Central Summands Research Articles

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].

Characterization of Lie-Type Higher Derivations of von Neumann Algebras with Local Actions

Let m and n be fixed positive integers. Suppose that A is a von Neumann algebra with no central summands of type I1, and Lm:A→A is a Lie-type higher derivation. In continuation of the rigorous and versatile framework for investigating the structure and properties of operators on Hilbert spaces, more facts are needed to characterize Lie-type higher derivations of von Neumann algebras with local actions. In the present paper, our main aim is to characterize Lie-type higher derivations on von Neumann algebras and prove that in cases of zero products, there exists an additive higher derivation ϕm:A→A and an additive higher map ζm:A→Z(A), which annihilates every (n−1)th commutator pn(S1,S2,⋯,Sn) with S1S2=0 such that Lm(S)=ϕm(S)+ζm(S)forallS∈A. We also demonstrate that the result holds true for the case of the projection product. Further, we discuss some more related results.

Nonlinear 2-local Lie n-derivations of von Neumann algebras

Let M be a finite von Neumann algebra with no central summands of type I 1. We show that each nonlinear 2-local Lie n-derivation δ : M → M with n ≥ 3 is of the form d + h, where d : M → M is a linear derivation and h is a homogeneous central-valued mapping which annihilates each ( n − 1 ) -th commutator of M .

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.

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.

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.

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.

$$*$$-Lie–Jordan-Type Maps on $$C^*$$-Algebras

Let $${{\mathfrak {A}}}\, $$ and $${{\mathfrak {A}}}\, '$$ be two $$C^*$$ -algebras with identities $$I_{{{\mathfrak {A}}}\, }$$ and $$I_{{{\mathfrak {A}}}\, '}$$ , respectively, and $$P_1$$ and $$P_2 = I_{{{\mathfrak {A}}}\, } - P_1$$ nontrivial projections in $${{\mathfrak {A}}}\, $$ . In this paper, we study the characterization of multiplicative $$*$$ -Lie–Jordan-type maps, where the notion of these maps arise here. In particular, if $${\mathcal {M}}_{{{\mathfrak {A}}}\, }$$ is a von Neumann algebra relative $$C^{*}$$ -algebra $${{\mathfrak {A}}}\, $$ without central summands of type $$I_1$$ then every bijective unital multiplicative $$*$$ -Lie–Jordan-type maps are $$*$$ -ring isomorphisms.

Nonlinear *-Jordan-type derivations on *-algebras

Let 𝒜 be a unital ∗-algebra with the unit I. Assume that 𝒜 contains a nontrivial projection P which satisfies X𝒜P=0 implies X=0 and X𝒜(I−P)=0 implies X=0. In this paper, it is shown that Φ is a nonlinear ∗-Jordan-type derivation on 𝒜 if and only if Φ is an additive ∗-derivation. As applications, the nonlinear ∗-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.

Non-global Nonlinear Lie Triple Derivable Maps on Finite von Neumann Algebras

Let $${\mathcal {M}}$$ be a finite von Neumann algebra with no central summands of type $$I_{1}$$ . Assume that $$\delta :{\mathcal {M}}\rightarrow {\mathcal {M}}$$ is a nonlinear map satisfying $$\delta ([[A,B],C])=[[\delta (A),B],C]+[[A,\delta (B)],C]+[[A,B],\delta (C)]$$ for any $$A,B,C\in {\mathcal {M}}$$ with $$ABC=0$$ . Then, we prove that there exists an additive derivation $$d:{\mathcal {M}}\rightarrow {\mathcal {M}}$$ , such that $$\delta (A)=d(A)+\tau (A)$$ for any $$A\in {\mathcal {M}}$$ , where $$\tau :{\mathcal {M}}\rightarrow {\mathcal {Z}}_{{\mathcal {M}}}$$ is a nonlinear map, such that $$\tau ([[A,B],C])=0$$ for any $$A,B,C\in {\mathcal {M}}$$ with $$ABC=0$$ .

Commuting maps on alternative rings

Suppose $\mathfrak{R}$ is a $2$,$3$-torsion free unital alternative ring having an idempotent element $e_1$ $\left(e_2 = 1-e_1\right)$ which satisfies $x \mathfrak{R} \cdot e_i = \{0\} \rightarrow x = 0$ $\left(i = 1,2\right)$. In this paper, we aim to characterize the commuting maps. Let $\varphi$ be a commuting map of $\mathfrak{R}$ so it is shown that $\varphi(x) = zx + \Xi(x)$ for all $x \in \mathfrak{R}$, where $z \in \mathcal{Z}(\mathfrak{R})$ and $\Xi$ is an additive map from $\mathfrak{R}$ into $\mathcal{Z}(\mathfrak{R})$. As a consequence a characterization of anti-commuting maps is obtained and we provide as an application, a characterization of commuting maps on von Neumann algebras relative alternative $C^{*}$-algebra with no central summands of type $I_1$.

The Second Nonlinear Mixed Lie Triple Derivations on Finite Von Neumann Algebras

Let $${\mathcal {M}}$$ be a finite von Neumann algebra with no central summands of type $${I}_{1}$$ . Suppose that $$L: {\mathcal {M}}\rightarrow {\mathcal {M}}$$ is the second nonlinear mixed Lie triple derivation. Then L is an additive $$*$$ -derivation. We also show that each local and 2-local second Lie triple derivation on finite von Neumann algebras with no central summands of type $${I}_{1}$$ is a $$*$$ -derivation. Besides, each local and 2-local second Lie triple derivation on factor von Neumann algebras $${\mathcal {M}}$$ with dim $${\mathcal {M}}>1$$ is also a $$*$$ -derivation.

The structure of 2-local Lie derivations on von Neumann algebras

In this article we characterize the form of each 2-local Lie derivation on a von Neumann algebra without central summands of type I1. We deduce that every 2-local Lie derivation δ on a finite von Neumann algebra M without central summands of type I1 can be written in the form δ(A)=AE−EA+h(A) for all A in M, where E is an element in M and h is a center-valued homogenous mapping which annihilates each commutator of M. In particular, every linear 2-local Lie derivation is a Lie derivation on a finite von Neumann algebra without central summands of type I1. We also show that every 2-local Lie derivation on a properly infinite von Neumann algebra is a Lie derivation.

Nonlinear Generalized Lie n-Derivations on von Neumann Algebras

Let $${\mathcal {M}}$$ be a von Neumann algebra without central summands of type $$I_1$$ . Assume that $$G:{{\mathcal {M}}}\rightarrow {{\mathcal {M}}}$$ is a nonlinear map. It is shown that G is a generalized Lie n-derivation ( $$n\ge 2$$ ) if and only if $$G(A)=\varphi (A)+\tau (A)$$ holds for all $$A\in {{\mathcal {M}}}$$ , where $$\varphi :{\mathcal M}\rightarrow {{\mathcal {M}}}$$ is an additive generalized derivation and $$\tau :{{\mathcal {M}}}\rightarrow {{\mathcal {Z}}}({{\mathcal {M}}})$$ is a central-valued map annihilating all $$(n-1)$$ th commutators. This generalizes some related known results.

Nonlinear ∗-Jordan triple derivations on von Neumann algebras

AbstractLetB(H) be the algebra of all bounded linear operators on a complex Hilbert spaceHand 𝓐 ⊆B(H) be a von Neumann algebra with no central summands of typeI1. ForA,B∈ 𝓐, define byA∙B=AB+BA∗a new product ofAandB. In this article, it is proved that a map Φ: 𝓐 →B(H) satisfies Φ(A∙B∙C) = Φ(A) ∙B∙C+A∙ Φ(B) ∙C+A∙B∙Φ(C) for allA,B,C∈ 𝓐 if and only if Φ is an additive *-derivation.

Strong skew commutativity preserving maps on rings with involution

Let R be a unital *-ring with the unit I. Assume that R contains a symmetric idempotent P which satisfies ARP = 0 implies A = 0 and AR(I − P) = 0 implies A = 0. In this paper, it is shown that a surjective map Φ: R → R is strong skew commutativity preserving (that is, satisfies Φ(A)Φ(B)−Φ(B)Φ(A) * = AB−BA * for all A,B ∈ R) if and only if there exist a map f:R→ZS(R) and an element Z ∈ ZS(R) with Z2 = I such that Φ(A) = ZA + f(A) for all A ∈ R, where ZS(R) is the symmetric center of R. As applications, the strong skew commutativity preserving maps on unital prime *-rings and von Neumann algebras with no central summands of type I1 are characterized.

Discrete cores of type III free product factors

We give a general description of the discrete decompositions of type III factors arising as central summands of free product von Neumann algebras based on our previous works. This enables us to give several precise structural results on type III free product factors.

Additive Lie derivations on the algebras of locally measurable operators

Let be a von Neumann algebra without central summands of type . We are studying conditions that an additive map on the algebra of locally measurable operators has the standard form, that is equal to the sum of an additive derivation and an additive center­-valued trace.

Strong commutativity preserving maps on rings

Suppose $\R$ is a unital ring having an idempotent element $e$ which satisfies $a{\R}e=0$ implies $a=0$ and $a{\R}(1-e)=0$ implies $a=0$. In this paper, we aim to characterize the map $f:{\R}\rightarrow {\R}$, $f$ is surjective and $[f(x),f(y)]=[x,y]$ for all $x,y\in {\R}$. It is shown that $f(x)=\alpha x +\xi (x)$ for all $x\in {\R}$, where $\alpha \in {\Z}({\R})$, $\alpha^2=1$, and $\xi$ is a map from ${\R}$ into ${\Z}({\R})$. As an application, a characterization of nonlinear surjective maps preserving strong commutativity on von Neumann algebras with no central summands of type $I_1$ is obtained.