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 II1-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.This work was supported by the Korea Science and Engineering Foundation (KOSEF) grant funded by the Korea government (MEST) (No. R01-2007-000-20064-0). A part of this research was carried out while he author was visiting the KIAS. He is grateful to professors and the staff in the KIAS for their hospitality