On notion of asymptotic derivations

In this paper we introduce some notion of asymptotic derivations of a C *‐ and W *‐dynamical systems which naturally arises from a one‐parameter group of automorphisms. We show that an asymptotic derivation on a unital simple C *‐algebra or a von Neumann algebra is asymptotically inner. Every asymptotic derivation on finite dimensional C *‐algebras is induced by an inner derivation. We prove that any asymptotic derivation on commutative von Neumann algebras have a strong limit zero. An asymptotic derivation on the hyperfinite II 1 ‐factor given by some one‐parameter group of automorphisms can be induced by a (inner) derivation. Finally, we show that every asymptotic Jordan derivation of a C *‐dynamical system is an asymptotic derivation and is also induced by a derivation if there exists a limit.