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In 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 α ψ is a linear *-isomorphism, and is a conjugate linear *-isomorphism only when η ∈ R . [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]