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Product states are always considered as the states that do not contain quantum correlation. We discuss here when a quantum channel sends the product states to themselves. Exact forms of such channels are proposed. It is shown that such a quantum channel is a local quantum channel, a composition of a local quantum channel and a flip operation, or such that one of the local states is fixed. Both finite- and infinite-dimensional systems are considered. [ABSTRACT FROM AUTHOR]

Abstract: Let be a von Neumann algebra with no central summands of type I1. If is a nonlinear Lie derivation, then is of the form , where is an additive derivation of and is a mapping of into its center which maps commutators into zero. [Copyright &y& Elsevier]

Abstract: Let , be von Neumann algebras with no central summands of type . For , define a new product of T and S. In this note, it is proved that a not necessarily linear bijective map satisfies for all if and only if Φ is the direct sum of a linear ⁎-isomorphism and a conjugate linear ⁎-isomorphism. [Copyright &y& Elsevier]

Abstract: Let , be factors. If is a multiplicative ∗-Lie isomorphism, then Φ is of the form , where σ is a linear ∗-isomorphism, or a conjugate linear ∗-isomorphism, or the negative of a linear ∗-anti-isomorphism, or the negative of a conjugate linear ∗-anti-isomorphism of onto and τ is a mapping of into which maps commutators into zero. [Copyright &y& Elsevier]

Let R be a prime ring with 1 containing a nontrivial idempotent E, and let R' be another prime ring. If Φ:R → R' is a multiplicative Lie isomorphism, then Φ(T + S) = Φ(T) + Φ(S) + Z'T,S for all T, S ∈ R, where Z'T,S is an element in the center Z' of R' depending on T and S. [ABSTRACT FROM AUTHOR]

Let H be an infinite-dimensional complex separable Hilbert space and  [image omitted] denote the algebra of all bounded linear operators acting on H. We show that an additive continuous surjective map Φ on  [image omitted] is asymptotic similarity preserving if and only if it is similarity preserving, and in turn, if and only if there exist a scalar c and an invertible bounded linear or conjugate linear operator A on H such that either Φ(T)=cATA-1 for all T or Φ (T)=cAT*A-1 for all T. [ABSTRACT FROM AUTHOR]