Homomorphisms between C*-algebras and linear derivations on C*-algebras

Abstract

It is shown that every almost unital almost linear mapping h: A -> 8 of a unital C*-algebra A to a unital C*-algebra B is a homomorphism when h(3 '' uy) = h(3 '' u)h(y) holds for all unitaries u is an element of A, all y is an element of A, and all n = 0, 1, 2,..., and that every almost unital almost linear continuous mapping h: A -> 8 of a unital C*-algebra A of real rank zero to a unital C*-algebra 8 is a homomorphism when h(3 '' uy) =h(3 '' u)h(y) holds for all u is an element of {upsilon is an element of A vertical bar upsilon = upsilon*, parallel to upsilon parallel to =1,and upsilon is invertible}, all y is an element of A, and all n = 0, 1, 2,.... Furthermore, we prove the Hyers-Ulam-Rassias stability of *-homomorphisms between unital C*-algebras, and C-linear *-derivations on unital C*-algebras. The concept of Hyers-Ulam-Rassias stability originated from the Th.M. Rassias' stability theorem that appeared in his paper: On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978) 297-300.