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

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.