On Homomorphisms between C*-Algebras and Linear Derivations on C*-Algebras

Abstract

It is shown that every almost linear Pexider mappings f, g, h from a unital C*-algebra $A$ into a unital C*-algebra ℬ are homomorphisms when f(2ⁿ _uy) = f(2ⁿ u)f(y), g(2ⁿ uy) = g(2ⁿ u)g(y) and h(2ⁿ uy) = h(2ⁿ u)h(y) hold for all unitaries u ∈ $A$, all y ∈ $A$, and all n ∈ ℤ, and that every almost linear continuous Pexider mappings f, g, h from a unital C*-algebra $A$ of real rank zero into a unital C*-algebra ℬ are homomorphisms when f(2ⁿ uy) = f(2ⁿ u)f(y), g(2ⁿ uy) = g(2ⁿ u)g(y) and h(2ⁿ uy) = h(2ⁿ u)h(y) hold for all u ∈ {v ∈ $A$ : v = v* and v is invertible}, all y ∈ $A$ and all n ∈ ℤ.

Furthermore, we prove the Cauchy-Rassias stability of *-homomorphisms between unital C*-algebras, and ℂ-linear *-derivations on unital C*-algebras.