On an approximate automorphism on a $C^{*}$-algebra

Abstract

It is shown that for an approximate algebra homomorphism $f : \mathcal {B} \rightarrow \mathcal {B}$ on a Banach $*$-algebra $\mathcal {B}$, there exists a unique algebra $*$-homomorphism $H : \mathcal {B} \rightarrow \mathcal {B}$ near the approximate algebra homomorphism. This is applied to show that for an approximate automorphism $f : \mathcal {A} \rightarrow \mathcal {A}$ on a unital $C^{*}$-algebra $\mathcal {A}$, there exists a unique automorphism $\alpha : \mathcal {A} \rightarrow \mathcal {A}$ near the approximate automorphism. In fact, we show that the approximate automorphism $f : \mathcal {A} \rightarrow \mathcal {A}$ is an automorphism.