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

AbstractIt is shown that every almost unital almost linear mapping h:A→B of a unital C∗-algebra A to a unital C∗-algebra B is a homomorphism when h(3nuy)=h(3nu)h(y) holds for all unitaries u∈A, all y∈A, and all n=0,1,2,…, and that every almost unital 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 when h(3nuy)=h(3nu)h(y) holds for all u∈{v∈A|v=v∗,‖v‖=1, and v is invertible}, all y∈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