋ Aτ1（B） = τ2（A）B ＋ Aδ（B） whenever AB = 0; （3） δ is a special generalized derivation; （4） δ is a generalized derivation. These results are then applied to nest algebras of Banach space
