Factor Von Neumann Algebra Research Articles

Nonlinear Mixed $\lambda$-Jordan Triple Derivation on $\ast$-algebras

Let $\mathcal{A}$ be a $\ast$-algebra with unit $I$ and $P_1$ and $P_2 = I - P_1$ includes a non-trivial projections, and let $\lambda \in \mathbb{C}\setminus\{0,-1\}$. In this paper, We aim to study the characterization of nonlinear mixed $\lambda$-Jordan triple derivation on $\ast$-algebras. As an application, we can also apply our results on prime $\ast$-algebras, factor von-Neumann algebras and standard operator algebras.

On 2-local derivations of von Neumann algebras

Let A be a semi-finite factor von Neumann algebra. We prove that if a map δ : A → A satisfies that for any A , B ∈ A there is a linear derivation δ A , B : A → A such that δ ( A ) B + Aδ ( B ) = δ A , B ( AB ) , then δ is a linear derivation. It is a positive answer to the problem on semi-finite factor von Neumann algebra posed by Molnár in the paper [A new look at local maps on algebraic structures of matrices and operators, New York J Math 28 (2022)].

Additive maps related to Lie structure on factor von Neumann algebras

Additive maps related to Lie structure on factor von Neumann algebras

The second nonlinear mixed bi-skew Lie triple derivations on factor von Neumann algebras

Let be a factor von Neumann algebra with dim . In this paper, it is shown that if a map satisfies for all , then ϕ is an additive ∗-derivation, where .

A note on skew Jordan triple derivations on ∗-rings

Let [Formula: see text] be a [Formula: see text]-torsion free unital ∗-ring containing nontrivial symmetric idempotent. In this article, it is shown that if a map [Formula: see text] : [Formula: see text] (not necessarily additive) satisfies [Formula: see text] for all [Formula: see text], then [Formula: see text] is additive. Moreover, if [Formula: see text] is self-adjoint, then [Formula: see text] is a ∗-derivation. As an applications, we apply our main result to some special classes of unital ∗-rings and ∗-algebras such as prime ∗-ring, prime ∗-algebra, standard operator algebra, factor von Neumann algebra and von Neumann algebra with no central summands of type [Formula: see text].

Nonlinear A*B+B*A Type Derivations on *-Algebras

Let [see formula in PDF] be a unital *-algebra with the unit [see formula in PDF] and a nontrivial projection [see formula in PDF]. Suppose that [see formula in PDF] satisfies [see formula in PDF] implies [see formula in PDF] and [see formula in PDF] implies [see formula in PDF] In this paper, we prove that [see formula in PDF] is a nonlinear [see formula in PDF] type derivation on [see formula in PDF] if and only if [see formula in PDF] is an additive *-derivation. This is then applied to prime *-algebra, von Neumann algebras with no central summands of type [see formula in PDF] factor von Neumann algebras and standard operator algebras.

Nonlinear Skew Lie-Type Derivations on ∗-Algebra

Let A be a unital ∗-algebra over the complex fields C. For any H1,H2∈A, a product [H1,H2]•=H1H2−H2H1* is called the skew Lie product. In this article, it is shown that if a map ξ : A→A (not necessarily linear) satisfies ξ(Pn(H1,H2,…,Hn))=∑i=1nPn(H1,…,Hi−1,ξ(Hi),Hi+1,…,Hn)(n≥3) for all H1,H2,…,Hn∈A, then ξ is additive. Moreover, if ξ(ie2) is self-adjoint, then ξ is ∗-derivation. As applications, we apply our main result to some special classes of unital ∗-algebras such as prime ∗-algebra, standard operator algebra, factor von Neumann algebra, and von Neumann algebra with no central summands of type I1.

Lie Centralizers and generalized Lie derivations on prime rings by local actions

Assume that be a unital prime ring with whose characteristic is not 2 and containing a nontrivial idempotent P. In this paper, it is shown that under certain conditions, where an additive map on and AB = P if and only if it has the form , where h is an additive map into its center vanishing at commutators with and . An application of our results we characterize generalized Lie derivations on . Using main result we apply several classical examples of unital prime rings with nontrivial idempotents such as Banach space standard operator algebras and factor von Neumann algebras.

NONLINEAR BI-SKEW LIE-TYPE DERIVATIONS ON *-ALGEBRAS

Let 𝒜 be a unital ∗-algebra. Under some mild conditions on 𝒜, we show that ϕ is a nonlinear bi-skew Lie-type derivation on 𝒜 if and only if ϕ is an additive ∗-derivation. As applications, nonlinear bi-skew Lie-type derivations on prime ∗-algebras, von Neumann algebras with no central summands of type I1, factor von Neumann algebras and standard operator algebras are characterized.

Triple derivable maps on prime algebras with involution

Let A be a unital prime *-algebra with characteristic not 2 and with a projection . Suppose that is a map satisfying for all , where denotes the symmetric Martindale algebra of quotients of A. It is shown that d is an additive triple derivation. Moreover, there exist with and an additive *-derivation such that for all . The analogous results for prime locally matrix *-algebras, prime *-algebras with nonzero socle, factor von Neumann algebras and standard operator *-algebras on Hilbert spaces are also described.

Maps preserving n-tuple A*B − B*A derivations on factor von Neumann algebras

Let A be a factor von Neumann algebra and ? preserve n-tuple new product derivations on A, that is, for every A1,A2,... ,An ? A, ?(A1 ? A2 ?...? An) = ?(A1) ? A2 ?...? An + A1 ? ?(A2) ?... ? An + ... + A1 ? A2 ? ... ? ?(An) where Ai ?Aj = A*iAj ?A*jAi for i,j ? N, then ? is additive *-derivation, on the condition that ?(?I/2) is self-adjoint operator for ? ? {1,i}.

Approximate equivalence in von Neumann algebras

Suppose $\mathcal{A}$ is a separable unital ASH C*-algebra, $\mathcal{R}$ is a sigma-finite II$_{\infty}$ factor von Neumann algebra, and $\pi,\rho :\mathcal{A}\rightarrow\mathcal{R}$ are unital $\ast$-homomorphisms such that, for every $a\in\mathcal{A}$, the range projections of $\pi\left( a\right) $ and $\rho\left( a\right) $ are Murray von Neuman equivalent in $\mathcal{R}% $. We prove that $\pi$ and $\rho$ are approximately unitarily equivalent modulo $\mathcal{K}_{\mathcal{R}}$, where $\mathcal{K}_{\mathcal{R}}$ is the norm closed ideal generated by the finite projections in $\mathcal{R}$. We also prove a very general result concerning approximate equivalence in arbitrary finite von Neumann algebras.

Additivity of nonlinear higher anti-derivable mappings on generalized matrix algebras

<abstract><p>In this article, we proved that each nonlinear higher anti-derivable mapping on generalized matrix algebras is automatically additive. As for its applications, we find a similar conclusion on triangular algebras, full matrix algebras, unital prime rings with a nontrivial idempotent, unital standard operator algebras and factor von Neumann algebras respectively.</p></abstract>

The nonlinear mixed bi-skew lie triple derivations on *-algebras

Let A be a unital *-algebra. In this paper, under some mild conditions on A, it is shown that a map ? : A ? Ais a nonlinear mixed bi-skew Lie triple derivation if and only if ? is an additive*-derivation. As applications, nonlinear mixed bi-skew Lie triple derivations on prime *-algebras, von Neumann algebras with no central summands of type I1, factor von Neumann algebras and standard operator algebras are characterized.

Nonlinear bi-skew Jordan-type derivations on *-algebras

Let A be a unital *-algebra. In this paper, under some mild conditions on A, it is shown that ? is a nonlinear bi-skew Jordan-type derivations on A if and only if ? is an additive *?derivation. As applications, the nonlinear bi-skew Jordan-type derivations on prime *-algebras, von Neumann algebras with no central summands of type I1, factor von Neumann algebras and standard operator algebras are characterized.

Multiplicative anti-derivations of generalized n-matrix rings

In this paper, we prove that every multiplicative anti-derivation map of an arbitrary generalized [Formula: see text]-matrix ring is additive. Even more, as consequence, we get the same conclusion for the class of triangular [Formula: see text]-matrix rings, unital prime rings with a nontrivial idempotent, standard operator algebras and factor von Neumann algebras.

-Lie-type maps on -algebras

Let and be two -algebras with identities and respectively, and P 1 and nontrivial symmetric projections in In this paper, we study the characterization of multiplicative -Lie-type maps. In particular, if is a factor von Neumann algebra then every complex scalar multiplication bijective unital multiplicative -Lie-type map is -isomorphism.

Nonlinear bi-skew Lie-type derivations on factor von Neumann algebras

Let be a factor von Neumann algebra with For any define and for all integers In this article, we prove that a map satisfies for all if and only if L is an additive *-derivation.

Mixed *-Jordan-type derivations on *-algebras

Let [Formula: see text] be an *-algebra with identity [Formula: see text] and [Formula: see text] and [Formula: see text] nontrivial symmetric idempotents in [Formula: see text]. In this paper we study the characterization of nonlinear mixed *-Jordan-type derivations. In particular, if [Formula: see text] is a factor von Neumann algebra then every unital nonlinear mixed *-Jordan-type derivations are additive *-derivations.

Nonlinear mixed Jordan triple ∗-derivations on factor von Neumann algebras

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