let $mathcal m$ be a factor von neumann algebra. it is shown that every nonlinear $*$-lie higher derivation$d={phi_{n}}_{ninmathbb{n}}$ on $mathcal m$ is additive. in particular, if $mathcal m$ is infinite type $i$factor, a concrete characterization of $d$ is given.

Let A be a factor von Neumann algebra with dimA ? 2. In this paper, it is proved that map : nonlinear mixed Jordan triple ?-derivation if and only an additive ?-derivation.

Let $ {\mathcal{A}} be a unital \ast -algebra containing nontrivial projection. Under some mild conditions on , it is shown that map \Phi:{\mathcal{A}}\rightarrow{\mathcal{A}} nonlinear mixed Jordan triple * -derivation if and only \Phi an additive -derivation. In particular, we apply the above result to prime -algebras, von Neumann algebras with no central summands of type I_{1} factor algebra...

Let A and B be two factor von Neumann algebras. In this paper, we proved that a bijective mapping Φ:A→B satisfies Φ(a∘b+ba∗)=Φ(a)∘Φ(b)+Φ(b)Φ(a)∗ (where ∘ is the special Jordan product on B, respectively), for all elements a,b∈A, if only Φ ∗-ring isomorphism. particular, algebras are type I factors, then unitary isomorphism or conjugate

Introduction Let Γ be a fuchsian subgroup of P SL(2, R). In this paper we consider the Γ-equivariant form of the Berezin's quantization of the upper half plane which will correspond to a deformation quantization of the (singular) space H/Γ. Our main result is that the von Neumann algebra associated to the Γ− equivariant form of the quantization is stable isomorphic with the von Neumann algebra ...

We prove that for all 1 ≤ p ≤ ∞, p 6= 2, the L spaces associated to two von Neumann algebras M, N are isometrically isomorphic if and only if M and N are Jordan *-isomorphic. This follows from a noncommutative L Banach-Stone theorem: a specific decomposition for surjective isometries of noncommutative L spaces.

We prove that for all 1 ≤ p ≤ ∞, p 6= 2, the Lp spaces associated to two von Neumann algebrasM, N are isometrically isomorphic if and only if M and N are Jordan *-isomorphic. This follows from a noncommutative Lp Banach-Stone theorem: a specific decomposition for surjective isometries of noncommutative Lp spaces.

We show that the notion of asymptotic lift generalizes naturally to normal positive maps φ : M → M acting on von Neumann algebras M . We focus on cases in which the domain of the asymptotic lift can be embedded as an operator subsystem M∞ ⊆ M , and characterize when M∞ is a Jordan subalgebra of M in terms of the asymptotic multiplicative properties of φ.

The purpose of this note is to show that any order isomorphism between noncommutative L2-spaces associated with von Neumann algebras is decomposed into a sum of a completely positive map and a completely copositive map. The result is an L2 version of a theorem of Kadison for a Jordan isomorphism on operator algebras.

A concept of orientation is relevant for the passage from Jordan structure to associative structure in operator algebras. The research reported in this paper bridges the approach of Connes for von Neumann algebras and ourselves for C*-algebras in a general theory of orientation that is of geometric nature and is related to dynamics.