Jianhua Zhang

Let [ ] be a factor von Neumann algebra with dim[ ]. In this paper, we prove that a map [ ] satisfies [ ] for all [ ] if and only if there exists a real number [ ] such that [ ], where [ ] is an additive ∗-derivation and [ ].

Let [see formula in PDF] be a unital [see formula in PDF]-algebra containing a nontrivial projection, [see formula in PDF] be the set of non-negative integers. Under some mild conditions on [see formula in PDF], it is shown that any nonlinear mixed Jordan triple [see formula in PDF]-higher derivation [see formula in PDF] is an additive [see formula in PDF]-higher derivation. In particular, we apply the above result to prime [see formula in PDF]-algebras and von Neumann algebras with no central summands of type [see formula in PDF].

Let U be a triangular operator algebra, and ?: U ? U be a linear map. In this paper, under some mild conditions on U, we prove that if ? satisfies ?([[U, V], W ]) = [[?(U), V], W ] = [[U, ?(V)], W] for any U, V, W ? U with UV = UW = P being the standard idempotent(resp. UV = UW = 0), then there exist ? ? Z(U) and a linear map ?: U ? Z(U) satisfying ?([[U, V], W ]) = 0 for any U, V, W ? U with UV = UW = P(resp.UV = UW = 0) such that ?(U) = ?U + ?(U) for U ? U. As an application, we give a characterization of generalized Lie triple derivations on U .

Let A be a unite prime *-algebra containing a non-trivial projection. Assume that ? : A ? A satisfies ?(A1 ?1 A2 ?2 ? ? ? ?n An+1) = ?n+1 h=1 A1 ?1 ? ? ? ?h?2 Ah?1 ?h?1 ?(Ah) ?h Ah+1 ?h+1 ? ? ? ?n An+1(n ? 2) for any A1,A2, ? ? ? ,An+1 ? A and ?r is ? or ? with 1 ? r ? n, where A ? B = AB* + BA* and A ? B = + BA. In this article, we prove that if n is even and ?2u?1 = ?, ?2u = ? with 1 ? u ? n/2, then there exists an element ? ? ZS(A) such that ?(A) = ?(A) + i?A, where ? is an additive *-derivation. Otherwise, ? is an additive *-derivation. In particular, the nonlinear mixed bi-skew Jordan-type derivations on factor von Neumann algebras and standard operator algebras are characterized.

Let A be a unite prime *-algebra containing a non-trivial projection. In this paper, we prove that a map ?: A ? A satisfies ?([A ? B, C]) = [?(A) ? B, C] + [A ? ?(B), C] + [A ? B, ?(C)] for all A, B, C ? A if and only if there exists an element ? ? Zs (A) such that ?(A) = d(A) + i?A, where d: A ? A is an additive *-derivation and A ? B = AB* + BA*. Also, we give the structure of this map on factor von Neumann algebras.