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Let ℳ, 𝒩 be von Neumann algebras with no central summands of type I1 or I2. If Φ : ℳ → 𝒩 is a multiplicative *-Lie isomorphism, then there is a central projection E ∈ 𝒩 such that modulo a centre-valued map which annihilates commutators, EΦ is a linear Lie *-homomorphism and (I − E)Φ is a conjugate linear Lie *-homomorphism.

In this article, we get a complete classification of distance preserving surjective maps Φ between nest algebras acting on separable Hilbert spaces.

In this article, we give a thorough discussion of additive maps between nest algebras acting on Banach spaces which preserve rank-one operators in both directions.

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.

Let H be an infinite-dimensional complex separable Hilbert space and denote the algebra of all bounded linear operators acting on H. We show that an additive continuous surjective map Φ on 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.

In this article, we discuss the positive definite solutions of the operator equation X m + A * X m n A = I , , on a Hilbert space. Necessary and sufficient conditions for the existence of a positive semi-definite solution are derived. Under some conditions, two effective iterative methods for getting a positive definite solution of this equation are proposed and error estimations are obtained.