J. Korean Math. Soc. 41 (2004), No. 3, pp. 461–477 HOMOMORPHISMS BETWEEN C ∗ -ALGEBRAS ASSOCIATED WITH THE TRIF FUNCTIONAL EQUATION AND LINEAR DERIVATIONS ON C ∗ -ALGEBRAS Chun-Gil Park ‡ and Jinchuan Hou ∗ Abstract. It is shown that every almost linear mapping h : A→ B of a unital C * -algebra A to a unital C * -algebra B is a homo- morphism under some condition on multiplication, and that every almost linear continuous mapping h : A→B of a unital C * -algebra A of real rank zero to a unital C * -algebra B is a homomorphism under some condition on multiplication. Furthermore, we are going to prove the generalized Hyers-Ulam- Rassias stability of *-homomorphisms between unital C * -algebras, and of C-linear *-derivations on unital C * -algebras. 1. Introduction Let X and Y be Banach spaces with norms ||·|| and ‖·‖, respectively. Consider f : X → Y to be a mapping such that f (tx) is continuous in t ∈ R for each fixed x ∈ X . Assume that there exist constants θ ≥ 0 and p ∈ [0, 1) such that ‖f (x + y) − f (x) − f (y)‖≤ θ(||x|| p + ||y|| p ) for all x, y ∈ X . Rassias [7] showed that there exists a unique R-linear mapping T : X → Y such that ‖f (x) − T (x)‖≤ 2θ 2 − 2 p ||x|| p Received March 10, 2003. 2000 Mathematics Subject Classification: Primary 47B48, 39B52, 46L05. Key words and phrases: homomorphism, C * -algebra of real rank zero, linear derivation, stability. ‡ Supported by grant No. R05-2003-000-10006-0 from the Basic Research Program of the Korea Science & Engineering Foundation. * Supported by NNSF of China and NSF of Shanxi Province.