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AbstractIn this paper, it is shown that every nonlinear bijective map ψ preserving mixed Jordan triple η-products with η≠−1 from a factor von Neumann algebra M with dim M>1 into another factor von Neumann algebra N with dim N>1 is additive. Also, for α∈{−1,1} , we establish that αψ is a linear *-isomorphism, and is a conjugate linear *-isomorphism only when η∈R . [ABSTRACT FROM AUTHOR]

Let be an ∗-algebra and k ∈ ℤ + . For A , B ∈ , the product defined by ∗ [ A , δ A (B) λ ∗ ] k = ∗ [ A , ∗ [ A , δ A (B) λ ∗ ] k − 1 ] 1 , where ∗ [ A , δ A (B) λ ∗ ] 0 = δ A (B) λ ∗ and ∗ [ A , δ A (B) λ ∗ ] 1 = A δ A (B) λ ∗ − δ A (B) λ ∗ A ∗ , is called λ - k -skew Lie product of A and B , where δ A (B) λ ∗ = A B − λ B A ∗ . This paper discusses some useful characteristics of λ - k -skew Lie products on prime ∗-algebras. Moreover, we investigate the effects of an additive map ϕ on with a λ - k -skew Lie product. [ABSTRACT FROM AUTHOR]

Let ℜ be a ring with center Z(ℜ). In this paper, we study the higher-order commutators with power central values on rings and algebras involving generalized derivations. Motivated by [A. Alahmadi, S. Ali, A. N. Khan and M. Salahuddin Khan, A characterization of generalized derivations on prime rings, Comm. Algebra 44 2016, 8, 3201–3210], we characterize generalized derivations and related maps that satisfy certain differential identities on prime rings. Precisely, we prove that if a prime ring of characteristic different from two admitting generalized derivation 𝔉 such that ([𝔉(sm)sn + sn𝔉(sm), sr]k)l ∈ Z(ℜ) or every s ∈ ℜ, then either 𝔉(s) = ps for every s ∈ ℜ or ℜ satisfies s4 and 𝔉(s) = sp for every s ∈ ℜ and p ∈ 𝔘, the Utumi quotient ring of ℜ. As an application, we prove that any spectrally generalized derivation on a semisimple Banach algebra satisfying the above mentioned differential identity must be a left multiplication map. [ABSTRACT FROM AUTHOR]

Motivated by Dar and Ali's [9] elegant work's, the goal of the paper is to study nth power *-property in rings with involution and its related problems. In fact, the main result of this paper gives conditions which ensure that an additive map d : R → R satisfying nth power *-property must be X-inner. As an application, we characterize additive map F : R → R such that F (x n) = F (x) (x ∗) n − 1 + ∑ i = 1 n − 1 x i d (x) (x ∗) n − i − 1 for all x ∈ R. where d : R → R is an additive map satisfying nth power *-property. These results provide the affirmative answer to the open problem (see Problem 3.3 [9]). [ABSTRACT FROM AUTHOR]

Let A be unital prime Banach algebra over ℝ or ℂ with centre and G1, G2 be open subsets of be a continuous linear generalized skew derivation, and be a continuous linear map. We prove that must be commutative if one of the following conditions holds: For each a ∈ G1, b ∈ G2, there exists an integer m ∈ Z>1 depending on a and b such that either. For each a ∈ G1, b ∈ G2, there exists an integer m ∈ Z>1 depending on a and b such that either. These results generalize a number of theorems of this type. In particular, as an application, we give an affirmative answer to some questions posed in [21]. [ABSTRACT FROM AUTHOR]

Let R be a ring with involution and let α and β be endomorphisms of R. In this paper we characterise generalised Jordan (α, β)-higher derivations and related maps on (semi)-prime rings with involution. As consequences of our main theorems, many known results can be either generalised or deduced. [ABSTRACT FROM AUTHOR]